ABSTRACT: Diatonic keyboard tunings in equal temperament, just intonation, meantone and well tempered scales are derived from Pythagorean tuning using algorithms whose operations involve combinations of pure fifths and syntonic commas. Graphic diagrams of the line of fifths are used to show the harmonic and mathematical relationships between common tunings and temperaments. Four modes of just intonation are derived from Pythagorean tuning by an algorithm that narrows each major third by one syntonic comma. Equal temperament is approximated with imperceptible error by algorithms that narrow Pythagorean and justly tuned enharmonic intervals by one or more syntonic commas.
[1.2] This article is not concerned with generating diatonic scale tuning from prime numbers or otherwise deriving its form from physical or mathematical principles. We will use the historic definition of a diatonic scale as two tetrachords plus one additional tone that completes the octave. The diatonic octave is divided into five whole tones and two semitones in all tunings and temperaments. The convention of assigning letter names to notes in a 12-note chromatic keyboard for diatonic scales will be observed. The chain of fifths will be introduced and used as the defining feature common to all scales without initially assuming any particular tuning.
[1.3] Several modern theorists, including Mark Lindley{2} and Easley Blackwood{3} have applied algebraic techniques in describing the tonal relationships which form traditional diatonic scales. Others have developed diagrams to compare the differences in pitch for a note in various tunings.{4} Although such techniques have their uses, I did not personally find them particularly helpful in visualizing the overall difference between one tuning system or temperament and another. For my own analysis, I prefer to diagram the chain of fifths and use rational fractions to express the size of the intervals.{5} Differences between tuning systems are indicated by placing commas or fractional commas between adjacent fifths. This provides a visual comparison between the separate diatonic tunings and temperaments of a complete scale and makes it easier to demonstrate the mathematical and musical relationships between them.
[1.4] Theorists classify tuning systems as either cyclic, generated by a reiterative sequence of fifths, or divisive, tunings that subdivide the octave.{6} Historically, this division is somewhat arbitrary. Euclid and Boethius derived the Pythagorean tuning that we now associate with a sequence of fifths by dividing a monochord into two octaves.{7} Methods based upon the overtone series or other systems which attempt to derive just intonation from acoustic phenomena are often considered to be divisive in nature. The typographical diagrams utilized in this article interpret tunings and temperaments in terms of the line of fifths.
[2.2] The chain of fifths has its origin at C and extends to the sharp notes on the right and the flat notes on the left. The chain is theoretically endless, neither closing nor repeating, so that each note in the chain is musically unique.{8} Figure 1 shows the central 21 links in the chain, with C at the origin. All the notes in the chain are considered to be within a single octave bounded by two C's. Since fifths and fourths are inversions, each note is a fifth above the note to its left and a fourth above the note to its right. As a matter of convention, it is useful to consider the links to the right of C to be ascending fifths and the those to the left to be ascending fourths.
The same seven letter names repeat themselves over and over in the same invariant sequence of the C-major scale-- F-C-G-D-A-E-B--with each repetition augmented by sharps to the right and flats to the left. Any group of six adjacent links defines a major diatonic scale whose tonic note is one link from the left end of the group, and a natural minor scale which starts two links from the right end. Horizontal movement of any six-link group is the equivalent of transposition of the scale into a new key as exemplified by Figure 2. A twelve-note chromatic keyboard commonly contained the notes from Eb to G#, before the acceptance of equal temperament. When all the fifths were pure, the keyboard would be suitable for the diatonic major and minor scales in the keys depicted in Figure 2. Other keys would be unavailable because the required sharp or flat notes would be outside the range of the chain of fifths.
Bb major, G-minor Eb-Bb-F-C-G-D-A
F major, D-minor Bb-F-C-G-D-A-E
C major, A-minor F-C-G-D-A-E-B
G major, E-minor C-G-D-A-E-B-F#
D major, B-minor G-D-A-E-B-F#-C#
A major, F#-minor D-A-E-B-F#-C#-G#
[2.3] The chain of fifths can be used to rationally order the harmonic structure of the diatonic scale. Figure 3 lists the harmonic intervals available on a twelve-note chromatic keyboard for each pairing of links in the chain of fifths. The pairs of notes listed are examples that start with C. Any interval can be horizontally transposed along the chain of fifths.
