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Miller, Willis Glen. The Effects of Unequal Temperament on Chopin's Mazurkas. 
N.p.: University of Houston, 2001.  

T1 - The Effects of Unequal Temperament on Chopin's Mazurkas 
A1 - Miller, W.G. 
UR - 
Y1 - 2001 PB - University of Houston 
ER -  

Dr. Willis G. Miller, III - The Effects of Unequal Temperament on Chopin's Mazurkas   

1. The Rise of Unequal Temperament 

Many present-day musicians may be surprised to read that pianists did not use Equal Temperament until after the beginning of the twentieth century. Beethoven, Chopin, Liszt, and Brahms lived in an era in which piano music contained infinite shades of color and expression; these colors were created largely through the tuning systems that were in use in their day. Equal Temperament destroys these colors, and causes the music to lose certain characteristics that could have been part of the composer's original musical intent. Chopin's music in particular is highly colorful, evidenced by his carefully-chosen key regions, expressive chromaticism, and unusual harmonies. It is hoped that, after an understanding of how unequal temperament was developed, the performer may be able to recover some of those expressive characteristics which have been lost to Equal Temperament. 

Throughout the classical and romantic era, many different types of unequal temperaments were used; these temperaments were based on models of published seventeenth- and eighteenth-century temperaments. The baroque temperaments are rather complex when viewed out of historical context; they were devised as sophisticated evolutions of simpler concepts. The earliest and simplest temperament systems were based on series of purely tuned fifths; these systems are called Pythagorean tunings and rendered a system with extremely wide thirds and one wolf fifth (out-of-tune to the point of being unplayable). Figure 1.1 demonstrates the Pythagorean system's eleven perfect fifths and one impure fifth. Note that the purity of Major thirds in this system has been tremendously sacrificed. 1  

Thus, while a keyboard tuned with this method would render pure fifths in most keys, it would eliminate the possibility of serving a piece written in Ab, Eb, or Db, where the wolf fifth would certainly appear. Moreover, there are eight keys containing Major thirds that are twenty-two cents wider than pure. This ancient temperament served well before the beginning of the Renaissance, when Major thirds were not used conspicuously. During the early 1500's, a new system was devised specifically so that keyboardists might use Major thirds; these systems were called meantone systems, and sacrificed pure fifths for pure thirds - the opposite approach of Pythagorean temperaments. Like Pythagorean tunings, meantone systems favored certain keys, and rendered others unplayable. 

Figure 1.2 gives keys with few accidentals completely pure thirds, but, in order to achieve that, the fifths must be flattened by six cents. The keys in the center of the graph above contain the most accidentals, and contain unplayable Major thirds that are forty-two cents wide. Ab Major is the least playable of all keys, containing both a wide Major third and fifth. 

In 1691, Andreas Werckmeister published a new system that compromised four fifths and gave the "common" keys relatively pure Major thirds. This system allowed all keys to be played in comfortably, and was thus named a "circular temperament" or "well temperament." His Temperament No. 1 was the first published circular temperament; it was an unequal temperament by today's terminology, because the distance between half- steps varied. However, because Werckmeister's invention allowed the performer to play in all keys without a wolf fifth, the contrast between it and the previously used mean-tone temperaments was so great that some writers referred to "circular" temperament as "equal." 

Compare Figure 1.3 to 1.1 and 1.2, observing how selected fifths were compromised, in order to obtain better Major thirds in the "common" keys. The chart above indicates that the keys with the farthest departure from pure Major thirds are those that are furthest away from C (F#, C#, and Ab). Their Major thirds are only twenty-two cents wide, but they also contain also pure fifths. Of all the keys, F Major is the purest, with its four- cent sharp Major third and perfectly pure fifth. 

Werckmeister's circular temperament served as a model for innumerable other circular temperaments. Almost all subsequent unequal temperaments took from Werckmeister's contribution the concept that the purer thirds should lie in keys with relatively few accidentals. By adjusting the spacing of the fifths, a temperament may be devised that contains pure or nearly pure Major thirds in the "common" keys and very sharp thirds in the "black-key" regions; or it may be devised so that the common keys contain only slightly impure thirds, and the black-key areas contain only moderately sharp thirds. The degree of color variance, then, is entirely manipulative. However, because most temperaments did place the purer thirds in the more common key areas, certain emotional characteristics became commonly ascribed to the keys themselves. The same year that Werckmeister published his first circular temperament, Jacques Ozanam recorded the following in his Dictionnaire mathématique:   

Whatever precaution is taken in tuning our instruments to render all the harmonies equal, there will always be some inequality; and this is what makes us notice an indefinable charactertisic of sadness, gaiety, melodiousness or hardness, which, with the help of our ears, makes us distinguish one key from the other. 4  

Ozanam's theories on key characteristics are echoed in Jean Phillippe Rameau's first publications, Traité de l'harmonie (1722), and Nouveau systéme de musique théorique (1726): 

We receive different impressions from intervals, in proportion to the amount of alteration. For example, the major third, which moves us naturally to joy, as we know from experience, impresses upon us ideas even of fury when it is too large. The minor third which brings us naturally to sweetness and tenderness, saddens us when it is too small. . . . Knowledgeable musicians know how to exploit these different effects of the intervals, and [they] value the alteration, which one might [otherwise] condemn, because of the expression they draw therefrom. 5 

At the time of the above writing, the temperament advocated by Rameau was a meantone modification (Figure 1.4); this temperament's extremely wide range of thirds clearly shows how this temperament created a broad spectrum of key-color. 

Eleven years after the publication of Nouveau systéme de musique théorique, Rameau became infatuated with the concept of Equal Temperament and completely dismissed the existence of key-characteristics. In a treatise that followed, Génération harmonique (1737), he included that in order to obtain true Equal Temperament, each pure fifth must be flattened by exactly one-twelfth of the Pythagorean comma; in doing so, he became the first modern advocate of Equal Temperament for keyboard instruments. However, his theories could not assist the actual tuning of the instruments - in order to obtain true Equal Temperament, a tuner would need to count and compare the beats of Major and minor thirds and Major sixths. 7 Though the sizes of Major thirds were being compared by 1892, most tuners did not begin comparing their progressive beats until after the beginning of the twentieth century. 8    

As mentioned previously, the term "equal temperament" was used often in the nineteenth century; however, it referred either to a circular, unequal temperament or to an approximation of Equal Temperament which would not be acceptable according to today's standards. James Alexander Hamilton (1785-1848) published a tuning treatise which included instructions for "equal temperament." In it, he directs the tuner to temper all fifths "equally and in a very small degree," to the end that this "depression, while it will not materially impair the consonancy of the fifths, will produce a series of somewhat sharp, though still agreeable and harmonious Major thirds." 9 While this is a theoretically correct description of Equal Temperament, he could not provide practical instructions on how to ensure that each fifth is tempered "equally," that is, to the same "very small degree." The only available instruction of that time was to tell the student to tune the fifth and then "flatten the G, so that, upon striking the notes C and G together, he hears two slow, distinct waves, terminating in one steady continuous sound; and the fifth will be properly tempered" 10 (see Figure 1.5). By applying the same method around the cycle of fifths, the result would be a tuning that was called "equal temperament." This "equal temperament" method has some obvious deficiencies when compared to today's standards: first, it does not require the tuner to compare the speed of the beats of thirds and sixths; second, the aural instructions, "count 'two slow, distinct waves, terminating in one continuous sound,'" cannot guarantee to the tuner that all fifths would yet receive the same degree of tempering. Hamilton further reveals the flaw in nineteenth-century "equal temperament" when he instructs the student to tune fifths through the cycle of fifths from C to G#, flattening each a "very small degree," and then writes that "it would be better to return to C, and tune the remaining fifths backwards"; that is, tempering the fifths C-F, F- Bb, and Bb-Eb. 11  

He does not direct the student to compare or temper the fifth G#-Eb (where the Pythagorean comma traditionally lay - see Figure 1.1). In later instructions, he recognizes this problem, describing this fifth as the "last and severest test" of his system, "as the two notes of which it is formed have been obtained by different series of fifths" 13 (italics added). He argues that if the student has been careful to observe his previous directions, "this fifth will be little, if all, inferior to the rest" 14 - a conclusion that is a far cry from declaring that this fifth and all the others have been flattened exactly the same amount.

Hamilton's manual is typical of many early nineteenth-century technicians' approach to Equal Temperament. A more detailed manual was written by A Merrick in 1811, which included a chart of monochord lengths that corresponded to an instrument tuned according to his own Equal Temperament tuning method. According to his figures, the temperament that was "nearly the equal" 15 produced the temperament given in Figure 1.6. Jorgenson reported that this temperament would have failed the entrance examinations for today's Piano Technician's Guild. 17

Though there were those nineteenth century musicians that preferred and attempted Equal Temperament, such as Merrick and Hamilton, a different group did exist that held onto the key-color (unequal temperament) tradition. The group that favored Equal Temperament began in the eighteenth century with Rameau's retraction of his key- characteristic theories; its opposition began six years after this retraction, with the publication of Rousseau's Dictionnaire de Musique. Rousseau defended his teacher's initial theories and wrote 

It is a fact of experience that the different musical keys all have a certain character of their own, which distinguishes each one in particular. A Major, for example is brilliant. F [Major] is majestic. Bb Major is tragic;  f minor is sad;  c minor is tender: and all the other keys have in the same way, by inclination, a certain aptitude to arouse this or that feeling, which clever masters know how to utilize well. 18 

Similar eighteenth-century lists of key color-descriptions are found in the writings of such Rousseau-followers as Castil-Blaze, Gervasoni, Gianelli, Laborde, and Valotti. 19 

Rousseau's writings shed much light on the evolution of Equal Temperament. He included in his Dictionnaire that Equal Temperament was a much older system than circular temperament, having been understood in theory and practiced in fretted instruments long before Werckmeister's "Correct" Temperament of 1691; a well temperament in his perspective was an evolution of, and improvement on, Equal Temperament. 20 Tuning instructions for his own well temperament rendered Figure 1.7.  

Theorists that embraced Rousseau's key-character argument devised well temperaments similar to the above model and published them throughout the eighteenth and nineteenth centuries. Johann Philipp Kirnberger (1721-1783) and Frederick Wilhelm Marpurg (1718- 1795) carried on the Rousseau/Rameau debate in Germany to the end of the eighteenth century, with Kirnberger leading the key-character side. Marpurg's theoretical temperaments approach Equal Temperament in design, and therefore were impractical and impossible to tune accurately on keyboard instruments in his day. Kirnberger's temperaments were more practical for keyboard musicians; they employ the same principles of the previous well temperament models. In 1806, Charles Earl Stanhope proposed an alteration to one of Kirnberger's temperaments, writing that Equal Temperament "destroys the difference of character which ought to exist in a well-tuned instrument, between the different Major keys. . . . dull monotony is substituted for pleasing and orderly variety. . . . modulation from key to key loses, in great measure, the very object of modulation, which is to relieve the ear." 22 Figure 1.8 represents his proposed solution.  

