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M U S I C T H E O R Y O N L I N E
A Publication of the
Society for Music Theory
Copyright (c) 1998 Society for Music Theory
+-----------------------------------------------------------+
| Volume 4, Number 4 July, 1998 ISSN: 1067-3040 |
+-----------------------------------------------------------+
All queries to: mto-editor@smt.ucsb.edu or to
mto-manager@smt.ucsb.edu
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AUTHOR: Scholtz, Kenneth P.
TITLE: Algorithms for Mapping Diatonic Keyboard
Tunings and Temperaments
KEYWORDS: Algorithm, chain of fifths, diatonic
scale, equal temperament, enharmonic, just
intonation, meantone, Pythagorean tuning,
schisma, syntonic comma
Kenneth P. Scholtz
2821 Anchor Ave.
Los Angeles, CA 90064-4605
kscholtz@earthlink.net
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] Introduction
[1.1] This article describes algorithms that map the traditional
harmonic tunings and temperaments for the keyboard: Pythagorean
tuning, equal temperament, just intonation, and historical
variations on meantone temperaments. Keyboard tuning is used
because it has been worked out in detail over hundreds of years
and is understood to give a reasonable approximation of the pitch
choices made by performers on instruments without fixed pitches.
I have excluded analysis of multiple divisions of the octave into
more than twelve notes because they were not commonly used
temperaments. Working from the chain of fifths, we will
demonstrate how the intervals in any tuning or temperament can be
mapped from Pythagorean tuning by algorithms that combine pure
fifths and syntonic commas.(1) The algorithms that map the
tunings and temperaments into one another are not purely
mathematical, but are derived from the harmonic structure of the
chain of fifths and enharmonic relationships between notes.
================
1. The syntonic comma is defined as the difference between the
Pythagorean tuning and just tuning of the major third. The
difference between the just third (5/4) and the Pythagorean third
(81/64) is 81/80, calculated as follows: 81/64 x 4/5 = 81/16 x
1/5 = 81/80. The syntonic comma is also the difference between
the Pythagorean and just tunings for all diatonic intervals other
than the fourth and fifth, which are the same in both tunings.
The reason for the repeated appearance of the syntonic comma will
be apparent from the discussion of the four modes of just
intonation in section 5.
================
[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.
==============
2. Mark Lindley and Ronald Turner-Smith, "An Algebraic Approach
to Mathematical Models of Scales," *Music Theory Online* 0.3
(1993), which is a commentary based upon their book,
*Mathematical Models of Musical Scales* (Bonn: Verlag fuer
Systematische Musikwissenschaft, 1993).
3. Blackwood, *The Structure of Recognizable Diatonic Tunings*,
(Princeton: Princeton University Press, 1985).
4. See, for example, L. L. Lloyd's diagrams in his articles in
the 1954 *Grove's Dictionary of Music and Musicians* on "Just
Intonation" and "Temperaments."
5. Intervals measured by rational fractions can be converted
into cents using the following approximate values: octave = 1200
cents, perfect fifth = 702 cents; Pythagorean comma = 24 cents;
syntonic comma = 22 cents.
===============
[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.
===============
6. See Lindley and Turner-Smith, "An Algebraic Approach,"
paragraph 4.
7. In modern terminology, dividing the string in half gives two
octaves, in thirds gives a fifth and an eleventh, and in fourths
gives a fourth, octave and double octave. The other diatonic
notes are then determined by calculating intervals of a fifth
from these intervals. This produces two diatonic octaves in
Pythagorean tuning.
===============
[2] The Chain of Fifths
[2.1] It is well known that every diatonic scale can be reordered
into a sequence of fifths. For the C-major scale, the sequence
of fifths is F-C-G-D-A-E-B, renumbering the notes of the scale in
the order 4-1-5-2-6-3-7. This is the sequence of notes produced
when one starts with C and then tunes a keyboard alternately down
a fourth and up a fifth until the last note of the scale is
reached. For consistency of metaphor, I call the sequence of
fifths a "chain" in which the fifths are "links," without
assuming any specific tuning system.
[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.
Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#
Fig. 1. A Section of the Chain of Fifths
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#
Fig. 2. Six Diatonic Scales in a Chromatic Chain
From Eb to G#
==============
8. Perfect fifths do not combine to produce a perfect octave.
The nth fifth in a sequence of n fifths is defined mathematically
by the expression (3/2)^(n), which can never be an exact multiple
of 2 since every power of 3 is an odd number. The mathematical
proof that no sequence of fourths can ever produce an octave is
less obvious since every even number can be expressed as a
fraction with an odd denominator, and every power of 3 can be
associated with an infinite sequence of even numbers, such as
18/9, 36/9, ...; 162/81,..., etc. However, an intuitive
musical proof can be deduced from the fact that fourths and
fifths are inversions of one another. No sequence of fourths can
generate an octave because the inverse sequence of fifths can
never do so.
================
[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.
Ascending Harmonic Intervals
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 Diatonic semitone
or minor second
6 C-F# Augmented Fourth Diminished Fifth
or tritone
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
Fig. 3. Harmonic Intervals Ordered From the Chain of Fifths
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.
[3] Pythagorean Tuning
[3.1] Tuning the diatonic scale with pure fifths, now known as
Pythagorean tuning, was the norm for nearly 2,000 years. The
Pythagorean tuning of twelve notes in a standard Eb-G# keyboard
is shown in Figure 4. The links are a sequence of powers of 3/2
to the right of C and powers of 4/3 to the left of C. Whenever
necessary, the fraction is divided by two to keep the pitch
within the compass of a single 2/1 octave. Thus, 3/2 x 3/2 =
9/4. 9/4 is divided by 2, making D = 9/8. Logarithms allow the
sequence to be calculated by addition instead of multiplication.
In logarithmic measure, each perfect fifth is approximately
702 cents. The tuning of D is equal to 204 cents, obtained by
subtracting a an octave of 1200 cents from 702 + 702 = 1404.
Since all the links are the same size, the harmonic intervals can
be transposed freely. All major seconds are 9/8, all major
thirds, 81/64, and etc.
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
Fig. 4 Pythagorean Tuning
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)
===============
9. In logarithmic measure, a perfect fifth is 702 cents.
Therefore twelve perfect fifths equals 8424 cents while seven
octaves is 8400 cents. The difference, the Pythagorean comma, is
therefore equal to 24 cents.
10. Euclid (attrib.), *Section of the Canon*, in Barker, *Greek
Musical Writings*, vol. 2 (Cambridge: Cambridge University
Press, 1989), 199. Twelve links on the chain of fifths can be
interpreted as 12 fifths, 6 whole tones, 4 minor thirds and 3
major thirds. Thus, the Pythagorean comma is also equal to the
difference between three major thirds or four minor thirds and an
octave. In Pythagorean tuning, a Pythagorean comma is the
interval between any two notes separated by 12 links on the chain
of fifths.
===============
[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.
==============
11. Blackwood, *Recognizable Diatonic Tunings*, 58.
==============
[4] Equal Temperament
[4.1] Equal temperament has been commonly used for the
past 150 years to tune pianos and organs and, before that, was
used for fretted instruments such as guitars and lutes. Equal
temperament flattens the fifths (and sharpens the fourths) by
1/12 of a Pythagorean comma, "p," as shown in Figure 5. This is
just enough to give a true octave from a chain of 12 links. In
musical terms, the 12 fifths in the chromatic line are made
exactly equal to 7 octaves. As can be seen from Figure 5, the
term "equal temperament" only applies to the fifths. The other
intervals are not tempered equally with respect to Pythagorean
tuning.
|<----------------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)
Fig. 5. Schematic of Equal Temperament
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)
============
12. The development of the 12-tone school of composition was a
logical consequence of accepting the 12-semitone model of equal
temperament in place of a chain of harmonic fifths.
============
[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
Fig. 6. Enharmonic Pairs in Equal Temperament
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.
===============
13. These stacked rows appear in Helmholtz's *On the Sensations
of Tone* (London: Longman, 1885; New York: Dover, 1954; orig.,
4th ed., Braunshweig: Verlag von Fr. Vieweg u. Sohn, 1877), 312,
where it is used to illustrate the enharmonic relation between
notes in Pythagorean intonation.
===============
[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
Fig. 7. Equal Temperament
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.
