Re: Welcome to Nikita Urazbaev -- one approach to 24-tET

Dear Nikita,

Please let me begin by saying what an honor and responsibility it is
to share some ideas about 24-tET with you, who have your own very
creative ideas about consonance and dissonance in this scale. In some
ways our approaches may be quite different, and in some ways very
similar. For example, we seem to share the concept of 24-tET as two
12-tET scales a quartertone apart.

In discussing some specifics, as you have requested, I would like to
emphasize that my own favorite progressions are only one set of
possibilities, and you know this well, since you have found some
equally impressive musical possibilities along some of the many other
lines.

Also, please let me apologize for any errors in either this article or
the MIDI examples below. May I mention, especially for people reading
this on the World Wide Web, that my examples of 24-tET musical
notation call for a monospaced or typewriter-like font?

As you comment, one of the main questions in using 24-tET is how to
combine notes from our two 12-tET scales or "keys," whether to seek
dramatic dissonance or pleasant consonance of various kinds.

To show the notes of the scale, I will borrow a notation from the
composer Nicola Vicentino, which he used in the 16th century, although
for a somewhat different kind of tuning. Here I will use an asterisk
sign or * to show a note raised by a quartertone, or 50 cents. In
Vicentino's 31-note meantone tuning, this same shows a note raised by
about a fifth of a tone, somewhere around 35-41 cents, so the sign may
not be too much out of place in 24-tET, although the musical style is
very different.

This kind of notation especially fits the kind of instrument I use, a
synthesizer with two 12-note keyboards, here tuned a quartertone
apart. The * sign tells us to play a note on the upper keyboard:

150 350 650 850 1050
C#* Eb* F#* G#* Bb*
C* D* E* F* G* A* B* C*
0 250 450 550 750 950 1150 1250
-------------------------------------------------------------
100 300 600 800 1000
C# Eb F# G# Bb
C D E F G A B C
0 200 400 500 700 900 1100 1200

Because many people are accustomed to diagrams based on an octave of
C-C, I have used this octave, with numbers showing intervals in cents
in terms of C.

My own favorite way of seeking beauty in 24-tET involves certain
progressions moving from more complex or tense intervals to more
simple or blending ones.

Let us first try some of these progressions in 12-tET, and then
explore some of the beautiful new versions that 24-tET offers.

In this kind of style, our basic concords are unisons, octaves,
fifths, and fourths. Other intervals are active or unstable, seeking
to resolve to our stable intervals.

More specifically, major thirds often expand to fifths, and major
sixths to octaves. Major seconds may expand to fourths. Minor thirds
often contract to unisons, and minor sevenths to fifths.

Here are a couple of three-voice progressions showing some of these
resolutions in basic 12-tET. Here I will a notation like that of MIDI,
with C4 as middle C, and with links to MIDI files for each of the
examples:

E4 F4 E4 F4 F4 E4 C#4 D4
B3 C4 D4 C4 Bb3 A3 A3 G3
G3 F3 G3 F3 G3 A3 F#3 G3

(M6-8 + M3-5) (M6-8 + M2-4) (m7-5 + m3-1) (m3-1 + M3-5)

MIDI example: <http://value.net/~mschulter/12tet001.mid>
MIDI example: <http://value.net/~mschulter/12tet002.mid>
MIDI example: <http://value.net/~mschulter/12tet003.mid>
MIDI example: <http://value.net/~mschulter/12tet004.mid>

Below each example I have shown in parentheses which of our two-voice
resolutions are involved in a given progression. A general pattern is
that major intervals (seconds, thirds, sixths) often tend to expand,
while minor intervals (especially thirds and sevenths) tend to
contract.

You may notice that in these 12-tET progressions, there is a common
pattern. in any of the two-voice resolutions I show above, one voice
moves by a whole-tone of 200 cents, and the other by a semitone of 100
cents.

To find new and beautiful versions of these same progressions in
24-tET using the notes of both keyboards, we can imagine that the
minor intervals such as thirds and sevenths want to be yet smaller, so
as more easily to reach stable unisons and fifths.

Major thirds and sixths, in contrast, want to be yet larger, so as to
be closer to the fifths and octaves to which they would like to
expand.

Let us see, and hear, what happens, when we let the minor intervals
grow smaller, and the major intervals larger. First we will try two
progressions with expanding major intervals:

E*4 F4
B*3 C4
G3 F3

(M6-8 + M3-5)

MIDI example: <http://value.net/~mschulter/24tet001.mid>

Here the usual major third G3-B3 at 400 cents has grown by a
quartertone to a large major third G3-B*3 at 450 cents, and the usual
major sixth G3-E4 to the larger G3-E*4 at 950 cents. I find these
intervals beautiful, and cosmic.

Since these intervals are larger, they have less far to expand in
order to reach the fifth F3-C4 and the octave F3-F4. Instead of usual
12-tET semitones of 100 cents, we can use small 24-tET semitones, or
actually quartertones, of 50 cents for the resolutions of the two
upper voices: B*3-C4 and E*4-F4.

Here is a new 24-tET version of a similar cadence, with the same
larger major sixth G3-E*4 at 950 cents, plus a large major second or
whole-tone of 250 cents between the upper voices, D4-E*4:

E*4 F4
D4 C4
G3 F3

(M6-8 + M2-4)

MIDI example: <http://value.net/~mschulter/24tet002.mid>

The lower two voices descend by usual 200-cent whole-tones, just as in
12-tET, but the highest voice again need move only a small 50-cent
step in order to complete the resolution.