In 1832, two years after Chopin's first Mazurkas were published, Jean Jousse approached the issue of temperament among his contemporaries, writing that "some incline for the equal temperament, others prefer the unequal temperament." Jorgenson also provides the following quotation from Jousse's An Essay on Temperament, which further reveals the quality of "equal temperament" in Chopin's day: 

. . . The advantage obtained by the equal temperament is that every interval and chord is produced so near perfection that none of them sound perceptibly imperfect; but it has the following disadvantages: first, it cannot be obtained in a strict sense, as may be proved, not only mathematically, but also by daily experience; therefore the best equally tempered instruments are still unequally tempered, and, what is worse, oftentimes in wrong places. 24 

Jousse followed these introductory statements with instructions for a well temperament and Equal Temperament. Jousse's well temperament (Figure 1.9) contains more pure fifths than Rousseau's, allowing a greater range of differences of Major thirds. The general pattern, though, remains the same.  

In 1885, thirty-four years after Chopin's death, Alexander John Ellis analyzed the work of four piano technicians from the English Broadwood factory. 26 One of the tuners was Ellis' personal technician; the other three were "Broadwood's best tuners," and were supervised by Alfred James Hipkins. According to Jorgenson, Hipkins had a close relationship with Chopin, and often tuned his piano. 27 The differences between these temperaments and true Equal Temperament prompted Jorgenson to name them "Victorian" temperaments (named after the era from which they were taken). 28 After further analysis and interpretation, Jorgenson created tables from Ellis' data which show the range of sizes of Major thirds. Based on his conclusions, Figures 1.10 and 1.11 may be created.  

The figures given in this chapter represent the pattern after which the majority of eighteenth and nineteenth century temperaments were fashioned. Augustus DeMorgan devised a tuning method that went against this pattern though his theoretical writings still supported unequal temperament. In 1843, he wrote, "We are for variety in the several keys, and against Equal Temperament; but we do not like variety without law." 31 He constructed a temperament that very evenly flattened each fifth, but paced purer fifths in the "common key" regions, and purer thirds in the "black key" regions (see Figure 1.12).  

Though well temperaments were in common use through 1885, there is also evidence of mean-tone and even pythagorean modifications in use until 1876. According to Jorgenson, the last published proposal for the adoption of a Pythagorean temperament was in 1808, 33 and the last published proposal for the adoption of a meantone temperament was in 1818. 34 However, a tuning manual written by Robert Holford Bosanquet in 1876 provides instructions for tuning pythagorean, meantone, and "perfect fifth" temperaments, among other unequal temperaments. It also includes musical examples that demonstrate the effectiveness of each temperament, specifically written to assist the performer in selecting a temperament for his own music. 35 History does not provide specific information as to which temperament Chopin - or any other composer - may have used. However, it is clear from these and previous dates mentioned that Chopin's music must have had, as its foundation, a system that was much more color-conscious than today's Equal Temperament.  

For the remainder of this paper, eight unequal temperaments have been chosen for application to Chopin's music. This choice is based either on the temperaments' dates of publication or on references to the temperaments in tuning manuals published during or after Chopin's lifetime. Pythagorean and mean-tone temperaments are eliminated as possibilities for the music because of their key restrictions; Merrick's Equal Temperament approximation is also eliminated because of its unreliability. The Rameau temperament is similar in structure to those mentioned in Bosanquet's 1876 manual; it is included to continue to show the reader the development from meantone to fully circular temperaments, and also to provide a temperament option for pieces which are written in those keys that the temperament has dictated as playable. The other well temperaments fill the remainder or the range of possibilities for application to Chopin's music and allow the reader to develop his own sense for the temperaments so that he might choose one "according to his own taste." 36 

2 The Effects of Unequal Temperament on the Architectural Structure of Chopin's Mazurkas  

Figure 2.1 and Figure 2.2 may be used for references throughout this chapter. 

Though the mazurka is not dependent on a specific structural form, many of Chopin's Mazurkas are written in rounded binary form. This chapter will observe mazurkas written in this simple form to illustrate the effects of unequal temperament on the pieces' architectural structure. The key areas of the A and B sections of these pieces are composed in a number of ways: they may be in a parallel Major or minor relationship, the relative Major or minor, a perfect fourth or fifth, or a non-related key. Of the five mazurkas in rounded binary form with the B section in the parallel Major, four are written in the key of A minor — Opus 59 no. 1; Opus 67 no. 4; Opus 68 no. 2; and the two A minor mazurkas without opus. 37 We shall now consider the steps a musician should take before selecting a temperament, and begin by examining three of the above mazurkas — Opus 67 no. 4; Opus 68 no. 2; and the A minor mazurka without opus dedicated to Èmile Gaillard. 

An understanding of the color qualities of the A minor and A Major triads in each temperament is a necessary first step before applying any of them to the pieces themselves. Figure 2.1 and Figure 2.2 indicate that all eight systems are a compromise of purity. Only the Werckmeister temperament contains a pure fifth, but its Major third is the second widest of the group. The most pure Major thirds are found in the Rousseau and DeMorgan temperaments; of these two, the DeMorgan temperament contains the purer fifth. Similarly, the reader may see that the purest minor third is found in the Werckmeister temperament (see Figure 2.2); the Stanhope and Rousseau sizes of minor thirds are very similar to those of the Werckmeister temperament, though their sizes of fifths differ significantly. 

As mentioned in Chapter I, most unequal temperaments of the eighteenth and nineteenth century were constructed so that the common keys contained the purest Major thirds. As a result of this common method of construction, certain emotional characteristics became attached to certain keys. A Major falls between the groups of "common" and "black-key" regions; it would seem logical, then, that any emotion attached to this key would lie between the extremes of the emotional qualities of C Major and those of C# or F# Major. Fortunately, there is a substantial collection of eighteenth- and nineteenth-century descriptions of key-characteristics. By observing these in the following sections, and comparing these descriptions to the sizes of the intervals of the primary triads, we may better understand how these sensations may be enhanced by the selection of a particular unequal temperament. 

From the unequal temperaments listed in Figure 2.1, the average size of a C Major third is five cents, with a standard deviation of 3.78 cents (this data excludes the DeMorgan temperament, due to its intentional construction against the normal form). Several eighteenth- and nineteenth-century writers and musicians commented on the "pure" character quality of this key, including the following descriptions: 38 

"Perhaps the key most fitting for . . . pure subjects." (George Joseph Vogler, 1778) 
"Complete pure." (C. F. D. Schubart, 1784) 
". . . virginal chastity and purity. . . Innocence, peace, and tenderness." (Johann Jakob Heinse, 1795) 
"Innocence and simplicity." (Pietro Lichtenthal, 1826) 
" . . . the key of simplicity, joy, gratitude, and rejoicing . . ."  (J. A. Schrader, 1827) 
"Cheerful and pure, innocence and simplicity." (Henri Weikert, 1827) 
"Innocence, simplicity, heroic greatness . . ." (Auguest Gathy, 1835) 
"Simple, unadorned." (Robert Schumann, 1835) 
" . . . this key is chosen for innocence, self-sufficient  simplicity, pure naturalness . . ." (Ferdinand Hand, 1837)  

Figure 2.3 outlines the range of sizes of Major thirds for the key of A Major. Excluding the DeMorgan temperament, the average size of these thirds is 15.29 cents — ten cents wider than C Major — with a standard deviation of 2.36 cents. The following is a sampling of key-character descriptions of this key: 39

"Cheerful and bright." (Justin Heinrich Knecht, 1792) 
"Brilliant." (Andre-Ernest-Modeste Grétry, 1797; also Pietro Gianelli, 1801) 
"Very penetrating. It is bright and shiny, but not as much so as E." (Gerog Vogler, 1812) 
"Clear as a bell and joyful." (F. L. Bührlen, 1825) 
"Sounds bright, cheerful. . . ." (Henri Weikert, 1827) 
". . . When the chord of A major is struck somewhat boldly, the childlike, innocent feeling suddenly becomes joyful cheerfulness. . . ." (Gustav Schilling, 1835-36) 
"Brilliant; elegant; joyous." (Hector Berlioz, 1843)    

Most writers immediately preceding and during Chopin's time share the concept that A Major is a "brighter" key than C, and include emotional descriptions of "cheer" and "joy," as opposed to C Major's emotional qualities of "peace" and "simplicity." It is concluded from these descriptions, and the fact that the average size of an A Major third was larger than a C Major third, that wider thirds more easily depict cheer and brightness, while purer thirds, such as those in C Major, are reserved for simpler sentiments, and sound muted by comparison. If the Major third gets abnormally wide, as in B Major, the emotional key-descriptions reach an extreme: 40 

"A harsh sound, more piercing than E, fit to express cries of  despair, howls, roars, and such like." (Francesco Galeazzi, 1796) 
"Keen and piercing. . . ." (William Gardiner, 1817) 
"Very noisy and hard; depicts wild passions, anger, rage, despair,  etc. . . ." (Henri Weikert, 1827) 
"Perhaps the most brilliant key; belongs to festivities and rejoicings." (T. S. R. , 1829) 
"Overstraining, overexcitement." (W. C. Müller, 1830) 
"wild passions . . . anger, rage, fury, jealousy, despair, and every burden of the heart which speaks out from its tones . . ." (Gustav Schilling, 1835-36) 
". . . It serves for the most violent passions, and expresses a defiant self-confidence, certain of its own strength." (Ferdinand Hand, 1837)  

Excluding the DeMorgan temperament, the average size of the B Major third from the data given in Figure 2.1 is twenty-one cents, with a standard deviation of 4.24 cents. The largest A Major thirds in Figure 2.3 easily approximate this average; it should be understood at this point that if a performer intends to convey emotional qualities such as the ones outlined in the descriptions of B Major above, he would choose the Stanhope, Werckmeister, or Jousse temperaments for a piece written in the key of A Major. 

A darker spectrum of emotions is depicted for minor key areas, though these ranging from purity (A minor) to despair (Bb minor). Figure 2.4 is a sampling of the many eighteenth- and nineteenth- century key-character descriptions of minor keys 41 (statistical data is given in cents and eliminates the DeMorgan temperament, as previously). 

By comparing the sizes of the minor thirds in Figure 2.5 with the list of descriptions in Figure 2.4, it is seen that a piece in A minor will reflect tenderness better in a Werckmeister or Stanhope temperament, where the minor thirds are only three or four cents flat respectively, than in a Rameau or Jousse temperament, where the third is eleven cents flat. The DeMorgan temperament, which gives A minor a third eighteen cents flat, will give A minor emotions more in keeping with those contained in the keys of C# minor or F minor — despair, deep grief, and tragedy.