| Links | Notes | Left to Right | Right to Left |
|---|---|---|---|
| 1 | C-G | Perfect fifth | Perfect fourth |
| 2 | C-D | Major second | Minor seventh |
| 3 | C-A | Major sixth | Minor third |
| 4 | C-E | Major third | Minor sixth |
| 5 | C-B | Major seventh or minor second | Diatonic semitone |
| 6 | C-F# | Augmented Fourth or tritone | Diminished Fifth |
| 7 | C-C# | Chromatic semitone | Diminished octave |
| 8 | C-G# | Augmented fifth | Diminished fourth |
| 9 | C-D# | Augmented second | Diminished minor seventh |
| 10 | C-A# | Augmented sixth | Diminished minor third |
| 11 | C-E# | Augmented third | Diminished minor sixth |
Figure 3 classifies harmonic intervals by the number of links and their direction on the chain of fifths. Each pair of intervals are inversions of one another. Chromatic intervals have more than six links, diatonic intervals have six or fewer. Generally speaking, major intervals and augmented intervals ascend from left to right, while minor intervals and diminished intervals ascend from right to left. The perfect fourth is the only consonant interval that ascends from right to left, which may reflect its harmonic ambiguity. Intervals that ascend to the right from a flat to a natural are always major, such as Bb-D, a four link ascending major third transposed from C-E. In the same manner, intervals that ascend to the right from a flat to a sharp are considered "augmented," as in Bb-D#, an 11-link augmented third transposed from C-E#. In the opposite direction, intervals that ascend to the left from a sharp to a natural are always minor, as in C#-E, a three link minor third transposed from C-Eb. Intervals that ascend to the left from a sharp to a flat are diminished, as in C#-Eb, a ten-link diminished third transposed from C-Ebb. Each pair of enharmonically equivalent notes is separated by twelve links. It is also well known that the progressions of diatonic harmony correspond to the chain of fifths, but this is a topic which will not be further detailed.
16/9 27/16 243/128 2187/2048
| | | |
Ab-w--Eb--Bb---F---C---G---D----A---E----B---F#----C#---G#
| | | | | | | |
32/27 4/3 1 3/2 9/8 81/64 729/512 6561/4096
The dashes between the letter names of the notes in Figure 4 indicate that the fifths are pure.
[3.2] Although D# (19683/16384) and Eb(32/27) are enharmonically identical in equal temperament, their pitches in Pythagorean tuning are different; D# is higher than Eb by 531441/524288 -- the Pythagorean comma(P). The more common method for deriving the Pythagorean comma is to calculate it as the difference between 12 consecutive fifths and seven octaves.{9} As Euclid knew, it is also the difference between six whole tones and one octave.{10}
[3.3] A wolf fifth, "w," occurs at the end of the line when Eb has to be used enharmonically because D# is unavailable; the "fifth," which is more precisely a diminished sixth, from G# to Eb(D#) will be too narrow by a Pythagorean comma. Blackwood describes such a narrow fifth as discordant and sounding badly out of place in a scale whose other fifths are pure.{11} It was known as the wolf fifth, because of the howling sound that it made on an organ.
|<----------------octave--------------(-1p)--------->|
| |------seventh----->| (-5/12P) |
| | -maj. Third-->| | (-1/3P) |
| fifth ----}|-->| |-->| (-1/12P) |
| | | | |
Eb-p-Bb-p-F-p-C-p-G-p-D-p-A-p-E-p-B-p-F#-p-C#-p-G#-p-D#
| | | |
fourth --}|<--|<--| | (+1/12P) maj. snd. |------>| | (-1/6P)
|-mj. sixth>| (-1/4P)
In Figure 5, the Pythagorean comma is indicated by the upper-case "P" and 1/12 of the Pythagorean comma by the lower-case "p." The direction of the arrow indicates the ascending harmonic interval. Reversing the arrow would invert the interval and change the sign of the fraction, as illustrated by the fifth (-1/4p) and the fourth (+1/4p).
[4.2] Since equal temperament makes every thirteenth note the same, it makes all enharmonic pairs equal; for example, G# becomes identical with Ab and D# becomes identical with Eb. This superposition is quite different from a Pythagorean keyboard in which substitution of enharmonically equivalent notes gives a fifth that is narrowed by a Pythagorean comma. Compressing the chromatic line of twelve pure fifths by a Pythagorean comma makes the tuning of Ab equal to the tuning of G#. Instead of an infinite series of links extending along the chain of fifths in both directions, equal temperament reduces the chromatic scale to exactly 12 notes which end on an octave and then repeat. In equal temperament, one does not have to choose between an available G# and an unavailable Ab; both are available and the tuning of the two notes is now the same. Equal temperament made possible modern keyboard music with full modulation between all keys. It has become common to consider equal temperament as being the division of an octave into twelve equal semitones, since the ratio of an octave is 2/1 and an equally tempered semitone multiplied by itself 12 times produces an exact octave.{12}
[4.3] We can easily demonstrate from the chain of fifths that equal temperament makes both the tuning and the musical function of each pair of enharmonically equivalent notes equal, using the 12 links from C to B#. After subtracting the Pythagorean comma, B# becomes the same musical note as C. C can be substituted for B#, which is indicated by placing it vertically below B# in Figure 6. E# is a fifth below B#, but since B# has been replaced by C, F must be placed below E# since F is a fifth below C. The substitutions continue to the left along the top row from B# to C until the bottom row is complete from C on the right to Dbb on the left. Figure 6 shows that the two rows created by these substitutions are the same as if the chain of fifths was split into two rows, with the sharp notes above the flat notes. The substitutions derived from equal temperament have made the tuning of the top row identical to that of the bottom row. Equal temperament combines each vertical pair of enharmonically equivalent notes into one single note.{13}
| C | G | D | A | E | B | F# | C# | G# | D# | A# | E# | B# |
| Dbb | Abb | Ebb | Bbb | Fb | Cb | Gb | Db | Ab | Eb | Bb | F | C |
This process can be extended out to infinity by vertically stacking each group of 12 notes in rows below Dbb-C and above C-B#. Each vertical stack of enharmonic notes will be identical. The profound consequence of equal temperament is that the chain of twelve tempered links absorbs the infinite chain into a single row of 12 chromatic notes that corresponds exactly to the 12 notes-to-the- octave (7 diatonic and 5 chromatic) equally-tempered keyboard.