==================
14. An irrational number, such as pi or the square root of two,
is one that cannot be expressed as the ratio of two integers. An
irrational number includes a nonterminating decimal written to
the number of decimals needed for practical accuracy.
==================
[5] The Four Modes of Just Intonation
[5.1] In Pythagorean tuning, only the fifths and fourths are
tuned as consonant intervals. The thirds and sixths generated by
a sequence of perfect fifths are not generally considered
consonant by comparison to the most consonant major third which
is 5/4, a minor third of 5/6 and their inversions, a major sixth
of 5/3 and a minor sixth of 8/5. The diatonic tuning that
includes both consonant thirds and pure fifths is known as just
intonation. One traditional method of forming a C-major scale in
just intonation is to combine consonant tonic, dominant and
subdominant triads, F-A-C, C-E-G and G-B-D. This tunes the just
diatonic scale shown in Figure 8. The D-A link is narrowed by a
syntonic comma indicated by the "k." Since the other links are
pure, the pitch of A, E and B will also be lower than Pythagorean
by a syntonic comma. Even though each note has its optimum
tuning relative to C, just intonation is a theoretical scale
which is unsatisfactory for tuning an instrument with fixed pitch
because the narrow D-A results in the unjust intervals shown in
Figure 8. The narrow D-A sounds much the same as a Pythagorean
wolf fifth. The Pythagorean minor third and its inversion, the
major sixth between D and F also sound badly out of place in an
otherwise just scale.
F----C----G----D--k--A----E-----B
4/3 1 3/2 9/8 5/3 5/4 15/8
D-A = 5/3 x 8/9 = 40/27, A-D = 27/20
D-F = 4/3 x 8/9 = 32/27, F-D = 27/16
Fig. 8. Dissonant Internal Intervals
in Just Intonation Scale
[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
Fig. 9. Just scales in Bb, F, C and G
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:
1. C is tuned to an arbitrary unit pitch of 1.
2. Tune the pitch of each note to be a perfect fifth (3/2)
from the notes immediately adjacent to it.
3. If the pitch of any note is greater than 2, divide the
pitch by 2 to keep it within the same octave.
4. Stop after the 12 notes from Eb to G# have been tuned.
The equal temperament algorithm is the same as the Pythagorean
algorithm except that the interval in step 2 is 1/12 of a
Pythagorean comma narrower than a pure fifth. Although justly
tuned scales often have been considered to represent a different
species from the Pythagorean or the equally tempered scale, we
will show in Figure 10 how an algorithm can be derived that maps
Pythagorean tuning into just chromatic scales.
[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#.(12)
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
Fig. 10. Just Intonation Compared to 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.
Bb-major: fifth of tonic triad narrowed by a syntonic comma.
F-major: major fourth broadened by a syntonic comma.
C-major: fifth of the II chord narrowed by a syntonic comma.
G-major: fifth of the dominant triad narrowed by a comma.
D-major: fifth of tonic triad narrowed by a syntonic comma.
A-major: major fourth broadened by a syntonic comma.
It is, thus, amply clear that just scales with consonant thirds
cannot be meaningfully used on a chromatic keyboard.
===============
15. Rossing, *The Science of Sound* (Reading, Mass: Addison Wesley,
1982), 161. This is a general textbook on acoustics. Rossing's
chart lists the equally-tempered, just and Pythagorean tunings for a
22-note chromatic scale in cents and decimals. The just scale
corresponds to the "C" scale in Figure 10.
===============
[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)
Fig. 11. The Four Modes of Just Intonation
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.