When choosing a temperament for any piece, the first consideration must be that the temperament will not render chords that the ear will deem exceedingly out-of-tune. Most theorists have agreed that the largest Major third that the ear can tolerate is twenty-two cents wider than pure 42 ; though Werckmeister's first published well temperament stayed within those bounds, most musicians would choose not to place heavy emphasis on a twenty-two cent wide Major third (such as C#, F#, or Ab in the Werckmeister temperament — see Figure 2.1). The widest third for the A Major triad, however, is only nineteen cents, found in the Stanhope temperament; the other temperaments render A Major thirds ranging from eleven to sixteen cents. Knowing that all Equal Temperament Major thirds are already fourteen cents wide, most twentieth-century pianists would accept any of the temperaments discussed here for the key of A Major. After observing the range of sizes of the minor third, the reader may see that all of them, save the DeMorgan temperament, render minor thirds purer than Equal Temperament (all Equal Temperament minor thirds are fifteen cents narrower than pure). Other than the DeMorgan temperament, which is eighteen cents narrow (by no means unplayable), the sizes range from eleven cents narrow to three cents narrow. Because the selected mazurkas from Opp. 67 and 68 stay within the A Major/A minor regions, we need to eliminate only the temperaments containing Major thirds that are larger than twenty-two cents and that may be found in A Major or minor. The Stanhope temperament contains both E Major and B Major triads with thirds twenty-three cents wide; the Rameau temperament contains a B Major triad with a third twenty-nine cents wide. All three of the A minor mazurkas use the E Major triad often. The mazurka without opus contains the uses of the V/V chord (B Major) in significant places, and the other two use it less conspicuously; but their slower tempi would allow the sharp third to be noticed enough to cause discomfort. These considerations would eliminate both the Rameau and the Stanhope temperaments as feasible performance options.

Having made these observations, the next step in selecting a temperament for the pieces is to examine each temperament for its color qualities, and, with regard to the performer's interpretation of the pieces, rate the effectiveness of these temperaments.  

An examining of the opening of the A and B sections (mm. 1-4 and 29-32) of Opus 68 no. 2 reveals not only a key change, but a character change as well (see Figure 2.6). There is a shift in dynamics from piano to mezzo-forte and forte; the tempo marking changes from Lento to Poco più mosso; and the lower voices clearly become more bound to the upper. Choosing a temperament with a relatively pure minor third, such as the Werckmeister temperament, will lend a character of purity and simplicity to the opening. Additionally, it should be noted that the Werckmeister temperament contains a pure fifth in A, especially suitable to a pure affect when considering the emphasis on the fifth on the downbeats of the first two measures. Should the performer choose to interpret the opening as having a characteristic of purity, this temperament may serve quite effectively. The Werckmeister temperament, however, will also have a very strong effect on the B section because of its relatively wide Major third. The third is actually the same size as an Equal Temperament third, and not at all unplayable, but the contrast in color between the sections can be very great indeed if voiced with heavy emphasis on the thirds. 

The Rousseau, Jousse, and Broadwood temperaments are very similar to each other in the keys of A Major and minor; the Rousseau has slightly purer fifths than do the others. Their minor thirds are considerably narrower than the Werckmeister temperament; choosing any one of them would lend a character of grief or deep sadness to the opening. Their Major thirds are also comparable to an Equal Temperament Major third. With these temperaments, the performer may create an atmosphere of deep sadness (A section) that contrasts with one of strength, courage, or cheer (B section). 

The A Major triad will have the purest Major third through the DeMorgan temperament due to its "reverse" construction, and the A minor triad will have the smallest third. Applying the DeMorgan temperament to this piece will result in an opening that sounds tragic due to the small minor third, and a B section that will sound purer and duller due to the relatively pure Major third. 

These five temperaments offer effective contrast in character to the A and B sections of the piece. Should the performer wish to exhibit a wide range of character, such as extreme grief and joy, he may select the Werckmeister temperament and emphasize the flat minor third in the first section and the wide Major third in the second. Should he wish to interpret the piece more conservatively, the Rousseau temperament may be chosen, in order to allow the music to express less drastic emotions — ones which are dictated simply by the nature of the composition. 

Opus 68 no. 2 and Opus 67 no. 4 are constructed similarly, not only in form, but also in character. In the latter (Figure 2.7), however, there is no change of tempo or character markings from the A section to the B section, nor is there the dynamic contrast of Opus 68 no. 2; but the B section seems to contain a sensation of hope, while the opening paints nearly the opposite affect. This change of affect is due both to the change of key and to the each sections' melodic direction — lowering in the A section (mm. 1- 2 and 3-4) and rising in the B section (mm. 33-34 and 35-36). Because the musical content is so similar between the two sections, the performer may choose to enhance the differences in character with a relatively strong temperament, or he may feel that a strong temperament would distort the relatively subtle character change, and apply a milder temperament. One factor to consider is Chopin's use of open fifths in measures one and five; the absolute purity of the Werckmeister fifth may startle some performers; moreover, this absolute purity creates a certain cold sensation, which may not appeal to the performer's interpretation for the opening of this piece. One tuning option that may serve this piece well is using the DeMorgan temperament; its Major third is only eleven cents wide, creating slight brightness without harshness, and its minor third is only three cents narrower than Equal Temperament. Moreover, the opening of this piece has the capabilities of portraying a "tragic" character, suitable to DeMorgan's very small minor third. Compared to the others, the Broadwoods and Rousseau temperaments are more similar to Equal Temperament for these particular keys, and would render sharper Major thirds than does the DeMorgan temperament. Knowing this, the performer must decide how bright he prefers the middle section, and make a choice based on that decision. 

The other A minor mazurka in consideration here, without opus number (Figure 2.8), is decidedly different in character than the other two. The tempo marking is Allegretto, and the melody that begins in the left hand incites a feeling of intensity and excitement. The change of key that marks the B section contains the same forward drive as the opening. Although both sections are marked piano and are to be played at a faster tempo than the previous examples, a carefully chosen temperament will still enhance the character change between the two sections. The relatively thin texture of the opening in comparison to the wider melodic range of the B section suggests the possibility of contrasting a colder color with a warmer one. The Werckmeister temperament may create too great a contrast, especially with the brightness of the Major third in juxtaposition with the absolute purity of the fifth. The Rousseau temperament will soften this effect somewhat, its slightly flattened fifth muffling the differences in color; the Broadwood temperaments take this effect one step further, with fifths four cents narrow. Either of the Broadwood temperaments will lend a slightly purer sound to the opening than will the Rousseau temperament, because their Major third (C-E) on the downbeat of the first measure is purer than in Rousseau's temperament. If too much emphasis is given to this opening diad, the DeMorgan temperament may be less effective; its C-E third is seventeen cents wide. However, the DeMorgan temperament, because of its purer A-C# third, gives the B section a warmer, gentler character than do the other temperaments.

Another mazurka that uses the general ABA form, with the middle section in the relative minor, is Opus 59 no. 3 in F# minor (Figure 2.9). Upon observing the range of sizes of the minor third F#-A, the reader will see that these nine temperaments offer only four different sizes: ten cents narrow (Rameau), fourteen cents narrow, (DeMorgan), eighteen cents narrow (Stanhope), and fifteen cents narrow (the remaining six temperaments). Although the ear is less sensitive to impurity in the minor third, the difference in color between the Rameau and the Stanhope temperaments can be heard easily: the Rameau third will sound "stronger" than the Stanhope third, and could be used for such character as anger or malevolence. As already seen, narrower minor thirds, such as the Stanhope third, may incite feelings of despair or sadness. 

This mazurka easily stays within the F# minor/Major regions; there is no serious emphasis on triads other than the relative major, the subdominant, and the dominant. This key range would again eliminate the Rameau and Stanhope temperaments as performance options — Rameau's Major thirds in B Major, F# Major, and C# Major all exceed the twenty-two cent limit, as does the B Major third in the Stanhope temperament. The Werckmeister temperament may not suit this particular piece well, due to its extremely wide Major third F#-A#; this third has a conspicuous role in the consistently chordal texture of the B section. The remaining temperaments offer very few differences in color: the Broadwoods, Jousse, and Rousseau temperaments are very similar in both F# minor and Major keys, though the Jousse temperament does push the B Major triad to its limits. If the reader compares any of these four temperaments to Equal Temperament, he will find only the slightest difference for the F# Minor and Major regions, but a more significant difference in the B Major areas (see mm. 47-48). The DeMorgan temperament, because of the nature of its construction, offers a much purer sound in the B section, implying characters of strength and solidity. In the A section, it does not offer a much different color from the other temperaments. Having considered these factors, the performer may base his choice of temperament on the F# Major section: if he want to begin the piece with a mood of grief or rage and lead it to strength and stability, he could apply the DeMorgan temperament; but if he wishes to turn the darker opening into a uninhibited celebration, he could choose the Jousse or Rousseau temperaments. The Broadwood temperaments will offer slightly less affectual extremes. 

The above examples concern themselves with the differences between significantly different key areas. The well temperaments were constructed so that the characters of closely-related keys differed only slightly, but moving around the cycle of fifths a full three steps to a key's parallel Major or minor produces noticeable changes in color. Though the color differences between closely related keys are less obvious, a carefully chosen unequal temperament can have a significant effect on the piece. The mazurka Opus 68 no.1 is in written in the key of C Major, in rounded binary form, with the middle section in the subdominant. 

The opening measures (see Figure 2.10) naturally convey militant or courageous expressions; the middle section is more placid, pensive, and distant in nature. Because the A and B regions of the piece are written with such few accidentals, any of the temperaments will apply without exceeding the Major third's twenty-two cent limit. Most performers may want to highlight the militant character of the opening, and choose a temperament with a very pure Major third. In this case, either the Rameau and Stanhope temperaments would serve well; the Stanhope temperament creates an greater sensation of strength than the Rameau temperament because of its pure fifth. Both of these temperaments also contain slightly wider thirds for the F Major region, creating a brightness that may lend itself well to this section's more carefree nature. The Werckmeister temperament, although it does contain a relatively pure Major third, somewhat softens the effect of strength because of its flattened fifth. The Broadwoods are only slightly purer than the Jousse temperament; these three temperaments will have a weaker portrayal of a militant character. Even though the C-E Major third of the Rousseau temperament is much wider than Werckmeister's, there is not much difference between each one's effect on the opening; this is due to the relative purity of Rousseau's fifth. In the B section, however, the Rousseau temperament creates a much brighter sound because of the wide third. The listener, having familiarized himself with these relatively pure sounds of Major thirds, will find a great shock when he returns to the fourteen-cent wide Equal tempered third, and an even greater shock when applying the DeMorgan temperament to this example. If the purity of the Major third is an effect that the performer finds attractive, he will undoubtedly want to choose either the Werckmeister, Rameau, or Stanhope temperament; or he may choose to couple this piece with one containing a large number of accidentals, and apply one of the Broadwoods, the Jousse, or the Rousseau temperament to the pair. 