[4.4] Calculating the tuning of an equally tempered scale generates irrational numbers,{14} because the rational fraction that measures the Pythagorean comma must be divided into twelve roots. One equally tempered semitone equals the twelfth root of 2, an irrational number whose value is normally given in a decimal approximation as 1.05946 (100 cents.) The tuning of the chain of equal tempered fifths in six place decimals is given in Figure 7.
| Note | Tuning | Cents | |
|---|---|---|---|
| G#/Ab | 1.587401 | 800 | |
| C#/Db | 1.059463 | 100 | (1300-1200) |
| F#/Gb | 1.414214 | 600 | (1800-1200) |
| B | 1.887749 | 1100 | |
| E | 1.259921 | 400 | (1600-1200) |
| A | 1.6817793 | 900 | |
| D | 1.122462 | 200 | (1400-1200) |
| G | 1.498307 | 700 | |
| C | 1.0 | 0/1200 | |
| F | 1.334840 | 500 | (1700-1200) |
| Bb/A# | 1.781797 | 1000 | |
| Eb/D# | 1.189207 | 300 |
Using the diagram in Figure 5, the 700-cent logarithmic tuning of an equally tempered fifth is narrower than a 702 cent perfect fifth by 2 cents, 1/12 of the 24-cent Pythagorean comma. The last column in Figure 7 displays this pattern; the pitch of each note in the vertical chain is 700 cents above the note below it before subtracting out the octave.
| F | C | G | D | k | A | E | B |
| 4/3 | 1 | 3/2 | 9/8 | 5/3 | 5/4 | 15/8 |
[5.2] Tuning a chromatic octave in just intonation creates further difficulties since alternate tunings are required for each note. A theoretical scale of just intonation can be calculated for four diatonic major scales in the chromatic octave by replicating the procedure used to form the scale of C-major. A scale of Bb-major would have triads in which Eb-G, Bb-D and F-A were pure major thirds. The pure thirds would be Bb-D, F-A, C-E in F-major and C-E, G-B and D-F# in G-major. Figure 9 shows the two alternative tunings for G, D and A that are required for these four scales. The multiplicity of alternate tunings makes just intonation even more impractical for the normal keyboard with twelve keys to the octave.
Eb----Bb----F----C--k--G-----D---A
32/27 16/9 4/3 1 27/20 10/9 5/3
Bb----F----C----G--k--D----A----E
16/9 4/3 1 3/2 10/9 5/3 5/4
F----C----G----D--k--A----E----B
4/3 1 3/2 9/8 5/3 5/4 15/8
C----G----D----A--k--E----B----F#
1 3/2 9/8 27/16 5/4 15/8 45/32
It can also be seen from Figure 9 that the absolute location of the syntonic comma moves but that its relative location is always on the fourth link of the scale. Each just scale could be considered to be virtually a separate entity whose tuning is fixed by the tuning of the triads that it contains.