Now it is time to discover one of the special surprises which 24-tET
shares with 29-tET also. Let us consider another resolution of exactly
the same unstable sonority we used in the last example, G3-D4-E*4:

E*4 E4
D4 E4
G3 A3

(m7-5 + m3-1)

MIDI example: <http://value.net/~mschulter/24tet003.mid>

While in the last example we treated the interval G3-E*4, or 950
cents, as a large major sixth seeking the octave, we can equally well
consider it as a small minor seventh seeking to contract to a fifth,
as happens here!

Similarly, we can consider the interval of 250 cents between the two
upper voices, D4-E*4, as either a very large major second often
expanding to a fourth, or, as here, a very small minor third seeking
to contract to a unison.

Although we need only one special sign, here the * symbol, to name all
24 notes, we might want sometimes to use another sign as well, in
order to show the different musical meanings of the note E*4 in these
two examples. If we use the sign "d" to show a note _lowered_ by a
quartertone, then we can also write the second progression like this:

Fd4 E4
D4 E4
G3 A3

(m7-5 + m3-1)

MIDI example: <http://value.net/~mschulter/24tet003.mid>

This notation tells us that G3-D4-Fd4 is here thought of as having a
small minor seventh G3-Fd4 and a small minor third D4-Fd4, as the
spellings suggest. Also, the "d" sign -- rather like the usual flat
sign -- may suggest that the note Fd4 might seek a resolution by
descending motion, as happens here (Fd4-E4).

In contrast, in this kind of notation with both * and d, the spelling
G3-D4-E*4 suggests that E*4 may move upward, as it does here (E*4-F4).
In this way, * is a bit like the usual sharp sign.

For 24-tET progressions of this kind, a useful general pattern is that
each voice moves either by a usual whole-tone of 200 cents (as in
12-tET), or by a small step of 50 cents. Here are two more examples
involving small minor thirds and sevenths, showing two resolutions for
the same unstable sonority. For each example, I will give two
notations: a simple notation using only the * sign; or the notation
with the "d" sign as well:

E*4 E4 Fd4 E4
A*3 A4 Bbd3 A3
G3 A3 or G3 A3

(m7-5 + m3-1) (m7-5 + m3-1)

MIDI example: <http://value.net/~mschulter/24tet004.mid>

E*4 D*3 Fd3 Ebd3
A*3 G*3 Bbd3 Abd3
G3 G*3 or G3 Abd3

(m7-5 + m3-1) (m7-5 + m3-1)

MIDI example: <http://value.net/~mschulter/24tet005.mid>

To this point, we have considered three-voice progressions. Maybe a
good place to conclude this article would by sharing a couple of
four-voice progressions, and adding a few words of friendship.

First, here is a 24-tET progression mostly with expanding intervals:
large major seconds, thirds, and sixths:

E*4 F4
D4 C4
B*3 C4
G3 F3

(M6-8 + M3-5 + m3-1 + M2-4)

MIDI example: <http://value.net/~mschulter/24tet006.mid>

Here is a progression with mostly contracting intervals, small minor
thirds and sevenths, given in both methods of spelling:

E*4 D*4 Fd4 Ebd4
D4 D*4 D4 Ebd4
A*3 G*3 Bbd3 Abd3
G3 G*3 or G3 Abd3

(m7-5 + m3-1 + M3-5 + m3-1) (m7-5 + m3-1 + M3-5 + m3-1)

MIDI example: <http://value.net/~mschulter/24tet007.mid>

- - -

Now for a few final comments. Timbre can be very important in shaping
perceptions of consonance and dissonance, and a gentle timbre can be
very helpful in making intervals such as large major thirds sound at
once new and pleasant. This can make the choice of instruments for a
piece, or of the timbres used on a synthesizer, very important.

From my perspective, 24-tET like 12-tET is a kind of variation on
Pythagorean tuning with a chain of pure fifths: fifths and fourths are
almost pure concords, while the usual thirds and sixths are rather
active, but a bit more "subdued" than in Pythagorean.

In 24-tET, these intervals are free to take on what I might call
almost an "ultramodern" quality, getting larger or smaller by a full
quartertone or 50 cents. At the same time, our nearly pure fifths and
fourths provide a welcome "home" or destination for these active
intervals, so that they can resolve in a very satisfying way.

What I have tried to do here is only to give examples of a few musical
patterns in 24-tET in one kind of style. Of course I would welcome any
questions and comments, and warmly invite you to use anything here in
your own music if it seems to fit. You might find yourself mixing
these progressions with more dissonant passages, or changing the
progressions to fit your own style.

Please understand that 24-tET is something of an "unusual" tuning for
me: I am much more familiar for this kind of style with Pythagorean
tuning, and with temperaments where fifths are larger than pure, such
as 29-tET.

The basic progressions I have described are common in 13th-14th century
Western European music, and I have heard similar things in Georgian
and Gurian music, although the rules may be a bit different, and
certainly each culture has its own traditions.

By the way, I have read that Georgian musicians were writing about
three-voice music in the 12th century, a bit earlier than in Western
Europe, so I tend to regard the Georgian tradition as in some ways an
elder sister. I wonder whether it is possible in Russia to find
recordings of traditional Georgian music; this is a heritage of many
centuries, and Igor Stravinsky, as I recall, wrote of his admiration
for it.

While polyphony based on stable fifths and fourths is common in many
world cultures, I tend to regard 24-tET as rather "new" and "modern."
The meeting of the old and the new is maybe a theme in much of my
music, and when playing in 24-tET I sometimes seem to be in kind of
dialogue with the 20th century.

In friendship,

Margo Schulter
mschulter@value.net