Those mazurkas that contain secondary key areas in the relative Major or minor allow us to observe different qualities of the triads without leaving the color-range of a single key. Opus 30 no.1 (Figure 2.13) is written in C minor; its middle section is in Eb major. The largest difference in the size of the C-Eb minor third is between the Werckmeister (twenty-one cents narrow) and Jousse (thirteen cents narrow) temperaments. The Jousse temperament, with its relatively narrow minor thirds, produces a mournful or yearning character created by the subtle presence of beats; this affect may be more desirable from an interpretative view. The other temperaments contain minor thirds ranging from fifteen to seventeen cents and will produce a milder atmosphere. For the B section, the range of Eb-G Major thirds is also wide, but the widest (Stanhope) is still only two cents wider than in Equal Temperament. The Stanhope and DeMorgan temperaments, however, do produce a significantly brighter tone than do the Rameau, Rousseau, or Jousse temperaments; this latter group produces a warmer, more approachable tone and may be more appropriate for this passage. For this mazurka, the performer may choose between temperaments that will either darken the opening or brighten the middle section. 

Similar moods are presented in Opus 63 no. 2 (Figure 2.14), which begins in F minor and moves to its relative major. The temperament with the narrowest F-Ab minor third (Rameau) produces feelings of absolute darkness and despair, extremely effective in the opening, but impractical for the middle section (the Ab-C Major third is twenty-five cents wide). The Werckmeister temperament, then, will offer the option with the most drastic color change, containing a minor third narrow by twenty-one cents and a Major third wide by twenty-two. The effect that the Werckmeister temperament has on the middle section is not as disagreeable to the listener as it may have been, even with its twenty-two cent wide third, because Chopin wrote almost exclusively in a texture that did not place a C directly above an Ab. By contrast, Opus 59 no. 3, contains a texture in its F# Major section that makes Werckmeister's twenty-two cent wide Major third more obvious. For Opus 67 no. 4, the other temperaments will render only subtly different shades of brightness in the middle section — the Rousseau, DeMorgan, and Jousse temperaments produce nearly the same warm effect, evoking a more tender aura than do the brighter Broadwood temperaments.

Opus 24 no. 1 (Figure 2.15) is in the relatively simple key of G minor; its secondary key area, Eb major, is only one accidental away. The opening contains a large number of dominant and mediant triads, while the middle section contains little more than the submediant and its dominant. None of the temperaments, then, present the problems of having Major thirds too extreme to play, and, among the temperaments discussed here, there is quite a range of available affects and colors. Although the first downbeat is a perfect fifth, the performer need not be too concerned that this interval be fairly pure: the narrowest fifth D-A (Stanhope) will produce only a gentle beating, less discernable due to the quiet dynamics and decay of the A from the pickup. However, the ten-cent narrow fifth G-D in the Stanhope temperament may cause an undesirable effect for the rest of the piece. The Rameau temperament has a relatively pure minor third G-Bb, and a perfectly pure Major third Bb-D. Because of the cold sensation of the pure Major third, however, the performer may want to choose one slightly wide, such as with the Rousseau or Jousse temperament. The Broadwood temperaments will offer wider Major thirds, and comparable minor thirds. The DeMorgan temperament's thirds for the A and B sections, in this case, are comparable to those of the Broadwood temperaments; but the DeMorgan temperament has a noticeably purer G-D fifth. Choosing a Broadwood temperament would give the piece a more mournful quality at the opening than in Equal Temperament; the middle section would sound simpler because of the purer Major third.

The mazurka Opus 6 no.3 (Figure 2.11) is written with a relatively large number of accidentals (E Major), and contains a middle section written mostly in A Major. Its opening is similar in character to the previous example; it is written with strong emphasis on the tonic, and contains chords rising in both register and dynamics. Most of the temperaments under consideration provide E Major with a majestically bright Major third; only the Stanhope is too wide and the DeMorgan is, by comparison, dull. Should the performer want to choose a temperament that would provide contrasts between the more militant A section and the more lyrical B section, he may select one that contains a brighter third in E and a purer third in A. For this interpretation, only the DeMorgan and Werckmeister temperaments would not apply. 

The DeMorgan temperament has a very soothing effect upon Opus 50 no. 2 (Figure 2.12), which contains outer sections in Ab Major, and an inner section in Db Major. The other temperaments will render much brighter colors, creating an atmosphere of intensity or angst, especially in the middle section. If the DeMorgan temperament does not suit the performer's taste, he may elect to apply the Jousse temperament, which has purer thirds than the others. Overall, for this mazurka, the performer must decide upon an interpretation that either begins brighter at the opening (the Broadwood "Best," Jousse, or Rousseau), or one that leads into a brighter middle section (DeMorgan). The Stanhope and Broadwood "Usual" temperaments offer little or no contrast, and the Werckmeister temperament, with its twenty-two cent-wide Major third, may be too bright for most performers. 

To this point, relatively small changes in character between sections have been considered, in looking at pieces that modulate to related keys. In considering those pieces whose middle section occupies a remote key, the changes in color from one section to another may be drastic. Because Chopin himself must have used some form of unequal temperament, we may assume that he intended a change of affect to accompany the change of key. As we have already seen, this intention is also evident in the dynamic and textural changes between a piece's opening and middle sections. Opus 33 No. 3 (Figure 2.16) presents a drastic change from its opening to middle section; observe the changes in dynamics, bass register, accompaniment style, and melodic range.

Most of the temperaments will provide a dramatic difference between the sizes of the C-E Major third and the Ab-C Major third. The DeMorgan temperament is the exception, offering only a two-cent difference between the two thirds; additionally, selecting this temperament would begin this piece with a seventeen-cent wide Major third, creating a brightness not at all in keeping with the Semplice character marking at the opening. The Rousseau temperament contains a much purer C-E third, but the contrast between it and the Ab-C third is not nearly as great as the other temperaments. A temperament with a pure or nearly pure C-E third, such as the Werckmeister, Rameau, or Stanhope temperaments, will indeed produce a "simple" effect on the opening; but, because purity in one key means sacrifice in another, the Ab-C thirds in these temperaments will also be the least pure. The Rameau Ab-C is, in fact, too wide to be used at all. For those performers that have accepted extremely wide thirds, the Werckmeister temperament is perfect for offering extreme contrasts; its C-E third is only four cents wide, suitable for a "simple" affect without coldness, and the twenty-two- cent-wide Ab-C third will bring affects of exuberance to the middle section. For those performers that require a slightly purer Ab-C third, the Stanhope temperament will produce comparable extremes in character. The Broadwood temperaments contain C-E thirds not too distant from purity; their change in color to the middle section will be less noticeable than the other temperaments. 

Opus 68 no. 4 (Figure 2.17), written in F minor, presents one of the most drastic key changes in all the mazurkas. It, too, is in rounded binary form, but the B section is not the point of change in color. In this piece, Chopin marked Dal segno senza fine at the end, a concept which works better in theory than in performance; most performers choose to superimpose a Dal segno al fine approach to the piece, placing the Fine on the second beat of measure twenty-three. The most interesting key-color change occurs in the A section, at measure fifteen. The change in texture between the two sections is observable in both the melodic and harmonic rhythm, and in the accompaniment style. Depending upon the performer's interpretation, the opening section could express feelings of deep yearning, grief, or solitude. The A Major section can bring a moment's relief from these darker emotions, expressing hope and encouragement. Knowing that the Major thirds of Equal Temperament are fourteen cents sharp, and the minor thirds are fifteen cents flat, let us consider which temperaments would offer greater extremes in color. 

The Rameau and DeMorgan temperaments have A Major thirds only eleven cents wide; pianists who are accustomed to Equal Temperament will find their quality slightly dull. The Rousseau, Jousse and Broadwood temperament contain A Major thirds that are comparable in size to Equal Temperament; however, the F minor thirds of these temperaments are purer than in Equal Temperament. In order to heighten a characteristic of grief, the F minor triad will need to be smaller than fifteen cents, as in the case of the Werckmeister and Stanhope temperaments. These temperaments also offer brighter thirds than does Equal Temperament, providing strong contrast for the A Major section. In this case, the Stanhope temperament will function without the offence of a Major third greater than twenty-two cents; the E Major triad that would serve as this section's dominant appears with a voicing that obscures the E-G# Major third (see mm. 16 and 18). Should the performer desire a slightly less bright A Major section, but a darker opening, the Rameau temperament would be an ideal candidate. The Broadwood temperaments offer purer A-C# Major thirds than in Equal Temperament, and F-Ab minor thirds only three cents narrower. In these temperaments, though the listener may be able to detect color changes between chords throughout the piece, the change of mood from A Major back to F minor (mm. 19-21) will be much less eventful. 

The content in this chapter was presented in order to acquaint the listener with basic differences between a variety of unequal temperaments. Becoming acquainted with the variation of color shades that each temperament gives to individual keys is the most fundamental part of choosing an unequal temperament for a performance setting. If the reader has not had the opportunity to acquaint himself with the different sounds that each temperament produces, he will have gained only a theoretical understanding of the constructions and qualities of the temperaments listed here. Although this understanding is necessary, sensitivity to key characteristics and color change at a performance level is developed only through aural training. Before continuing to the next chapter, the reader should be fully acquainted with the aural examples for Chapters I and II.    