[5.3] It is helpful at this point to introduce the concept of an algorithm, which is a set of instructions for making a series of calculations. Each of the tunings we have discussed so far can be derived from an algorithm for tuning the chain of fifths into one 12-note chromatic octave on a keyboard. The Pythagorean tuning algorithm has four steps:
[5.4] The scale in Figure 10 has a syntonic comma at every fourth link. This produces a chromatic scale in which the diatonic scale of C-major is just and the maximum number of major thirds are a consonant 5/4. This scale is generated by an algorithm in which (1) the fourth link of a Pythagorean C-major scale is narrowed by a syntonic comma and (2) every fourth link above and below that link is also narrowed by a syntonic comma. The tuning of the just chromatic scale from Eb to G# depicted in Figure 10 corresponds to the tuning given by Rossing for these same 12 chromatic notes within a larger chromatic scale of just intonation from Fb to B#.{15}
Just Intonation
6/5 9/5 4/3 1 3/2 9/8 5/3 5/4 15/8 45/32 25/24 25/16
| | | | | | | | | | | |
Eb---Bb--k--F---C---G---D--k--A----E----B----F#--k--C#----G#
Eb---Bb----F--C--G---D----A-----E------B-------F#------C#--------G#
| | | | | | | | | | | |
32/27 16/9 4/3 1 3/2 9/8 27/16 81/64 243/128 729/512 2187/2048 6561/4096
Pythagorean Tuning
We can use the graphic diagram in Figure 10 to evaluate the difference between the just and Pythagorean pitch of each note without having to multiply out the 81/80 syntonic comma. In this just chromatic scale, Bb and Eb are each a syntonic comma higher in pitch while justly tuned A, E, B and F# are each a syntonic comma lower in pitch. Just C# and G# are lower than Pythagorean by two syntonic commas. Examining Figure 10 demonstrates still more reasons why a chromatic scale of just intonation is unusable on a standard keyboard. Narrowing every fourth link by a comma does not leave all the thirds and fifths consonant. The fifths that are narrowed (which means that the fourths are inversely broadened) by a full syntonic comma are obviously dissonant. Thus, the just chromatic scale has two more dissonant links than a Pythagorean chromatic scale. The wolf fifth from enharmonic D#(Eb) to G# will be narrowed by three syntonic commas. Any group of three adjacent links that does not include a comma will form a Pythagorean minor third and a Pythagorean major sixth. Whole tones in this scale that include a syntonic comma are 10/9, while the other whole tones are in the just and Pythagorean proportion of 9/8. To assess the severe impact of these tuning discrepancies we summarize the most significant consequences when a keyboard tuned in just intonation as in Figure 10 is used to play the six diatonic major scales listed in Figure 2.
[5.5] Since the scale in Figure 10 was created by an algorithm that narrowed every fourth link by one syntonic comma, one can generate four chromatic modes of just intonation, one for each of the scales shown in Figure 9. These are the only just chromatic scales that are possible when C has the relative tuning of 1, since the pattern of the algorithm repeats after the fourth link.{16} Figure 11 presents the tuning of the four modes of just intonation along a chromatic scale from Dbb to B# for a vertical chain of fifths in Pythagorean tuning and the four just chromatic modes in Bb, F, C, and G. The number of syntonic commas by which each note has been altered from Pythagorean tuning is indicated in parentheses, the pitch of the notes above C being lowered by the number of syntonic commas indicated and the ones below C being raised. The remaining notes are Pythagorean.
Nt Pythagorean Just (G) Just (C) Just (F) Just (Bb) B# 531441/524288 125/64 (-3k) 125/64 (-3k) 125/64 (-3k) 125/64 (-3k) E# 177147/131072 675/512(-2k) 125/96 (-3k) 125/96 (-3k) 125/96 (-3k) A# 59049/32768 225/128(-2k) 225/128(-2k) 125/72 (-3k) 125/72 (-3k) D# 19683/16384 75/64 (-2k) 75/64 (-2k) 75/64 (-2k) 125/108(-3k) G# 6561/4096 25/16 (-2k) 25/16 (-2k) 25/16 (-2k) 25/16 (-2k) C# 2187/2048 135/128(-1k) 25/24 (-2k) 25/24 (-2k) 25/24 (-2k) F# 729/512 45/32 (-1k) 45/32 (-1k) 25/18 (-2k) 25/18 (-2k) B 243/128 15/8 (-1k) 15/8 (-1k) 15/8 (-1k) 50/27 (-2k) E 81/64 5/4 (-1k) 5/4 (-1k) 5/4 (-1k) 5/4 (-1k) A 27/16 27/16 5/3 (-1k) 5/3 (-1k) 5/3 (-1k) D 9/8 9/8 9/8 10/9 (-1k) 10/9 (-1k) G 3/2 3/2 3/2 3/2 40/27 (-1k) C 1 1 1 1 1 F 4/3 27/20 (+1k) 4/3 4/3 4/3 Bb 16/9 9/5 (+1k) 9/5 (+1k) 16/9 16/9 Eb 32/27 6/5 (+1k) 6/5 (+1k) 6/5 (+1k) 32/27 Ab 128/81 8/5 (+1k) 8/5 (+1k) 8/5 (+1k) 8/5 (+1k) Db 256/243 27/25 (+2k) 16/15 (+1k) 16/15 (+1k) 16/15 (+1k) Gb 1024/729 36/25 (+2k) 36/25 (+2k) 64/45 (+1k) 64/45 (+1k) Cb 2048/2187 48/25 (+2k) 48/25 (+2k) 48/25 (+2k) 256/135(+1k) Fb 8192/6561 32/25 (+2k) 32/25 (+2k) 32/25 (+2k) 32/25 (+2k) Bbb 32768/19683 216/125(+3k) 128/75 (+2k) 128/75 (+2k) 128/75 (+2k) Ebb 65536/59049 144/125(+3k) 144/125(+3k) 256/125(+2k) 256/125(+2k) Abb 262144/177147 192/125(+3k) 192/125(+3k) 192/125(+3k) 1024/675(+2k) Dbb 524288/531441 128/125(+3k) 128/125(+3k) 128/125(+3k) 128/125(+3k)
Figure 11 provides an insight on the relationship between Pythagorean and just tuning.{17} Instead of presenting just intonation as a series of pitches that are independently derived, Figure 11 shows that the pitch of the sharp and flat notes are modified in a regular progression as syntonic commas are sequentially added to or subtracted from Pythagorean tuning. The logarithmic tuning of each note can be easily calculated using the fact that a pure fifth is 702 cents, a pure fourth, its inversion, is 498 cents, and a syntonic comma is 22 cents. As an example, 27/25, the tuning of Db in the just G-major mode is 5 links plus 2 syntonic commas left of C: 5 x 498 = 2490 + 44 = 2534. After subtracting 2400, Db + 2k = 134 cents.
