Greetings all,
I am now reminded twice (by Bob and Richard) of the shortcomings of cut
and paste. Yes, I should have kept the "plus a M3" on the end of this
sentance:
>> A syntonic comma is the difference between four just fifths and two
>Just octaves ......{plus an M3}.
This is the descrepancy that is aborbed by the first four fifths in
Aaron's meantone (1/4 comma). What confuses many people is that the
Pythagorean comma is found by the difference between 7 octaves and 12 fifths,
the Syntonic comma is found by comparing the result of four Just fifths to
its first resultant third, which is, as Ric said, is a very wide Pythagroean
third. At least, very wide to our ears, but there is a historical value to
this harshness. I enclose some writings by Margo Schulter on this subject.
For those that want it all, her web site is
http://www.medieval.org/emfaq/harmony/pyth.html
Regards,
Ed Foote RPT
4.4. The two commas: bugs or features?
Tuning systems, like musical styles, have their characteristic qualities and
quirks, and Pythagorean intonation is no exception. Two small intervals known
as "commas" define some of the distinctive features of a Pythagorean tonal
universe.
One quirk of Pythgorean tuning is the "Wolf" fifth or fourth which results
between the extreme notes of our tuning chain in fifths, g#-eb' or eb-g# in a
standard scheme with Eb at one end of the chain and G# at the other. The
amount by which this Wolf falls short of a pure fifth or exceeds a pure
fourth is known as a Pythagorean comma, equal to about 23.46 cents.
Another trait of the tuning is its rather wide major thirds and sixths, and
its correspondingly narrow minor thirds and sixths. In a Gothic context, this
is a feature rather than a bug, since it gives these intervals an active
quality inviting very effective resolutions . A measure of this distinctive
quality is the difference between a Pythagorean third or sixth and the same
interval in its simplest ratio - for example, the Pythagorean M3 (81:64, 408
cents) vis-a-vis the ideal Renaissance M3 (80:64 or 5:4, 386 cents). This
difference of 81:80 (about 21.51 cents) is known as the syntonic comma, or com
ma of Didymus.
4.4.1. The Pythagorean comma: mostly a bug
Our experiment in building a chromatic scale revealed that although all notes
are tuned in perfect fifths, we get only 11 out of the 12 potentially perfect
fifths in a full chromatic octave. Thus in our standard tuning
Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, each fifth in the chain is perfect but the two
notes at extremes of the chain do not quite mesh. Rather the interval g#-eb'
or eb-g# is a Wolf fifth or fourth, about 23.46 cents smaller than a pure
fifth or larger than a pure fourth.
Similarly, if we extended our chain by a 12th fifth in the sharp direction,
Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#, we would find that our final D# was not
precisely at a unison or octave with Eb, but rather a Pythagorean comma
sharper or closer to the nearest E.
One way to explain this apparent anomaly is to show that 12 perfect
fifths do not quite equal any even octave: rather they exceed it by this same
Pythagorean comma. We can demonstrate this point in two ways.
Taking the ratio of the fifth, 3:2, we can calculate the interval generated
by 12 fifths as (3:2)^12, an impressive 531441:4096 - or, factoring out seven
octaves, 531441:524288. To avoid the complication of multiple octaves, we can
also measure the Pythagorean comma as six whole-tones or (9:8)^6, or
531441:262144, which when we subtract an octave gives the same result.
Using cents, we can easily calculate that 12 fifths of about 702
cents each will yield an interval of (702 x 12) or 8424 cents, and likewise
six whole-tones of about 204 cents yield an interval of (204 x 6) or 1224
cents. Subtracting seven octaves (8400 cents) in the first case, and one
octave (1200 cents) in the second, we get an approximation of 24 cents for
the Pythagorean comma. Since in fact a 3:2 fifth is closer to 701.955 cents,
and a 9:8 major second to 203.91 cents, a closer approximation is 23.46 cents.
Another way to demonstrate the size of this comma is to note that an octave
is equal precisely to five Pythagorean whole-tones and two diatonic
semitones, as can be seen in each of the modal scales in Section 4.3. Thus we
have, using rounded values in cents:
5 whole-tones (9:8) = 204 x 5 = 1020 cents
2 diatonic semitones (256:243) = 90 x 2 = 180 cents
----------
1200 cents
Each whole-tone may be divided into a diatonic semitone of 256:243 plus an
apotome of 2187:2048 - about 90 and 114 cents respectively. Adding these five
diatonic semitones and five apotomes to our other two diatonic semitones, we
have:
7 diatonic semitones = 90 x 7 = 630 cents
5 apotomes = 114 x 5 = 570 cents
----------
1200 cents
In contrast, if we take six whole-tones (12 fifths minus six octaves), we
find that they contain in rounded cents:
6 diatonic semitones = 90 x 6 = 540 cents
6 apotomes = 114 x 6 = 684 cents
----------
1224 cents
Again, if we used more precise values in cents, we would find that an
interval built from six whole-tones exceeds an octave by 23.46 cents. This
value is identical to the difference between a diatonic semitone and an
apotome, i.e. (113.6850... - 90.2249...) cents, or again very close to 23.46
cents.