3 The Effects of Unequal Temperament on Harmonic Progressions in the Mazurkas  

After the reader has become acquainted with the variety of colors that unequal temperaments create, he should increase his aural sensitivity to the level of counting beats. Beats are a physical phenomenon created when any two sound waves are out-of- sync, or out-of-tune, with each other. Figure 3.1 is a graphical portrayal of a pure (sine- wave) tone at Concert A (440Hz). Because this sound is cycling at 440 cycles per second, there are ten full cycles of the wave in .022727272 seconds. This is seen in the figure at the right, and explained mathematically with the following simple formula:  

In Figure 3.2, a pure sine-wave cycling at 482Hz (red) is graphically overlaid with the previous figure. Because the wave at 482Hz (red) is cycling faster than the one at 440Hz (blue), it will complete a higher number of cycles in the same amount of time-space. Its exact amount of cycles may be calculated as follows: 

   482.0 Cycles / second 
 x   0.22727272 Seconds 
    10.95 Cycles 

Figure 3.2 is only a graphical comparison of the speeds of both waves. Figure 3.3 is the aural result of both tones being sounded simultaneously. It shows that the resulting sound begins with the highest amplitude (loudest volume), lowers to near silence at .012 seconds, and then returns back to the original volume just before .022727272 seconds. Thus, the above is a graphical representation of nearly a single beat. Figure 3.3 is mathematically explained by subtracting the two tones' speeds, and then multiplying that speed by the containing time-space: 

  482.0  Cycles / second 
- 440.0  Cycles / second 
   42.0  Cycles / second 

  42.0  Cycles / second 
x  0.22727272 Seconds  
    .954545424 Cycles 
 or approx. 1 beat in .022727272 seconds 

Beats occur when two tones at impure relationships to each other are sounded simultaneously. However, if two tones that share the same base frequency are sounded together, they are "in tune," and no beats are present. For example, tones cycling at 300Hz and 600Hz do not produce beats when sounded simultaneously, because the latter is exactly twice the speed of the former. Nor do tones cycling at 200Hz and 300Hz produce beats when sounded simultaneously, because they have a fundamental of 100Hz in common. If, then, the frequency in hertz of any given note is known, then the frequency of pure intervals relating to it can be obtained by multiplying the frequency of the known note by the ratio of the pure interval. For example, to calculate the frequency for a pure fifth above A440, the following equation would be applied:

  440.0  Cycles / second 
 x  1.5  (ratio of a pure fifth) 
  660.0  Cycles / second 

If the frequency of any two out-of-tune notes are known, and the ratio representing what would have been a pure interval between the notes is also known, then impurity of the actual interval between the notes may also be calculated. For example, to determine the number of beats that the impure fifth A440 - E663 produces, the following formula could be applied:  

  663.0  Cycles / second 
 - 440 x 3.2  (660 — pure fifth above A440) 
    3.0  Cycles / second OR 3.0 beats / second 

To determine the number of beats in a triad containing a slightly impure fifth and Major third, the impurity of each interval in the triad is calculated and then totaled.  

  C = 263.441125299 Hz 
  E = 330.005634324 Hz 
  G = 393.809677039 Hz  

  330.005634324 Hz (given E)  
- 329.301406623 (263.441125299 x 5/4; pure E above given C)  
     .704227701 beats per second (E is sharper than a pure C-E third)  

   393.809677039 Hz (given G)  
 - 396.006761189 (330.005634324 x 6/5; pure G above given E) 
    -2.197084150 beats per second (G is flatter than pure E-G third)  

   393.809677039 Hz(given G)  
 - 394.742531923 (263.441125299 x 3/2; pure G above given C) 
    -0.932854884 beats per second (G is flatter than pure) 

    0.704227701 beats between C and G 
    2.197084150 beats between C and E 
  + 0.932854884 beats between E and G  
    3.834166735 total beats  

Using beat speeds as a method of comparison between various temperaments can be much more effective than using only sizes in cents. However, in most literature on historical temperaments, data is given only in cents. The following method is used to convert cents to ratios: 

cents = log(ratio) x 1200 / log(2) 

To convert cents to ratios, then, the reverse of this formula is applied: 

ratio = 10^(cents / (1200 / log(2))) 

By applying the above formulas and methods to Figure 2.1, Figure 3.4 may be constructed for realizing the beat speeds of the Major triads in the nine temperaments at hand.  

After the listener has acquainted himself with the color changes that each unequal temperament provides, he may then re-listen to individual triads in these temperaments and better understand how these colors were perceived. Although the above chart will not show if a triad is composed of either a wide third or a narrow one, it will show more clearly the severity of the temperament's imperfection. 

In Figure 3.4, observe the Equal Temperament row (last row). Beginning on column "g", trace the beat speed for each triad, going upwards chromatically. Column "ab" has a slightly higher beat speed, and column "a" even higher. Even though these triads are composed of exactly the same intervals, because the frequency of the notes in the chords increase logarithmically as they ascend the chromatic scale, the distance between them in Hertz is greater, producing faster beats. When one observes each triad's beat speed in the unequal temperaments' rows, starting at column "g" and ascending chromatically, the pattern is much different, because these temperaments were constructed so that the relative purity of the triads would increase or decrease through the cycle of fifths. A definite contour of beat speed can been seen, however, in the unequal temperaments by tracing the beat speeds beginning on c' and moving by whole step (every other column) across the row. 

The beat speed for minor triads are much less predictable. Observe Figure 3.5 and notice the lack of pattern in the beat-speeds of all unequal temperaments. Because the beats produced by impure minor thirds are more difficult to discern than those produced by Major thirds, the reader may find it helpful to listen to the minor triads in each temperament while referring to this figure.  

In the nineteenth century, professional tuners counted the beat of fourths and fifths, but not the beats of Major or minor thirds; the color quality of the thirds, however, was compared and adjusted. In the following sections, in order to make more concrete distinguishments between the different temperaments, we will refer to each chord's beat speed, rather than color quality, as a method of comparison.  

The graphics in the musical examples given in the remainder of this chapter have been constructed with the methods outlined above. Figure 3.6 shows a contour that has been created based on the beat speeds of the first five chords of Op. 41 no. 2. The example is a simple chord progression (V7/iv-iv-V7-iv6/4-i) in a common key. It is therefore to be expected that a definite similarity of contour would exist between most of the temperaments; but notice that the contour of the DeMorgan temperament runs against the others, and that the extremities of Equal Temperament lie between those of every standard unequal temperament. Also, observe that the overall contour of the beat speeds, which are scientifically calculated, reflect what many may agree conforms to the flow of musical tension and resolution through the phrase. That is, the first chord (V7/iv), would have a strong tendency toward the second (iv); the third chord (V7) would likewise be drawn to the last (i) chord. To a large degree, all of the temperaments reflect this scheme; some temperaments reflect it with more contrast. The Werckmeister, Rameau, and Stanhope temperaments offer the widest ranges of tension and resolution between the first four chords. Of these, only the Stanhope temperament creates fewer beats in the final chord (i) than in the penultimate one (iv); the Rousseau temperament does the opposite of this. A slightly more dramatic contour is seen in the Broadwood "Usual" temperament than in the "Best," but these are much more subdued than in the earlier temperaments. The beat speeds of the Equal and DeMorgan temperaments differ very little from chord to chord; this will result in a much less dramatic effect on the forward motion of the phrase than in the other temperaments.  

The second phrase of Opus 6 no. 2 (Figure 3.7) demonstrates the differences between temperaments in a phrase containing only dominant-tonic movement. In this case, the temperaments do not follow the same general contour: the Rousseau, Broadwood, and Equal Temperaments are particularly flat; but the Jousse and Stanhope temperaments share contours opposed to the Werckmeister, Rameau, and DeMorgan temperaments. This latter group strongly reflects the standard dominant (tension) to tonic (release) sensation — in these temperaments, the dominant-seven chords beat faster than do the tonic chords; moreover, in the Werckmeister and Rameau temperaments, the secondary dominant is considerably purer than the dominant, creating an even stronger pull toward the tonic. While the effect of these temperaments has a "theoretically correct" appeal, the other temperaments may have an effect more suitable to the forward motion of the phrase. Noticing a dramatic change from a relatively impure dominant to a purer tonic may cause the performer to rush through the dominant chords and linger more on the tonic chords; but having tonic chord with a higher beat speed than the dominant chords, as in the Jousse and Stanhope temperaments, the tendency for forward motion will be strong both in the dominant measures and in the tonic measures. The choice remains contingent upon the interpretation of the performer. 

The middle section of Opus 33 no. 2 (Figure 3.8) has nearly the same chord progression as in the previous example, but the contours of the beat speed of the temperaments are much more consistent here. Perhaps the most striking effect of the graph above is the rapid and drastic change of beat speeds between chords eight through thirteen. The Rameau, Stanhope, and Werckmeister temperaments clearly stand out from the other temperaments, but the strongly similar shape of the other temperaments allow us to venture to discuss what may have been composer's reasons for writing the way he chose. If the temperament that Chopin had in mind when this piece was conceived had a similar effect as the majority of these temperaments, one could see that since the chosen texture for the dominant chord (8) is considerably purer than that of its resolution (9), lingering for two beats on the dominant and resolving to the tonic only for a moment may be preferable to the other way around. Between chords 10 and 11, the effect in beats is exactly the same, and the harmonic rhythm is duplicated, though movement from chord to chord is only a 6-5 resolution within the dominant. The general effect between chords 12 and 13 is moving from a relatively impure chord to a purer one; lingering then on this last chord, even though it is a dominant, helps bring the phrase to a close and prepare the listener for what is to follow. 

A similar pattern to this example is seen in Opus 41 no. 2 (Figure 3.6) — perhaps Chopin recognized the purity of the A minor chord (chord 2) in contradiction to its dominant (chord 1) and chose the rhythm to heighten that effect; if so, the rhythms of the two measures that follow could only have been chosen for the same reasons.  

For the opening of Op. 30 no. 2 (Figure 3.9), the Rameau temperament certainly stands out. Remembering that the Rousseau temperament could be considered unusable in this key because of the 36-cent wide Major third in F# Major (see Figure 2.1), the performer may choose to immediately eliminate it as an option. A few things about this phrase are interesting to note: firstly, the DeMorgan, and Equal Temperaments move against the others from chord one to two. The latter group will produce a gentle increase of beat speed, causing the listener to anticipate the resolution to the tonic to a greater degree. Secondly, the average beat speed of the minor dominant chord (4) approximates that of the Major dominant chord (2), and will again pull the motion of the phrase forward. As the dynamics shift from forte to piano, Chopin moves to a key (G) that is one accidental closer to C, which naturally results in purer sounds. Though G is by nature a purer key than D in most of the unequal temperaments, the D chord (6) sounds purer than the G chord (5) because of the texture Chopin chose — eliminating the third in the bass drastically reduces the number of beats (refer again to Figure 2.1 to compare the relative purities of fifths and thirds). Thirdly, Chopin returned to the more impure key (B minor) for the dynamic shift back to forte, but lowered the number of beats at the end of the phrase by dropping down to a lower register. Lastly, the beat speed of the iio6 chord produces significantly more tension than does the V7; giving it two quarter notes to the V7's one will prolong the drive to the tonic. 

In the opening of Op. 67 no. 2 (Figure 3.10), the beat speeds of the Rameau and Werckmeister temperaments clearly stand out against the other, later temperaments; this is largely due to the differences in purity between the C minor and G minor triads (see Figure 2.2). Considering the score, the performer may wish to select a temperament in which the G minor chord (Chord 2) does not produce significantly more beats than the dominant seven which precedes it. If this be the case, he would eliminate the DeMorgan and Stanhope temperaments. But notice that throughout the phrase, hardly any of the temperaments render a dominant/tonic resolution in which the tonic is purer than the dominant. The Werckmeister, Stanhope, DeMorgan, both Broadwoods, and Equal Temperament all produce less pure resolutions between chords 1-2, 4-5, and 6-7. The Rousseau and Jousse temperaments do little better, producing a slightly purer resolution only between chords 4-5. Only the Rameau temperament consistently produces dominant-tonic resolution in which the tonic contains lower beat speeds than the dominant. For the end of the phrase, however, all of the temperaments, DeMorgan included, render purer tonic chords in the iv-i resolutions; how each temperament affects these chords may then be the performer's deciding factor when selecting a temperament. The Werckmeister and Rameau temperaments offer the most contrast between the chords; the Stanhope and DeMorgan temperaments offer the least. Since Chopin repeats this plagal cadence, one may surmise that he considered it more significant than simply the end of the phrase. 

