Re: For Joseph Pehrson (three questions)

Hello, there, Joseph Pehrson, and everyone.

Please let me quickly address three recent questions, some of which
have been nicely covered by others over the last couple of days.

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1. Pythagorean tuning and accidentals
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To what Dan Stearns and others have contributed regarding the spelling
of Pythagorean schisma thirds, I might mainly a general rule which can
help in working out such intervals.

In Pythagorean tuning, each flat or sharp lowers or raises a note by a
large or chromatic semitone, 2187:2048 or ~113.69 cents -- often
convenient to treat as a rounded 114 cents.

For example, let's consider the interval C-E#. Here we have a regular
major third C-E at 81:64, a rounded 408 cents, plus the chromatic
semitone E-E# or 114 cents, giving around 522 cents in all. This is a
Pythagorean comma larger than a regular 4:3 fourth, around 498 cents,
often known as an augmented third or "Wolf fourth."

In contrast, the regular fourth C-F has a major third C-E plus the
small or diatonic semitone E-F, 256:243 or a rounded 90 cents -- here
408 + 90 cents gives 498 cents, a normal 4:3 fourth.

Thus C-E#, for example, is a Pythagorean comma _wider_ than C-F; this
comma is equal to the difference between the two semitones,
531441:524288 or ~23.46 cents -- following convention, let's call it a
rounded 24 cents.

Likewise a Pythagorean diminished fourth or schisma major third,
e.g. C-Fb in the example from Dan Stearns, subtracts a chromatic
semitone Fb-F from the regular fourth C-F -- 498 - 114 = 384 cents.
This is a 24-cent comma smaller than the usual major third C-E at
81:64, equal to the fourth C-F less the diatonic semitone E-F -- about
490 - 90 = 408 cents.

A Pythagorean augmented second or schisma minor third, e.g. Fb-G in
Dan's example, is equal to a usual whole-tone F-G at 9:8, a rounded
204 cents, plus the chromatic semitone Fb-F -- 204 + 114 or 318
cents. This is a 24-cent comma larger than the usual minor third E-G,
a whole-tone F-G plus a diatonic semitone E-F -- 204 + 90 or 294 cents
(32:27).

Thus one method for estimating the size of an unfamiliar Pythagorean
interval is to start with the size of the unaltered interval (no
sharps or flats), and then to add or subtract 114 cents for each sharp
or flat.

For example, how about C-D##? Without accidentals, we would have a
whole-tone C-D of about 204 cents. Each sharp adds 114 cents, or 228
cents in all, so we have 204 + 228 = 432 cents. This is a "septimal
schisma major third" a 24-cent comma larger than the regular C-E, or
about 3.8 cents smaller than 9:7.

How about A#-Db? Our unaltered interval would be a regular fourth A-D,
or 498 cents, and both the sharp and the flat in this case subtract
114 cents each by raising the lower note and lowering the upper one,
so we have 498 - 228 or 270 cents, a "septimal schisma minor third" a
24-cent comma smaller than the usual 32:27 minor third A#-C# or Bb-Db
at around 294 cents, or about 3.8 cents larger than 7:6.

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2. Conversities, inversions, and chromaticism
---------------------------------------------

In reply to your response, maybe I should begin by observing that
while the concept of conversity applies specifically to multi-voice
sonorities with three or more voices and intervals, the concept of
inversion can also apply to simple two-voice intervals.

A conversity is defined a relationship between multi-voice sonorities
sharing the same set of intervals in different arrangements. Since
there is only one arrangement of a single interval, conversity does
not apply.

In contrast, inversion can apply to a two-voice interval, for example
the major second C3-D3, which has the inversion of D3-C4, a minor
seventh; more generally, we can say that major seconds and minor
sevenths are inversions. I like to describe such intervals as "octave
complements" -- thus M2 and m7 together make an octave.

Octave complements often have interesting affinities in various
musics, but not always and in all ways: thus in 13th-century Western
European theory, M3 is often classified as an "imperfect concord"
(relatively blending but mildly unstable), but m6 as a "perfect
discord" (acutely tense).

Mostly for the sake of simplicity, I used mainly diatonic note
spellings in my examples, but the concept of conversity can equally
apply to the many medieval and Renaissance sonorities involving
accidentals -- used either to form regularly spelled intervals of the
usual types, or less "usual" intervals ranging from early 15th-century
Pythagorean schisma thirds to the "enharmonic" intervals of Vicentino
(1555) altered by a meantone diesis of about 1/5-tone.

Whatever the merits of a 20th-century "interval class" theory, which
treats octave complements as equivalent -- a useful system for some
purposes, but not a universal assumption -- I would emphasize that the
general concept can be applied to tunings with fewer or more than 12
pitch classes per octave. Thus we have 53 such classes in a
quasi-cyclical Pythagorean tuning or 53-tone equal temperament
(53-tet), 31 basic classes in Vicentino's system or 31-tet, etc.

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3. Pythagorean "septimal schisma" intervals
-------------------------------------------

To answer a basic historical question quickly, I'm not aware of any
medieval sources describing or advocating the use of these intervals.
Rather, their use is a feature of the modern "Xeno-Gothic" tuning, a
24-note Pythagorean scheme discussed in a post I made to the Tuning
List around the first week of July 1998 -- when the list was at Mills
College. I would be glad to supply copies by e-mail.

Around 1325, Jacobus of Liege does actually describe a _tonus maior_
or "larger whole-tone" formed by two chromatic semitones, as opposed
to the normal whole-tone at 9:8, formed by a diatonic and a chromatic
semitone. As it happens, this _tonus maior_ (~228 cents) is in modern
terms a septimal schisma interval about 3.80 cents smaller than 8:7;
but Jacobus regards it as an ill-formed and highly discordant
variation on the normal whole-tone, not a practical interval.

In that same era, 1318, Marchettus of Padua describes the use of very
narrow semitones in common 14th-century cadential progressions.
However, his tuning mathematics for these semitones are not
conventional Pythagorean, and additionally leave lots of room for
differing interpretations.

A modern tuning system, Xeno-Gothic, takes one possible interpretation
of Marchettus (for example, very wide major thirds and sixths in
directed resolutions of M3-5 and M6-8 to stable fifths and octaves by
stepwise contrary motion), and implements this in a Pythagorean tuning
by using major thirds near 9:7 and sixths near 12:7 -- a comma larger
than usual.

However, this is my neo-Pythagorean implementation of one possible
reading of Marchettus's suggestions regarding cadences -- as opposed
to a medieval source describing the use of such a tuning.

Most respectfully,

Margo Schulter
mschulter@value.net