[5.6] F. Murray Barbour listed 22 historical scales of just intonation dating from 1482 to 1776 in his historical survey of tuning and temperament.{18} Barbour defined just intonation more broadly than we have done, including within the concept any 12-note chromatic scale that possessed some pure fifths and at least one pure third. Barbour described a late 18th-century tuning by Marpurg which is the same as Figure 10 as "the model form of just intonation."{19} Two scales which correspond to the F-major mode shown in Figure 11 were described by Barbour as "the most symmetric arrangement of all." The first was an early 17th century scale by Salomon de Caus which started with Bb; the second was by Mersenne and started on Gb.{20} An even earlier tuning by Fogliano in the F-major mode starting with Eb was also included.{21} None of the other tunings are precisely in the form of Figure 11. A complete listing of these historical just tunings is given in Appendix I.
[5.7] It is also interesting to compare the actual tuning of notes in Figure 11 with Ellis's "Table of Intervals not exceeding One Octave" in his appendix to Helmholtz.{22} Figure 11 gives new significance to intervals whose names as given by Ellis imply a diatonic origin. Thus Ellis's list includes the following varieties of fourths: "acute," 27/20, a "superfluous," 25/18 and another "superfluous," 125/96. Figure 11 shows that the "acute" fourth is the just fourth in the G-major mode, the first "superfluous" fourth is an augmented fourth, F# in the F-major and Bb-major modes, and the second "superfluous" fourth is an augmented third, E#, in the C, F and Bb modes. In all, Ellis lists 33 intervals obtained by combining one or more perfect fifths with one or more just thirds, only 3 less than the 36 tunings in Figure 11 that are altered by one or more Pythagorean commas. All but two of Ellis's 33 intervals are included in Figure 11; one is a well-tempered interval and the other is not part of a chromatic scale in which C is tuned to 1. The complete list of Ellis's intervals and their notation in terms of Pythagorean tuning and syntonic commas is contained in Appendix II.
[5.8] We can now appreciate why the chain of fifths is useful for evaluating the harmonic consequences of alternate tunings of a diatonic scale embedded in a chromatic keyboard. It is not possible to arbitrarily change the pitch of one note without altering its relationship with all other notes in the chromatic space. The interdependence of tuning is not limited to instruments with fixed pitch. For example, a string quartet could not play passages containing a sequence of triads in just intonation without altering the melodic intervals and, possibly the overall level of pitch.{23} The problems of tuning harmonic intervals can only be solved by dynamic ad hoc adjustments of pitch to obtain optimum consonance while preserving the melodic line and stability of pitch.{24} This is one of the reasons why it is so difficult for inexperienced musicians to achieve good ensemble intonation.
[5.9] The syntonic comma is not an independent variable; the independent variables are the tuning ratios for the consonant fifth and major third as determined by psychoacoustic measurements, which are 3/2 and 5/4. The syntonic comma, even though it has been known and separately named for two thousand years is merely derived from the difference between the consonant or just major third and the Pythagorean major third generated by four consecutive fifths. However, one would not intuitively expect that this single dependent variable could be used to measure all the differences in tuning between Pythagorean and just scales, including intervals that are not generated by major thirds.{25} The just intonation algorithms provide a coherent framework for this ancient and well respected dependent variable known as the syntonic comma. As was noted above, just intonation and Pythagorean tuning are commonly thought of as different species of tuning. The just intonation algorithms demonstrate that just intonation can more completely be understood in terms of the chain of fifths and syntonic commas than if considered independently.