Should the performer adopt this mentality and choose to capitalize on this cadence, the Rameau temperament's gross differences between the final chords may neutralize any advantage that it may have had over the other temperaments in the beginning of the phrase. The Rousseau and Jousse temperaments may offer the best compromise; they both produce only slightly impure dominant-tonic resolutions, and convincing iv-i cadences at the end of the phrase. 

Figure 3.11 is a striking example of the differences between the temperaments in a historical context; Chapter 1 showed that the general development of unequal temperament was from strong purity in "common" keys and noticeable impurity in "black" keys to only slightly purer common keys than black keys; Equal Temperament finally made no distinguishments in purity. Opus 68 no. 3 is written in the key of F Major — a key that was often made nearly as pure as (if not purer than) C Major. The graph in Figure 3.11 shows that the earliest unequal temperaments, the Werckmeister and Rameau temperaments, form contours that are consistently below those of the other temperaments; the Stanhope temperament, being based on a Kirnberger model, also shares a place at the bottom of the graph. But the DeMorgan temperament, constructed so that F Major is one of the least pure keys, creates a beat contour that sits above all the others, indicating the most out-of-tune of all the temperaments at hand. The contour for Equal Temperament lies just below the DeMorgan, indicating that all of the other unequal temperaments create purer sounds. 

The reader may observe for himself the particular differences in contour among the temperaments. It will be mentioned here only that the general direction of each line is downward, indicating a general direction of impurity to purity. This is due not to a modulation from an impure key to a purer one — for even the DeMorgan temperament follows the pattern — but rather from a higher register to a lower one. The range of purity for Chord 17 is far lower than that of Chord 3, even though they are comprised of the same notes, simply because of the range in which they fall. By moving from a higher register to a lower one, Chopin created a gentle move to slower-beating chords, regardless of the temperament chosen; this direction will certainly lead the listener into the piano section that is to follow. 

The effects that the temperaments have on chord progressions are as unique as the temperaments themselves. Though the exact beat speeds in any given chord are difficult to predict when choosing a temperament, it is nevertheless not necessary for the reader to attempt to reconstruct "contour charts" similar to the ones shown here; if the factors discussed in this chapter are taken into consideration — the relative purity of the triads, voicing, range, number of notes, etc. — approximations of beat speeds can be made quickly by constructing charts similar to Figure 2.1-Figure 2.2. 

Neither is it necessary for the audience to realize the exact beat speeds in a chord progression in order to add to (or detract from) the performer's interpretation. The beat speeds have been scientifically calculated for this chapter in order to fully distinguish the characteristics of the temperaments, and to provide the reader with visual aid as he may desire to sharpen his own aural sensitivity to the differences between the temperaments. Should the reader also be patient enough to train his ear to the level of recognizing differences in temperaments over a single melody line — which will be covered in the next chapter — he will, like most nineteenth century pianists, be able to order a precise tuning that has been tailored to his own preferences and in every way enhances his own realization of the music.

4 The Effects of Unequal Temperament on Melodic Lines in the Mazurkas 

The structures of the temperaments discussed in this paper have been drawn from a number of sources. Generally, these sources rely on one of two different methods of describing unequal temperaments: either by comparing them to Equal Temperament, or by listing the notes in chromatic order and giving each one's distance to C or A. All known sources give data based on the cents method. Owen Jorgenson described Rousseau's Equal-Beating temperament (Figure 1.7) as seen in Figure 4.1. This method is especially useful to tuning technicians who wish to duplicate a historical temperament. 

The second method described above is slightly different than Jorgenson's approach; Barbour used this method in his Tuning and Temperament to describe Werckmeister's 1/4-Comma Temperament (Figure 1.3). Barbour's method also indicates the position of the comma.  

In order to compare the effects of unequal temperament on a single melodic line, and in order to demonstrate the impurity of Equal Temperament, the author devised a system that compares each tempered melodic interval to its corresponding pure interval; the basis of comparison in this system is the cent. 

Obtaining the sizes of tempered intervals is relatively simple, given a system such as Barbour's or Jorgenson's. With Barbour's system, interval sizes based on C are seen immediately: a Major sixth up from C is simply the number of cents given to A, since the system is based on C. A Major sixth up from C# is obtained by subtracting 96 (Db) from 1002 (Bb), resulting in 906 cents — the C#-A# Major sixth is therefore 6 cents wider than the C-A Major sixth. 

Obtaining the sizes of pure intervals requires the ratio-to-cents conversion method described in Chapter 3. Using the method  
   log(ratio) x 1200 / log(2) = cents 
the size of a perfect fifth (ratio 3:2) is obtained as  log(1.5) x 1200 / log(2) = 701.955 cents or approximately 702 cents. Figure 4.3 shows the sizes of pure intervals within the scope of an octave.  

Listings such as Jorgenson's and Barbour's are more concerned with describing the structure of a temperament for the purposes of reproduction than with comparing intervallic size; for this purpose, one-dimensional (linear) portrayals suffice. In order to fully capture detail when presenting intervallic comparison, a two-dimensional grid is necessary. Based on the data given in any linear description of a temperament, intervallic data may be drawn and mapped. Interval size may be displayed in either absolute or relative terms. The system that the author has devised produces the a parallel to Figure 4.2.  

The above method 49 combines Jorgenson's use of relativity (cents away from Equal Temperament) and Barbour's inclusion of the comma (comparison to purity).  

The musical examples for this chapter will demonstrate the different effect each temperament has on a melodic line. Numbers between notes indicate the relative purity of each interval in the method described above.    

Figure 4.5 is an example from the middle section of the Mazurka Op. 67 no. 2, in which all harmonies are dropped and the right hand is given this single melodic line. The minor second D-Eb takes a prominent part throughout this excerpt. Since the melody is presented without accompaniment, changes in temperament are more obvious. An Equal- tempered semi-tone is twelve cents narrower than a pure one; the Jousse and Rameau temperament thus render a significantly purer (and wider) interval that Equal Temperament does to the D-Eb semitone. After being acquainted with the different sensations each temperament creates in this minute interval, the performer may choose a temperament that minimizes this interval, to go along with an interpretation that renders little importance to it; or, he may choose a temperament that renders it as wide as possible, to support an interpretation that gives it melodic significance. 

Two other significant intervals in this selection, and ones that occur more than once, are between G-F# and F#-Eb. The DeMorgan and Equal Temperaments are similar in size in both of these intervals; together they represent one extreme for both intervals — the smallest G-F# distance and the largest F#-Eb distance. The other extremes are also found in a single temperament, Rameau's modified meantone, rendering the largest G-F# and smallest F#-Eb distances. Since the widest intervals cannot be presented in a single temperament, most performers may choose to place melodic significance on the G-F# semitone, concluding that the size of the minor third will naturally demand attention. Given this decision, the performer would select the Rameau, Stanhope, Werckmeister, or a Broadwood temperament. The Rameau, Stanhope, and Werckmeister also render wider D-C minor sevenths (mm. 37). 

The two most prominent intervals in Figure 4.6 (Op. 7 no. 2) occur in measures 2 (minor second) and 4 (Major second). None of the temperaments produce purity in one of these interval without considerable sacrifice to the other: the Stanhope temperament produces a pure minor second, but a narrow Major second; the Major second in the DeMorgan temperament is nearly pure, but the minor second is sixteen cents narrow. Because the minor second also opens the piece, the performer may choose the side of purity in the minor second. An acceptable compromise may be selecting a Broadwood temperament. With this option, both the minor and Major seconds are only 5 and 6 cents narrow, respectively. However, the compromise will also be heard between measures 2 and 3; the D-E Major second is narrower in the Broadwood temperaments than in some of the others. Only the ear and an interpretation in accordance with the selected temperament will be the final determinants.  

Figure 4.7 was chosen to demonstrate the differences in sizes between Major seconds (mm. 1 and 3) and the Major and minor thirds that create a minor triad (mm. 2 and 4). Concerning the triad, recall that the C#-G# fifth will be pure in most unequal temperaments, since the purity of thirds is given to the "common" keys, and the purity of fifths is comprised for the purity of thirds. This is further shown in that the first five temperaments in m. 2 contain Major thirds that are as wide as the minor thirds are narrow; that is, the purity of the fifth is calculated be the sum of the purity of the thirds, since 3:2 (fifth) = 5:4 (Major third) x 6:5 (minor third). In the DeMorgan temperament, the sum of the sizes of the Major third indicates a three-cent narrow fifth — one that has been sacrificed of purer thirds. The Broadwoods begin the compromise to Equal Temperament. It was seen in Chapter 2 that the general principle for affects in the minor mode is that as the minor third decreases in size, the grief becomes more poignant. This principle is reinforced when the extremely narrow third follows the wide Major third, as in the Stanhope or Jousse temperaments. 

Due to the nature of temperaments, minor seconds are generally further from purity than Major seconds. When a minor second follows two consecutive Major seconds, as in the cases above (see mm. 1-2 and m.3), the distance between the groups can be softened by contrasting particularly large Major seconds with smaller minor seconds. In m.1, the Werckmeister temperament maximizes this effect, going from a slightly impure Major second to a perfectly pure one, and then falling to an extremely narrow minor second. In the third measure, this effect is seen again in the Werckmeister and, in varying degrees, the Rameau, Rousseau, and Jousse temperaments. The DeMorgan temperament, when contrasted with these particular other temperaments, sounds flat due to the second's similarity in size. In the Broadwood and Equal Temperaments, all color changes from one Major second to another are virtually lost.  

Figure 4.8 demonstrates different sizes of non-consecutive minor seconds. Observe how each temperament affects the three groups of minor seconds in mm.1-2 (G-Ab, B-C, and D-Eb). With the exception of the DeMorgan temperament, all of the unequal temperaments begin with a narrow minor second, are followed by a wider one, and return to a narrower one. Even the Broadwood temperaments present this effect in considerable clarity. To what degree this effect will be carried out is to be determined by the performer.  