==============
16. It is possible to generate an infinite number of chromatic
just scales by allowing the four adjacent links in Pythagorean
tuning to freely slide along the chain of fifths. Whenever C is
not part of the Pythagorean notes, it will necessarily be raised
or lowered by one or more syntonic commas and no note in the
chromatic scale will be tuned to 1. However, changing the
absolute tuning of the notes in this manner does not change their
relative tuning. Therefore, all of the just chromatic scales
generated in this manner can be transposed into one of the four
modes listed in Figure 11. The major and minor scales available
in each of the four modes are the following:
G-mode: Cb, Eb, G, B, D# E-minor mode: Fb, Ab, C, E, G#
C-mode: Fb, Ab, C, E, G# A-minor mode: Bbb, Db, F, A, C#
F-mode: Bbb, Db, F, A C# D-minor mode: Gb, Bb, D, F#, A#
Bb-mode: Ebb, Gb, Bb, D, F# G-minor mode: Cb, Eb, G, B, D#
17. The algorithm used to construct just chromatic scales can be
used to illustrate why intervals generated by the prime 7 cannot
be systematically included in diatonic keyboard tunings. Pleasant
sounding intervals can include sevens. Examples which are
commonly cited are a 7/4 minor seventh, a 7/5 diminished fifth,
both of which are part of a diminished seventh chord in which the
notes C, E, G, Bb are in the proportion of 4:5:6:7, or
1:5/4:3/2:7/4. A single scale in which these intervals are used
can be contrived.
Eb- Bb-s-F--C--G-s-D----A
7/6 7/4 4/3 1 3/2 7/4 21/16
However, temperaments based on 7's or factors of 7 do not produce
a usable family of chromatic keyboard scales from the chain of
fifths, which would have the desired ratios for the diminished
seventh chord. The "septimal" comma which produces a 7/4 minor
seventh and an 8/7 major second on the second link of the chain
of fifths is 64/63 (27 cents). Therefore, reiteration of the
septimal comma, "s", in an algorithm which tunes every minor
seventh to 7/4 produces two modes of septimal chromatic scales in
which E is 64/49, not the desired 5/4.
Mode 1 (F) Mode 2 (G)
G# 4096/2041 (+4s) 4096/2041 (+4s)
C# 8192/7203 (+4s) 348/343 (+3s)
F# 512/343 (+3s) 512/343 (+3s)
B 2048/1029 (+3s) 96/49 (+2s)
E 64/49 (+2s) 64/49 (+2s)
A 246/147 (+2s) 12/7 (+1s)
D 8/7 (+1s) 8/7 (+1s)
G 32/21 (+1s) 3/2
C 1 1
F 4/3 21/16 (-1s)
Bb 7/4 (-1s) 7/4 (-1s)
Eb 7/6 (-1s) 147/128 (-2s)
Ab 49/32 (-2s) 49/32 (-2s)
Db 49/48 (-2s) 1029/1024 (-3s)
Gb 343/256 (-3s) 343/256 (-3s)
Cb 343/192 (-3s) 7203/4096 (-4s)
Fb 2401/2048 (-4s) 2401/2048 (-4s)
These septimal modes are worse than the just modes because the
septimal comma is broader than the syntonic comma by five cents.
The deviation from just intonation is further aggravated because
septimal commas increase the pitch of notes to the right of C and
decrease the pitch of notes to the left of C, contrary to the
action of the syntonic comma. Thus, compared to just intonation,
the pitch of notes to the right of C will be painfully sharp and
notes to the left will be dismally flat. For example, the major
third necessary for the diminished seventh chord will be 2s+1k
(76 cents) broader than in a just scale while the minor sixth
will be 2s + 1k (49 cents) narrower.
==============
[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.
===============
18. Barbour, *Tuning and Temperament: A Historical Survey*
(East Lansing, Michigan: Michigan State College Press, 1953),
90-102.
19. Ibid., 100.
20. Ibid., 97-98.
21. Ibid., 94.
===============
[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.
==============
22. Helmholtz, *On the Sensations of Tone*, 453. A similar
list of "Extended Just Tuning" is found in Blackwood,
*Recognizable Diatonic Tunings*, 116-119.
===============
[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.
===============
23. Blackwood, *Recognizable Diatonic Tunings*, 74, demonstrates
that in a common progression of C-major triads from II to V, just
intonation would require that D as the root of the II chord be
one syntonic comma lower in pitch than D as the fifth of the V
chord. In *The Science of Musical Sound* (*Scientific American
Books*, 1983, 67), John Pierce shows a five chord
progression, I, IV, II, V, I, in which the just tuning of C drops
by a syntonic comma from the first chord to the last.
24. It is for this reason that Lloyd in his article on Just
Intonation in the 1954 Grove Dictionary adopted the position that
instruments without fixed pitch and vocalists use a flexible
scale in which the size of the intervals vary according to the
context and part of the reason that Lindley and Turner-Smith
introduced the concept of "leeway" into their algebraic tuning
theory.