[6.2] The quarter-comma (4q = 1k) narrowing of each link in the chain of fifths is shown in Figure 12, which also depicts the resulting alteration of other intervals from Pythagorean tuning. Compared to Pythagorean tuning, it is evident that the fifths to the right of C are all tuned a quarter-comma flat, while the fourths to the left of C become a quarter-comma sharp.
|---mj. seventh -5/4k--->| |
|---mj. third -1k-->| |--->|{-fifth -1/4k |
| | | |
Eb--q-Bb-q--F--q-C--q-G--q-D--q-A--q-E--q-B--q-F#-q-C#-q-G#
| | | |
fourth +1/4k -}|<---| |-second->|{---(-1/2k)
|-sixth -3/4k->|
As before, Figures 10 and 12 allow us to measure the difference between the just and meantone scales without using arithmetic. As an example, a mean-tone chromatic semitone is narrowed by 7/4 syntonic commas compared to Pythagorean tuning and 3/4 of a syntonic comma compared to just intonation.
[6.3] Meantone temperament and its variations was the established mode for tuning keyboards for nearly three centuries. English pianos and organs were tuned this way until the middle of the nineteenth century.{27} As Thurston Dart explained in his treatise on early music:
[6.4] Since a syntonic comma is 81/80, a quarter-comma, its fourth root, is an irrational number. However, rational fractions that closely approximate the size of a quarter-comma and which cumulatively equal a full syntonic comma can be derived by arithmetically dividing a syntonic comma of 324/320 into four parts as follows:
324/320 = 321/320 x 322/321 x 323/322 x 324/323
Therefore, a quarter-comma fifth may calculated as being:
3/2 x 320/321 = 160/107
Rational fractions approximating each of the other meantone
intervals may be calculated in a similar manner, using the other
fractions in the expansion. A chromatic meantone tuning for a
keyboard is shown in Figure 13 for a chain of fifths tempered by
a quarter-comma (q).{31}
323/270 107/80 180/161 540/323 225/161 25/16
| | | | | |
Eb-q-Bb--q-F--q-C--q-G--q-D--q-A--q-E--q-B--q-F#--q-C#--q-G#
| | | | |
161/90 1 160/107 200/107 675/626
Logarithmic values for a meantone scale can be easily calculated by subtracting one-fourth of a syntonic comma (22/4 = 5.5 cents) from every pure fifth. Thus each meantone fifth will be 696.5 cents.
|------ C-E ------>| (-1k)
|<--- G-B>| (-3/4k)
|-------F-A------>| (-3/4k)
|-----Bb-D----->|---- D-F# ---->| (-1/2k)
|-----Eb-G---->| | |------A-C#-- >| (-1/4k)
| | | | |
Ab--Eb--Bb--F--C--q-G--q-D--q-A--q-E--B--F#--C#--G#
| |
|<----------(p-k)--------chromatic Octave------>|
[7.2] Figure 14 illustrates a simplified form of well temperament in which all but the central four links are Pythagorean and the major thirds vary from just (C-E) to fully Pythagorean (Ab-C and E-G#). No major triad will be exactly consonant. The wolf fifth between Ab(G#) and Eb has been essentially eliminated, and notes which are enharmonic to Eb, Bb, F#, C# and G# will differ in pitch only by a schisma. Therefore, use of these notes enharmonically increases the number of available scales from 6 to 11. The tuning for this version of well temperament are given in Figure 15.
32/37 180/161 5/4 45/32 405/256
| | | | \
Ab---Eb---Bb---F---C--q--G--q--D--q--A--q--E---B---F#---C#---G#
\ | | | | | | |
128/81 16/9 4/3 1 160/107 540/323 15/8 135/128
A number of historical well temperaments were devised that separated the quarter commas by one or more links to improve playability in desired keys. Other temperaments subdivided the syntonic even further into 2/7 and 1/6 commas. The more that the commas were divided and dispersed, the more that well temperament approached equal temperament.
[8.2] Schismatic and syntonic equal temperament is derived by extending a well-tempered scale eleven links to the left and right of C. When the eleventh note to the left of C is raised by one syntonic comma and the eleventh note to the right of C is lowered by one syntonic comma the results are tunings for E# and for Abb that are almost exactly the same as the equally tempered tunings for F and G respectively.
E#-1k = 10935/8192 = 1.334839 F(ET) = 1.334840
Abb+1k = 16384/10935 = 1.498308 G(ET) = 1.498307
This strategy works because a syntonic comma is very nearly 11/12
of a Pythagorean comma, expressed as k = 11/12P. Therefore a
schisma, defined as sk = P-k = 32805/32786, is very nearly 1/12
of a Pythagorean comma "p," the amount by which each link is
narrowed in equal temperament. The pitch of E# in Pythagorean
tuning is higher than its enharmonic note, F, by a Pythagorean
comma(P). Therefore,
E#-k ~ F+P-k ~ F+sk ~ F+p.