Modern tuners occasionally choose to brighten the extremities of the piano by widening the octaves. If this technique is not used, the sizes of perfect fourths and fifths will complement each other perfectly, as seen in m.3 of Figure 4.9 — the perfect fourth will always be exactly as wide as the fifth is narrow, regardless of the temperament. Mathematically, this is because the pure octave is the sum of the pure fifth and fourth (2:1 = 3:2 * 4:3). Aurally, this can mean a significant difference between temperaments such as the Werckmeister and Stanhope when fourths and fifths are played consecutively (as in m.3 above). The same principle does not apply, however, to Major seconds and Major thirds. That is, the sum of the purities of the perfect intervals will always equal zero — the purity of a perfect unison or octave (m.4); but the sum of the purities of Major seconds will never be the purity of a pure Major third, because 5:4 (pure Major third) is not 9:8 (pure Major second) * 9:8. Mathematically, the two pure Major seconds produce a Major third with a 81:64 ratio (a Pythagorean Major third), which is 22 cents wider than a pure (5:4) Major third. Aurally, this can be demonstrated perfectly in the Werckmeister temperament of Figure 4.9: measure 2 contains two consecutive pure Major seconds — C-Bb and Bb-Ab — measure 4 contains a Major third (Ab-C) that is 22 cents wide. In this situation, the performer should realize that purity in one of the intervals will necessitate impurity in the other; his interpretation will influence his selection of temperaments. 

Figure 4.10 demonstrates the different effects of the temperaments on the size of the tritone. Though the exact size of a pure tritone has been arguable (see Appendix A), the numbers here at least provide a method of comparison. The size chosen — 600 cents — is that of an Equal-tempered tritone, and is equated with the ratio sqr(2):1. With this size as the basis of a "pure" tritone, an octave divided by the tritone will mathematically behave in the same manner as division into perfect fourths and fifths; that is, the purity of the lower tritone will be complemented by the purity of the upper tritone. This is heard in the initial beats of measure one and two in the example above; the first tritone is always exactly as wide as the second is narrow. Because of this, the listener should familiarize himself with the different qualities of these intervals; the Rameau temperament presents the largest contrast between the two measures. In this temperament, the tritone from G-C# is ten cents narrow, leaving the C#-G tritone to be ten cents wide. However, this temperament also sets the D-Ab tritone that spans the last two measures at only four cents flat; the difference between the rising tritones is therefore only 6 cents. The Stanhope temperament produces an even greater contrast between these two tritones: G-C# is eight cents narrow, and D-Ab is two cents sharp — a difference of ten cents. 

Figure 4.11 includes the longest portion of any Chopin mazurka that contains a melodic line with no supporting harmonies. Every interval between a minor second and an octave is presented at least once in these measures. It is interesting that the opening Major second, which is given a prominent role not only at the beginning but also throughout the selection, is pure in four unequal temperaments, and impure by only a cent in the Rameau temperament. For contrast, observe the range in sizes in the other Major seconds in the second measure (F#-E) and those in the third measure (B-C# and C#-D#). In the Werckmeister temperament, the perfect fourth that spans the second and third measures is wide by any standard, even to listeners accustomed to Equal Temperament's already-widened fourths. Compared to Equal Temperament, the Stanhope fourth here is as narrow as the Werckmeister fourth is wide. All of the unequal temperaments, except the DeMorgan, render minor thirds that are narrower than an Equal-tempered one; of these, the Rameau third is particularly narrow, partially due to the unusually wide Major second that precedes it. 

The melodic significance of the minor seventh in the second measure is not realized until it is presented again, spanning the fifth and sixth measures of this selection; this is due to the significance of the upbeat in this section. With this particular interval, only the Stanhope temperament deviates strongly from an Equal-tempered minor seventh, but the Stanhope is also closer to purity than five of the other temperaments (equal tempered included), and just as close as the other three. Concerning the two sets of perfect fifths that occur in the last two measures of the first line of this selection, only the Rameau temperament presents any strong inconsistencies in size. Its seven-cent sharp G#-D# fifth is as wide as any fifth reaches in the temperaments discussed here.

There are three significant minor seconds in the second line of Figure 4.11: between the third and fourth measures, F#-G; the fifth measure, C#-D; and in the final two notes, A#-B. Each of the unequal temperaments create three differently sized minor seconds for these three intervals; the Rameau temperament has the widest span — a pure F#-G second and a 36-cent narrow A#-B second.    

5 Conclusions and Additional Issues  

Several unanswerable questions arise when considering the history of temperament. Only one of the many that concerns most historians is the issue of which temperament a composer may have preferred. All the major historians and researchers to this day agree that no written record exists that reveals the tuning preference of any one composer before 1900, save a few noncommittal comments that J. S. Bach made to his eldest son. In order to speculate which temperament Bach may have used, many authors have observed general preferences or avoidances of intervals or key areas in his music, and then gathered statistical data about these decisions, concluding that he used a tuning that fit that data well. John Barnes is one of the few authors that took the aural tuning instructions Bach gave his son as a departure point for determining his tuning preference 50 , providing perhaps a more convincing argument than solely on statistical data. Some documents offer newly devised temperaments that would fit the statistical data better than an eighteenth-century temperament. 51 James Constable used sophisticated statistical methods and arrived at his own conclusions for the keyboard works of J. K. F. Fischer. 52 All authors that venture into this controversial subject are careful to disclaim that their conclusions are by any means decisive. Any composer almost certainly used a variety of tuning methods during his lifetime. Moreover, because published tuning methods were taken as departure points from which each musician would tailor a system to his own taste 53 , it is even more improbable that anyone will fully uncover which temperament Chopin did use. What is left to us is to consider his music itself alongside several different unequal temperament models, such as the ones discussed in detail in this paper, and develop an aural sense of the effects of these temperaments on his music. In doing so, we as performers may be able to resurrect some of Chopin's original intents. 

Having considered the effects of these temperaments on his music at a number of different depths, we may benefit by considering a few more questions. For performers in particular, an important question to consider is the following: did the temperaments affect the composition of the music, or was the music composed to fit the character of the temperament? It is a question worthy of some thought, and it would behoove the reader to contemplate it himself before continuing here. Perhaps the question may be more completely answered when applied first to a specific composer and then to the general musical realm. Bach, for example, was familiar with tuning methods enough to tune his instrument completely in fifteen minutes; certainly he would understand the idiosyncrasies of various systems. While composing, he would have the capability to disassociate the limitations of the temperaments from the musical work until it was completed, and then create a temperament to match the work. However, Liszt, who spent much time improvising and composing a work at the keyboard itself, may have penned compositions that would have been affected by whatever temperament in which the instrument may have been during that time. For the greater scheme of history, the question may be rephrased to the following: did the rise of Equal Temperament in keyboard music at the end of the nineteenth century lead to more chromaticism and eventually atonality, or did atonal writing demand atonal (not key-preferring; i.e., Equal) temperaments? 

The same question could be asked regarding any other turning period in music history: was meantone temperament devised to accept the increased use of the Major third in the early Renaissance, or did keyboard composers begin to include Major thirds in celebration of the new capabilities of meantone? Was circular temperament devised for pieces written in several keys, or were the pieces written for demonstrate circular temperament? The Well-Tempered Clavier is perhaps an unusual example; its title indicated that it was written to demonstrate the effectiveness of circular temperaments. But for the majority of works, the general music history observation that "theory follows practice" may apply here. Outside of keyboard music, pure thirds were being used long before the meantone system was developed. When one considers that Major thirds were used in English choir music and Franco-Flemish motets during the Middle Ages, the logical conclusion would be that meantone was developed for keyboardists to reproduce those sounds as well; keyboard music at that time generally followed the development of vocal music. The same can be said for the emergence of the first published circular temperament: before 1691, there is a clear increase in the tonal range of keyboard music, suggesting that this temperament was published to fit the music. Moreover, 1691 may not have marked the first use of this temperament or another like it.  

These considerations lead us to the final question: since we do not know which temperament Chopin used, should we choose a temperament to fit his music, or choose music to fit a temperament? The question is necessary for most twenty-first century performers; frequently, modern recital programs are devised that demonstrate the performer's ability to play literature from a wide range of styles and time periods, including the twentieth century. Much twentieth-century music can only be played satisfactorily in Equal Temperament, due to the nature of the music, and not to mention the applicable performance practice argument that music should be played in a temperament that belonged to the period in which the music was composed. Programs containing twentieth-century music have, therefore, only one tuning option, unless the performer has multiple instruments at his disposal. Conversely, if the program does not include twentieth-century works, other temperaments may be chosen. 

It has been stressed in the musical discussions throughout the preceding chapters that the performer should choose a temperament that creates a character in keeping with his interpretation of the piece. All of the temperaments discussed in this paper may be suitable for Chopin's music, with the exception of the Rameau and Stanhope temperaments in pieces containing a large number of accidentals. Indeed, Chopin could have used at any point in his life a temperament similar to either the Rameau or the Stanhope temperaments, given the dates of tuning manuals discussed in Chapter I. The author has found that his most effective performances of Chopin's music have been under circumstances in which he was able to develop an interpretation at the same time of choosing and modifying an unequal temperament as needed. The historical evidence of Chopin's using an unequal temperament is absolutely conclusive; it seems only logical, then, that the ideal performance environment is for the performer to choose a temperament that was in use in Chopin's time, and learn the music through the temperament, allowing the temperament to naturally fashion the interpretation along with the performer's own musical experience.

Perhaps the only question that remains is the following: given the entire spectrum of unequal temperaments, which one may be most suitable for the music? The answer lies largely in the repertoire itself. As mentioned above, some unequal temperament will not be suitable to music written in a large number of accidentals. If the repertoire is chosen for the performance such that all pieces are in relatively simple keys, and the program contains only similarly chosen repertoire or repertoire from the preceding eras, then a temperament with a wide range of sizes of Major thirds — such as the Werckmeister or Rameau temperaments — may be selected as a starting point. The performer should have a piano tuned in that temperament, and become as accustomed as possible to the colors that that temperament gives the music. For performers who have little or no previous exposure to unequal temperament, it may take several weeks to become accustomed to a temperament like Werckmeister or Rameau in order to understand how to manipulate it to create a new temperament devised to their own taste. The author has found that most performers, after having tuned their instrument to any unequal temperament, and practiced on it for only a few days, fully embrace the color changes and purer intervals and develop such a sensitivity to them that returning to Equal Temperament, with its relatively colorless quality, is unpleasant at best.  

The later unequal temperaments discussed here — the Jousse and Broadwoods — lend to Chopin's music colors that are significantly different from Equal Temperament, and can also be applied with great effectiveness to all romantic and classical era music. All composers before the twentieth century — Beethoven, Schubert, Weber, Mendelssohn, Schumann, Brahms, and Liszt — were accustomed only to unequal temperament, and wrote music written for unequally tempered instruments. As modern performers, we do not limit ourselves to performing modern music; neither should we place restrictions of temperaments created in the twentieth century to performances of music created in previous centuries. If we do, the original spirit and character of the music will be disfigured, and the original intentions of the composer thus ignored. We cannot pinpoint the exact emotional quality that a composer would have attached to a composition, just as we cannot pinpoint which temperament he was using for the piece. But by coupling historical facts with an intelligent choice of temperament, not only may we come closer to the composer's intention, but we may also present to the audience a new work of art — the performance itself — that contains infinite levels of expression, character, and color. As scholars, it is our duty to recreate environments as close as possible to those belonging to the composer. As performers, it is our privilege to relive them.    