Jacobus of Liege notes this discrepancy, describing the hexatone or interval
of six whole-tones as a rough discord not equivalent to a pure octave.
>From a medieval perspective, the Pythagorean comma might be regarded as a
minor "bug" in the tuning system. As long as we stick to a chain of fifths
from Eb to G#, and the Wolf fifth or fourth between these two notes rarely
occurs in actual polyphony, the bug is mostly of academic interest. There
remain eleven perfect fifths or fourths per octave, happily the eleven most
likely to be used in practice.
Our title for this section refers to the Pythagorean comma as "mostly" a bug,
because it appears that some musicians of the epoch around 1400 were cleverly
taking advantage of this quirk to adjust another aspect of the tuning system
a bit less congenial to an emerging "modern" style than to traditional
polyphony from Perotin to Machaut.
4.4.2. The syntonic comma: "One era's feature ..."
While the Pythagorean comma seems to be a "bug," since it limits us to 11
perfect fifths out of 12 per octave, the syntonic comma might more justly be
called an artistic feature of Gothic music and tuning: the active and
unstable quality of thirds and sixths. As Carl Dahlhaus has eloquently
stated, the tuning of these intervals is "to be understood as a musical
phenomenon rather than a mathematically imposed acoustic blemish"
(translation by Mark Lindley).
Comparing the sizes of Pythagorean thirds and sixths with their counterparts
in Renaissance theory having the simplest possible ratios, we find in each
case a difference of 81:80 or about 22 cents, the syntonic comma:
-----------------------------------------------------------
Interval Pythagorean ratio Simplest ratio
-----------------------------------------------------------
M3 81:64 (408 cents) 5:4 (386 cents)
m3 32:27 (294 cents) 6:5 (316 cents)
M6 27:16 (906 cents) 5:3 (884 cents)
m6 128:81 (792 cents) 8:5 (814 cents)
-----------------------------------------------------------
Using more precise measures for these intervals, we would find, for example,
that the Pythagorean M3 is roughly 407.82 cents, and the simplest M3 of 5:4
roughly 386.31 cents, giving a syntonic comma of about 21.51 cents.
This differential can serve as a kind of index of the degree of acoustical
tension in thirds and sixths. In a standard Pythagorean tuning, they are a
full syntonic comma (21.5 cents) wide or narrow, producing a considerable
degree of tension which fits nicely with the active role of these intervals
in Gothic polyphony.
In later styles, where thirds and sixths take on a quality of stable euphony
and rest, this feature of Pythagorean tuning becomes more of a "misfeature,"
if not an outright "bug." Thus the just intonation and meantone systems of
the Renaissance aim to present thirds and sixths - or at least as many as
possible - in their simplest ratios, a differential of 0 cents. The
"well-tempered" tunings of the 18th century place these intervals on a kind
of sliding scale of tensions, with differentials ranging in one scheme from
2/11 of a syntonic comma to a full syntonic comma, the modes or keys
considered more remote having the greater acoustical tension.
In modern 12-tone equal temperament, major thirds at 400 cents and minor
sixths at 800 cents have a differential of about 13.69 cents (or about .64 of
a syntonic comma); minor thirds at 300 cents and major seconds at 900 cents
have a differential of about 15.64 cents (or about .73 of a syntonic comma).
As in Pythagorean tuning, M3 and M6 are wide while m3 and m6 are narrow.
>From an acoustical or mathematical viewpoint, both the Pythagorean and
syntonic commas reflect basic facts of musical geometry. It is impossible to
tune 12 pure fifths so as to arrive at an even octave; and it is impossible
in any fixed 12-tone tuning to achieve pure fifths and also to obtain thirds
(and sixths) in their simplest ratios.
In a Gothic setting, the Pythagorean comma and the resulting Wolf fifth or
fourth between Eb and G# is only a minor inconvenience or "bug," since these
accidentals are rarely combined. The syntonic comma, in contrast, is a
congenial feature: stable fifths and fourths in their ideal ratios, and
active thirds and sixths, both mesh nicely with the harmonic style.
>From another perspective, the syntonic comma also represents a fact of
musical geometry noted by Mark Lindley: the expressively narrow and incisive
Pythagorean diatonic semitone of 90 cents (256:243) is necessarily associated
with wide M3 and M6, and narrow m3 and m6.