As shown in Figure 5, F+p is the pitch of F in equal temperament.
Since the precise value of a schisma is 1.954 cents and 1/12 of a
Pythagorean comma is 1.955 cents, the error is only 1/1000 of a
cent. One can then obtain decimal tunings for equal temperament
by reiterating the proportions for E#-1k to a full chromatic
chain of 12 links that would exceed an exact octave by a ratio of
only 2.000018/2 or 0.012 cents.
[8.3] We will now derive and apply two algorithms which map Pythagorean tuning into equal temperament. The algorithm based upon the schisma produces schismatic ET. As with ordinary equal temperament, this procedure equates the tuning of enharmonic pairs, substituting G for Abb + k, D for Ebb + k + 1sk, and so on. Every 10th link to the left of Abb + k and to the right of E#-1k is tempered by one schisma. The five schismatic notes tuned to the right of Abb+k are enharmonically equivalent to the notes normally located to the right of G, while the five schismatic notes tuned to the left of E#-k are enharmonically equivalent to the notes normally located to the left of F. The steps of the algorithm are as follows.
| Tuning | Note | Schismatic ET | Syntonic ET |
|---|---|---|---|
| 1.000009 | Dbb | C - k + 11sk | |
| 1.498295 | Abb | G - k + 10sk | |
| 1.22454 | Ebb | D - k + 9sk | |
| 1.683681 | Bbb | A - k + 8sk | |
| 1.261336 | Fb | E - k + 7sk | |
| 1.887739 | Cb | B - k + 6sk | |
| 1.414207 | Gb | F# - k + 5sk | |
| 1.059459 | Db | C# - k + 4sk | F######## - 5k |
| 1.587396 | Ab | G# - k + 3sk | D###### - 4k |
| 1.189205 | Eb | D# - k + 2sk | B#### - 3k |
| 1.781795 | Bb | A# - k + 1sk | G### - 2k |
| 1.334839 | F | E# - k + 0sk | E# - 1k |
| 1.0 | C | C | |
| 1.498308 | G | Abb + k | Abb + 1k |
| 1.22464 | D | Ebb + k - 1sk | Fbbb + 2k |
| 1.681797 | A | Bbb + k - 2sk | Dbbbbb + 3k |
| 1.259925 | E | Fb + k - 3sk | Bbbbbbbb + 4k |
| 1.887756 | B | Cb + k - 4sk | Gbbbbbbbb + 5k |
| 1.41422 | F# | Gb + k - 5sk | Ebbbbbbbbbb + 6k |
| 1.059468 | C# | Db + k - 6sk | |
| 1.58741 | G# | Ab + k - 7sk | |
| 1.189215 | D# | Eb + k - 8sk | |
| 1.781811 | A# | Bb + k - 9sk | |
| 1.334851 | E# | F + k - 10sk | |
| 1.000009 | B# | C + k - 11sk |
[8.5] We can now go a step further and derive the algorithm that maps just intonation into syntonic ET from the algorithms previously used to map Pythagorean tuning into just intonation. For example, E# in the just mode of C-major is equal to Pythagorean E#-3k. Adding back the three syntonic commas, E#-1k in schismatic and syntonic ET is enharmonically equivalent to E# + 2k in the just C-major mode. The algorithm for just schismatic ET parallels the algorithm given in section 8.3, adding back to each interval the number of commas listed in Figure 11. In syntonic ET, Bb is enharmonically equivalent to G###-2k. If the C-mode just scale were extended to G###, the Pythagorean tuning would be narrowed by 5 syntonic commas. Replacing the 5 commas, G### + 3k maps just G### into Bb in just syntonic ET. We have not set out the full just chromatic scale for the 121 links necessary in Figure 11, but it can be done simply by adding 1 syntonic comma for every four links to the left and subtracting 1 syntonic comma for every four links to the right as many times as is necessary. Applying these algorithms in Figure 17 gives scales of just schismatic and just syntonic ET for the C-major scale.