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APPENDIX A : Derivation Methods and Naming Conventions of Ratios

Seconds and Sevenths              Ratio    Size    Derivation Method

 Octave                           2:1      1200   
  Just chromatic Major seventh    48:25    1129    2:1 - 25:24
   Limma Major seventh            243:128  1110    2:1 - 256:243
    Galileian Major seventh       17:9     1001    2:1 - 17:9
     Just diatonic Major seventh  15:8     1088    2:1 - 16:15
      maximum minor 7th           20:11    1035    2:1 - 11:10
       major minor 7th            9:5      1017.5  2:1 - 10:9
        minor minor 7th           16:9     996     2:1 - 9:8
         minimum minor 7th        7:4      969     2:1 - 8:7
         maximum tone             8:7      234    
        major tone                9:8      204    
       minor tone                 10:9     182    
      minimum tone                11:10    165    
     Just diatonic semitone       16:15    112    
    Galileian semitone            18:17    99     
   Limma                          256:243  90      4:3 - 9:8 - 9:8
  Just chromatic semitone         25:24    74     
 Fundamental                      1:1      0      

Thirds and Sixths - primary derivation  Ratio   Size  Derivation Method

Octave                                  2:1     1200 
 small Major 6th                        12:7    933   2:1 - 7:6
  just Major 6th                        5:3     884   2:1 - 6:5
   minor 6th                            8:5     814   2:1 - 5:4
    Pythagorean minor 6th               128:81  792   2:1 - 81:64
    Pythagorean Major 3rd               81:64   408   9:8 + 9:8
   Major 3rd                            5:4     386  
  just minor 3rd                        6:5     316  
 small minor 3rd                        7:6     267  
Fundamental                             1:1     0    

Thirds and sixths - secondary derivation   Ratio      Size   Derivation Method

maximum Major 6th                          12:7       933    3:2 + 8:7
 major Major 6th                           27:16      906    3:2 + 9:8
  minor Major 6th                          15:9       884    3:2 + 10:9
   minimum Major 6th                       33:20      867    3:2 + 11:10
    Just diatonic minor 6th                8:5        814    3:2 + 16:15
     Galileian minor 6th                   27:17      801    3:2 + 18:17
      Just chromatic minor 6th             75:48      773    3:2 + 25:24
       Perfect 5th                         3:2        702    
        Major Just chr. tritone            36:25      631    3:2 - 25:24
         Major Just dia. tritone           64:45      610    4:3 + 16:15
          Galileian major tritone          17:12      603    3:2 - 18:17
           ET tri-tone                     sqr(2):1   600    
          Galileian minor tritone          24:17      597    4:3 + 18:17
         minor Just dia, tritone           45:32      590    3:2 - 16:15
        minor Just chr. tritone            25:18      569    4:3 + 25:24
       Perfect 4th                         4:3        498    
      Just chromatic Major third           96:75      427    4:3 - 25:24
     Galileian Major third                 34:27      399    4:3 - 18:17
    Just diatonic Major 3rd                5:4        386    4:3 - 16:15
   maximum minor 3rd                       10:33      333    4:3 - 11:10
  major minor 3rd                          18:15      316    4:3 - 10:9
 minor minor 3rd                           32:27      294    4:3 - 9:8
minimum minor 3rd                          7:6        267    4:3 - 8:7

Tritone                     Ratio     Size  Derivation Method

Perfect 5th                 3:2       702  
 Major Just chr. tritone.   36:25     631   3:2 - 25:24
  Major Just dia. tritone.  64:45     610   4:3 + 16:15
   Galileian major tritone  17:12     603   3:2 - 18:17
    ET tri-tone             sqr(2):1  600  
   Galileian minor tritone  24:17     597   4:3 + 18:17
  minor Just dia, tritone   45:32     590   3:2 - 16:15
 minor Just chr. tritone    25:18     569   4:3 + 25:24
Perfect 4th                 4:3       498  

Appendix B : Selected Bibliography for Derivation Methods and Naming Conventions of Ratios 

Backus, John. The Acoustical Foundations of Music. New York: W. W. Norton and Company, Inc., 1969. 

Barbour, J. Murray. "Bach and the Art of the Temperament," Musical Quarterly, XXXIII (January, 1947), 64-89. 

_______________. History of Unequal Temperaments. Recordings and liner notes. Jackson Heights, NY: Misurgia Records, 1958(?). 

_______________. "Irregular Systems of Temperament," Journal of the American Musicological Society, I (Fall, 1948), 20-26. 

________________ and Fritz A. Kuttner. Meantone Temperament in Theory and Practice. Jackson Heights, NY: Misurgia Records, 1958. 

_______________. Tuning and Temperament. New York: Da Capo, 1972. 

Barnes, John. "Bach's Keyboard Temperament: Internal Evidence from the Well-Tempered Clavier," Early Music VII (April, 1979), 236 - 249. 

Blood, William. "'Well-Tempering' the Clavier: Five Methods." Early Music VII (October, 1979), 491 - 495. 

Bobbitt, Richard. "The Physical Basis of Intervallic Quality and Its Application to the Problem of Dissonance." Journal of Music Theory III (November 1959): 173-207. 

Kellner, Herbert Anton. "A Mathematical Approach Reconstituting J. S. Bach's Keyboard Temperament," Bach X (1979), 2-9. 

_______________. "Das Wohltemperierte Clavier: Tuning and Musical Structure," English Harpsichord Magazine II (April, 1980), 137-140. 

Helmholtz, Hermann. Sensations of Tone. Alexander Ellis, trans. New York: Dover Publications, 1954.

Lindley, Mark. "Early Sixteenth-Century Keyboard Temperaments," Musica Disciplina XXVIII (1974), 129-151.

_______________. "Equal Temperament," "Just Intonation," "Meantone," "Pythagorean Intonation," "Well-Tempered Clavier," The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie. London: Macmillan, 1980.

Lloyd, Llewellyn S. "Temperaments," Grove's Dictionary of Music and Musicians, ed. Eric Blom. London: Macmillan, 1954.

Mackenzie of Ord, Alexander C. N. Keyboard Temperament in England during the Eighteenth and Nineteenth Centuries. University of Bristol, 1983.

Meffen, John. The Temperament of Keyboard Instruments in England from the Virginalists to the middle of the Nineteenth Century. University of Leeds, 1977.

Mekiel, Joyce. "The Harmonic Theories of Kirnberger and Marpurg." Journal of Music Theory IV (November 1960), 169-193.

Sargent, George. "Eighteenth-century Tuning Directions: Precise Intervallic Determinations," The Music Review XXX (February, 1969), 27-34.


1. All figures and tables give deviation from purity in cents (1/100 of a semitone).

2. Ian Henderson, Strobe Tuner Settings for the Historic Scales (New York, 1983), p. 78.

3. Based on Murray Barbour, Tuning and Temperament: A
Historical Survey (East Lansing: Michigan State College Press, 1953), p. 162.

4. Rita Steblin, A History of Key Characteristics in the Eighteenth and Early Nineteenth Centuries (Ann Arbor, Michigan: UMI Research Press, 1983), p. 41.

5. B. Glenn Chandler, "Rameau's Nouveau systéme de musique théorique: An Annotated Translation with Commentary" (Ph.D. diss., Indiana University, 1975), p. 100.

6. Based on J. Murray Barbour, Tuning and Temperament: A Historical Survey (East Lansing: Michigan State College Press, 1953), p. 135.

7. Owen Jorgenson, Tuning: Containing the Perfection of Eighteenth- Century Temperament; The Lost Art of Nineteenth Century Temperament; and The Science of Equal Temperament. East Lansing: Michigan State University Press, 1991, p. 5.

8. Ibid, p. 8.

9. James Alexander Hamilton, Hamilton's practical introduction of the art of tuning the pianoforte; written for the use of persons desirous of tuning their own instruments, with a mathematical demonstration of the theory of Equal in Temperament.. a list of authors who have written on temperament... also... instructions for the maintenance and preservation of the pianoforte (London: R. Cocks and Company, 18--?), p. 8.

10. Ibid, pp. 8-9.

11. Ibid, pp. 9-11.

12. Ibid, p. 9

13. Ibid, pp. 12-13.

14. Ibid, p. 13.

15. Owen Jorgenson, Tuning, p. 362.

16. Ibid.

17. Ibid.

18. Thomas Webb Hunt, "The Dictionnaire de Musique of Jean-Jacques Rousseau," Ph.D. diss., North Texas State University, 1967, p. 240.

19. See Rita Steblin, A History of Key Characteristics in the Eighteenth and Early Nineteenth Centuries (Ann Arbor, Michigan: UMI Research Press, 1983), pp. 66-67.

20. Owen Jorgenson, Tuning, p. 152.

21. Based on Owen Jorgenson, Tuning, p. 150.

22. Owen Jorgenson, "In Tune with Old Tunings" (Clavier: November 1978: 26-28), p. 26.

23. Based on Owen Jorgenson, Tuning, pp. 291-292.

24. Owen Jorgenson, Tuning, p. 417.

25. Ibid pp. 418-419.

26. Hermann Helmholtz, On the Sensations of Tone (New York: Dover Publications, 1954), p. 485.

27. Owen Jorgenson, Tuning, p. 535.

28. Ibid, p. 536.

29. Ibid, p. 558.

30. Owen Jorgenson, Tuning, p. 551.

31. Ibid, p. 455.

32. Ibid, p. 463.

33. Ibid, p. 328.

34. Ibid, p. 384.

35. See Robert Holford Bosanquet, Musical Intervals and Temperament (London 1876); Rudolf Rasch, ed. Tuning and Temperament Library, Vol. IV (Utrecht: The Diapason Equal Press, 1987).

36. Owen Jorgenson, "In Tune with Old Tunings" (Clavier: November 1978: 26- 28), p. 28.

37. See Frédéric Chopin, Mazurkas, Rafael Joseffy, ed. (New York: G. Schirmer, Inc., 1943), pp. vii, 144-151.

38. See Rita Steblin, A History of Key Characteristics in the Eighteenth and Early Nineteenth Centuries (Ann Arbor, Michigan: UMI Research Press, 1983), Appendix A, pp. 222-226.

39. Ibid, pp. 288-292.

40. Ibid, pp. 303-305.

41. Ibid, pp. 227-308.

42. Ed Foote, lecture given at the shed University of Houston, May 13, 2000.