===============
[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.
=============
25. Inspection of the chain of fifths tells us that a major
third can only generate one-quarter of the infinite series of
chromatic notes that are generated by the fifth. This is why the
tuning of the major third is a subsidiary factor in the
generation of diatonic scales.
=============
[6] Meantone Temperament
[6.1] Meantone temperament is a keyboard tuning that makes a
chromatic scale with consonant major thirds playable in diatonic
scales in a few closely related keys. It does so by equally
dividing the syntonic comma over the four links of each
major third. This results in a whole tone that is exactly half-
way between the alternate 9/8 and 10/9 found in just scales. The
name "meantone" is derived from the resulting mid-size interval
for the whole tone even though the division of the whole tone was
only a by-product of distributing the syntonic comma. Dividing
the comma into four "quarter commas" and distributing the
quarter-commas so that every fifth was tempered by one
quarter-comma also smooths out the three narrow fifths created by
the undivided comma in the just scale, replacing them with a
sequence of tempered fifths that are far more tolerable to
the ear. The earliest generally accepted meantone scale was
described by Pietro Aron in the early sixteenth century for a
chromatic scale from Eb to G#.(26)
==============
26. Barbour, *Tuning and Temperament*, 26.
==============
[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->|
Fig. 12. Schematic of Quarter-Comma Meantone Temperament
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:
Thus mean-tone provides the player with a group of
about a dozen 'central' keys in which all the important
chords are more in tune than they are in the modern
piano. ... Mean-tone is admittedly imperfect as a
tuning for chromatic music; for diatonic music,
however, it cannot be bettered, as the musicians of
earlier times knew very well.(28)
For a time, organs were built with more than twelve keys per
octave in order to utilize more of the extended chromatic scale.
The keys could be split or other devices used to enable the
performer to play either note of an enharmonic pair such as Ab/G#
or Db/C#.(29) There was a practice, apparently, of raising or
lowering the location of the wolf in the tuning of harpsichords
to fit the composition being played at the moment.(30) However,
manufacturers refrained from constructing instruments with more
than 12 keys per octave and performers refrained from learning
how to play such instruments, probably because the standard
keyboard was suitable for most uses. As musical development led
to the use of more keys outside its central compass, meantone
temperament became increasingly impractical.
===============
27. *Grove's Dictionary of Music and Musicians* (1954),
"Temperaments" (380).
28. Dart, *The Interpretation of Music* (New York: Harper & Row,
1963), 47.
29. Barbour, *Tuning and Temperament*, 108-9. Since the first
reference to split keys found by Barbour goes back to 1484, this
device must have been used for both Pythagorean and meantone
tunings.
30. *Grove's Dictionary of Music and Musicians*,
"Temperaments" (379).
===============
[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
Fig. 13. Quarter-Comma Meantone Temperament
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.
===============
31. Barbour's table 24 gives a monochord mean-tone tuning derived
by Gibelius in 1666 by arithmetic division of the comma which is
the same tuning shown in Figure 12. Gibelius's monochord is
divided into an octave between 216000 and 108000, in which G =
144450, D = 193200 and A = 129200. The equivalence of these
monochord tunings to Figure 12 is calculated as follows:
G = 216000/144450 = 4320/2889 = 480/321 = 160/107
D = 216000/193200 = 540/483 = 180/161
A = 216000/129200 = 540/323
Barbour states that these approximations "check closely with
numbers obtained by taking roots," with the G being off by
0.000003. Barbour, *Tuning and Temperament*, 29.
===============
[7] Well Temperament
[7.1] The term well temperament includes a family of temperaments
that modified meantone temperament to eliminate wolves and to
expand the range of playable keys by taking advantage of the
small difference between the Pythagorean comma and the syntonic
comma. The difference between the two commas is an interval of
32805/32786 (2 cents), which is called the schisma. If only four
links of a chromatic keyboard scale are tempered by a
quarter-comma, with the remainder being tuned pure, the chromatic
scale will exceed an acoustic octave by only a schisma and the
wolf fifth will be thereby minimized to the point of
nonexistence.
|------ 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------>|
Figure 14. Schematic of Well Temperament
[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
Figure 15. Well Temperament
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] Algorithms For Schismatic and Syntonic Equal Temperament
[8.1] We will develop and expand upon the method advocated by
Kirnberger in the eighteenth century for deriving a scale of
equal temperament from Pythagorean intonation.(32) A scale
obtained by the first procedure is schismatic equal temperament
and a scale obtained by the second procedure is syntonic equal
temperament. The tuning of schismatic ET is exactly equal to the
tuning for syntonic ET, the only difference being in their
algorithms. Both match the tuning of equal temperament to
several decimal places as shown in Figure 16.