| Tuning | Note | Just Schis. ET | Just Synt. ET |
|---|---|---|---|
| 1.059459 | Db | C# + 1k + 4sk | F######## + 9k |
| 1.587396 | Ab | G# + 1k + 3sk | D###### + 7k |
| 1.189205 | Eb | D# + 1k + 2sk | B#### + 5k |
| 1.781795 | Bb | A# + 1k + 1sk | G### + 3k |
| 1.334839 | F | E# + 2k | E# + 2k |
| 1.0 | C | C | |
| 1.498308 | G | Abb - 2k | Abb - 2k |
| 1.22464 | D | Ebb - 2k - 1sk | Fbbb - 3k |
| 1.681797 | A | Bbb - 1k - 2sk | Dbbbbb - 5k |
| 1.259925 | E | Fb - 1k - 3sk | Bbbbbbbb - 7k |
| 1.887756 | B | Cb - 1k - 4sk | Gbbbbbbbb - 9k |
| 1.41422 | F# | Gb - 1k - 5sk | Ebbbbbbbbbb - 10k |
[8.6] Of course, neither syntonic nor schismatic ET are precisely equal. They do not merge all enharmonically equivalent notes, as shown by the outer sections of Figure 16 which contains notes from Gb to F# in schismatic ET. The enharmonic value for Db is not exactly equal to the value of C# as it is in equal temperament and C is not exactly equal to Dbb and B#. Therefore, the number of possible notes is still endless and does not collapse to twelve. The divergences between the tunings produced by the two equal temperaments are of no audible or acoustic significance either to the tuning of individual notes or to the overall sound of the temperament. A keyboard instrument tuned to syntonic ET would not sound distinguishable from one tuned with unlikely mathematical perfection to traditional equal temperament. The twelve chromatic links tuned to syntonic ET would have to be replicated nearly 163 times before the divergence in tuning C would exceed even one schisma.
81. Ramis, 1482. Ab-F# Ab-G, 8=0, 4=-1k.
82. Erlangen, 15c. Ebb, Bbb, Gb-B, 2=+1k, 8=0, 2=-1.
83. Erlangen revised, Eb-G#, 7=0, 3=-1k.
84. Fogliano 1529 Eb-G#, 1=+1k, 4=0, 4=-1k, 3=-2k.
85. Fogliano 1529 Eb-G#, 2=+1k, 4=0, 3=-1k, 3=-2k.
86. Fogliano 1529 Eb-G#, 1=+1k, 1=+1/2k, 3=0, 1=-1/2k,
3=-1k, 3=-2k.
88. Agricola 1539. Bb-D#, 8=0, 4=-1k
89. De Caus 17c. Bb-D#, 4=0, 4=-1k, 4=-2k
90. Kepler 1619 Eb-G#, 2=+2, 5=0, 5=-1
91. Kepler 17c. Ab-C#, 3=+1, 5=0, 4=-1
92. Mersenne 1637 Gb-B, 4=+1, 4=0, 4=-1
93. Mersenne 1637 Bb-D#, 4=0, 4=-1, 4=-2
94. Mersenne 1637 Gb-B, 5=+1, 3=0, 4=-1
95. Mersenne 1637 Gb-B, 5=+1, 4=0, 3=-1
96. Marpurg 1776 Eb-G#, 2=+1, 4=0, 4=-1, 2=-2
97. Marpurg 1776 Eb-G#, 1=+1, 6=0, 2=-1, 2=-2
98. Marpurg 1776 Eb-G#, 2=+1, 3=0, 4=-1, 3=-2
99. Malcolm 1721 Db-F#, 3=+1, 5=0, 4=-1
100. Rousseau 1768 Ab-C#, 3=+1, 4=0, 3=-1, 2=-2
101. Euler 1739 F-A#, 4=0, 3=-1, 5=-2
102. Montvallon 1742 Eb-G#, 1=+1, 5=0, 6=-1
103. Romieu 1758 Eb-G#, 1=+1, 5=0, 4=-1, 2=-2
24:25 Small semitone C#-2k
128:135 Larger limma C#-1k
15:16 Diatonic or just semitone Db+1k
25:27 Great limma Db+2k
9:10 The minor tone of just intonation D-1k
125:144 Acute diminished third Ebb+3k
108:125 Grave augmented tone D#-3k
64:75 Augmented tone D#-2k
5:6 Just minor third Eb+1k
4:5 Just major third E-1k
25:32 Diminished fourth Fb+2k
96:125 Superfluous fourth E#-3k
243:320 Grave fourth (F-1k)
20:27 Acute fourth F+1k
18:25 Superfluous fourth F#-2k
32:45 Tritone, augmented fourth F#-1k
45:64 Diminished fifth Gb+1k
25:36 Acute diminished fifth Gb+2k
27:40 Grave fifth G-1k
16:25 Grave superfluous fifth G#-2k
256:405 Extreme sharp fifth (G#-1k)
5:8 Just minor sixth Ab+1k
3:5 Just major sixth A-1k
75:128 Just diminished seventh Bbb+2k
125:216 Acute diminished seventh Bbb+3k
72:125 Just superfluous sixth A#-3k
128:225 Extreme sharp sixth A#-1k
5:9 Acute minor seventh Bb+1k
27:50 Grave major seventh B-2k
8:15 Just major seventh B-1k
25:48 Diminished octave Cb+2k
64:125 Superfluous seventh B#-3k
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prepared by
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7/10/1998