Happily, from a Gothic viewpoint, both incisive melodic semitones and active
vertical thirds and sixths concord nicely with the artistic style. As
discussed, these dimensions together contribute to the expressiveness of many
cadences of the period.
In other periods, the tradeoffs between acoustical necessity and
musical style may perhaps be somewhat less happy. While favorite meantone
tunings of the Renaissance continued to accept an Eb-G# Wolf relegated to a
lair on the remote periphery of the modal system, by the 18th century it had
become a stylistic imperative to domesticate this creature. Schemes of
well-temperament and equal temperament, compromising many intervals slightly
rather than one or a few intolerably, are one approach to this problem;
keyboards with more than 12 notes per octave, proposed as early as the 15th
century, are another.
Similarly, in styles where thirds and sixths serve as restful concords, there
will be an inevitable compromise between vertical euphony and the desire for
incisive diatonic semitones. Renaissance tunings optimize thirds and sixths,
accepting the consequence of diatonic semitones considerably wider than 100
cents, typically in fact rather close to the Pythagorean apotome of 114
cents. Equal temperament yields acoustically somewhat more tense thirds
differing from the Renaissance ideal by the better part of a syntonic comma,
as we have seen, and semitones all measuring an even 100 cents.
The inexorable mathematics of the two commas remains constant, but stylistic
parameters and artful tuning solutions change. It would seem that indeed one
era's feature can be another era's bug.
4.5. Pythagorean tuning modified: a transition around 1400
By the early 14th century, keyboards with all 12 chromatic notes had become
common, and the full set of accidentals had become integral to the modern
practice and theory of the Ars Nova. Such accidentals served, for example,
the increasingly clear preference for resolutions by contrary motion where
one voice moves by a whole-step and the other by a half-step, e.g. m3-1,
M3-5, M6-8 - and, for Jacobus, also m7-5. Thus:
f#'-g' f#'-g' c#'-d'
d' -c' c#'-d' g# -a
b -c' a -g e -d
5 5 M6 8 M6 -8
M3 1 M3 5 M3 -5
(M3-5 + m3-1) (M3-5 + M6-8) (M3-5 + M6-8)
In the first two progressions, taken from a motet by Petrus de Cruce (c.
1280?), the accidentals f#' and c#' facilitate motion from an unstable 5/M3
or M6/M3 sonority to a stable fifth or trine (see Sections 3.2.1, 3.3) by way
of these resolutions. In the third example, typical of the 14th century, g#
(the final accidental to be added) and c#' likewise facilitate resolutions of
M3-5 and M6-8.
Another way of stating this preference is to say that a third contracting to
a unison should be minor, while a third expanding to a fifth or a sixth to an
octave should be major. Accidentals applied to unstable intervals and
combinations assist in fulfilling this preference articulated by various
theorists of the early 14th century, including even the conservative Jacobus.
In such progressions, the Pythagorean accidentals facilitate the "closest
approach" of an unstable interval to its stable goal even on a microtonal
level Let us consider again the progression:
f#'-g'
c#'-d'
a -g
The sharps raise each upper note of the penultimate sonority by a full
apotome of 114 cents (c-c#, f-f#), placing it only a 90-cent diatonic
semitone from its cadential goal. Vertically, the Pythagorean M3 at 408 cents
(a-c#') and M6 at 906 cents (a-f#') need expand only 294 cents each to attain
the stable fifth and octave respectfully, This expansion is brought about as
the lowest voice descends by a 204-cent whole-tone and each upper voice
ascends by a 90-cent semitone.
Artistically speaking, the unstable M3 and M6 are at once about 21.5 cents wid
er than their ideal Renaissance counterparts, adding a bit of extra dynamic
tension, and 21.5 cents closer to their directed goal, faclitating the
efficient and expressive release of this tension.
Indeed, Marchettus of Padua (c. 1318) proposes a variation on
Pythagorean tuning based on a subtle division of the whole-tone designed in
part to permit the smallest possible semitones in the "perfection" of
intervals such as M3-5 and M6-8. One reading of his system would actually
stretch the major sixth so far as to make it a minor seventh! - although
another interpretation would result in a cadential M6 not far from the
traditional Pythagorean ratio and a resolving semitonal motion not far from
the usual 90 cents.
For much music of the 14th century, including the famous Mass of Guillaume de
Machaut, Pythagorean tuning appears to provide an excellent solution in
practice as well as theory. However, musical styles change, and the period
around the end of the 14th century is no exception.
Composers of this epoch such as Matteo de Perugia and the
contributors to the Faenza Codex of keyboard music, Mark Lindley suggests,
may have exploited a rearrangement of the traditional Pythagorean tuning in
order to explore the possibilities of more blending thirds and sixths.
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