==============
32. Barbour, *Tuning and Temperament*, 64.
==============
[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.
1. Starting with C, move 10 links to the left to Abb+1k.
Substitute G enharmonically for Abb+1k.
2. Move 1 link to the right 5 times, adding 1 schisma to
the width of each interval. Rename each note enharmonically.
3. Returning to C, move 10 links to the right to E#-1k.
Substitute F enharmonically for E#-1k.
4. Move 1 link to the left 4 times, subtracting 1 schisma
from the width from each interval. Rename each note
enharmonically.
[8.4] The algorithm that produces syntonic equal temperament
also starts with the intervals C-Abb, ten links to the left of C,
and C-E#, ten links to the right of C. Substituting G, which is
enharmonic to Abb+1k in syntonic ET, and reiterating the process,
D in Syntonic ET would be 10 more links to the left of Abb plus
one more syntonic comma, giving D enharmonically equivalent to
Fbbb +2k. At the other end of the chain, enharmonic Bb in
syntonic ET would be 10 links minus one syntonic comma to the
right of E#-1k(F), written as Bb = G###-2k. Syntonic ET is based
upon substitution of enharmonic pairs just as is equal
temperament. The tuning of each note is exactly the same in both
syntonic and schismatic ET. The notation and tuning for
schismatic and syntonic ET are given in Figure 16. The algorithm
for syntonic ET is as follows.
1. Starting with C, move 10 links to the left to Abb + k.
Substitute G enharmonically for Abb+1k.
2. Reiterate the action 5 more times, adding 1 syntonic
comma to the width of each 10-link interval. Rename each note
enharmonically.
3. Returning to C, move 10 links to the right of C to E#-1k.
Substitute F enharmonically for E#-1k.
4. Reiterate the action 4 more times, subtracting 1
syntonic comma for each 10-link interval. Rename each note
enharmonically.
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
Fig. 16. Schismatic and Syntonic Equal Temperament
[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
Fig. 17. Just (C-Major mode) Schismatic and Syntonic Equal
Temperaments
[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.
[9] Concluding Thoughts
[9.1] This article has explored the relations between keyboard
tunings based upon the chain of fifths. We began by accepting
the historical definition of a diatonic scale embedded in a
12-note chromatic keyboard. We demonstrated that the terms used
to name diatonic and chromatic intervals can be rationally
ordered according to the number of links between the intervals on
the chain of fifths. We then derived algorithms that map
Pythagorean tuning into just intonation and equal temperament and
map just intonation into equal temperament. Pythagorean tuning
and just intonation have traditionally been classified as
distinct tuning strategies, while Pythagorean tuning and equal
temperament have been related only by the Pythagorean comma.
This article has shown that the three tunings can be related
using algorithms that utilize combinations of pure fifths and
syntonic commas.
======================
APPENDIX I Historical Scales of Just Intonation Listed by
Barbour.
The numbers in the left hand margin correspond to the number
used by Barbour. The table lists the author of the
temperament, the date of its publication, the chromatic
compass of the tuning on the chain of fifths, and the number
of intervals which are pure or are altered by one or more
commas, starting at the left end of the chain.
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
APPENDIX II. Ellis's Table of Intervals not Exceeding an
Octave
This chart lists the intervals identified by Ellis as being
formed from pure fifths and major thirds. It provides the
inverted proportions used by Ellis and his name for the interval
along with the Pythagorean equivalent of the interval from Figure
11. F#-1k was not part of Figure 11 because it is not included
in a scale in which C = 0. G#-1k is a well-tempered interval
from Figure 14. The remainder of the intervals are from Figure
11.
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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