13-based voice-leading (MIDI files)

Hello, everyone, and thanks to everyone for that thread on "13-limit
counterpoint," which inspired me to write a short composition using
tempered approximations of 13-based intervals in Peppermint. The style
is a genre popular in Western Europe around 1200 called a conductus,
which might be used to "conduct" a procession, with voices often more
or less in note-against-note motion. Here are a few different MIDI
registrations:

<http://www.bestii.com/~mschulter/huscond3.mid>
<http://www.bestii.com/~mschulter/huscond6.mid>
<http://www.bestii.com/~mschulter/huscond7.mid>
<http://www.bestii.com/~mschulter/huscond8.mid>

The piece is in a Near Eastern maqam mode, Maqam Husayni, which in
contemporary Arab theory based on 24 equal quartertones per octave
would be 0-150-300-500-700-850-1000-1200 cents. Scott Marcus, however,
finds that practical performers often tune string instruments in pure
fifths or fourths, favoring minor thirds around Pythagorean (32:27, or
about 294.135 cents), and recognize various nuances of intonation to
fit different modes or musical situations.

Here Maqam Husayni is realized in Peppermint as an approximation of a
13-based just version, shown first followed by the Pepperment tuning
with signed numbers indicating differences from just:

Just: 1/1 13/12 13/11 4/3 3/2 13/8 39/22 2/1
0 138.573 289.210 498.045 701.955 840.528 991.165 1200
13:12 12:11 44:39 9:8 13:12 12:11 44:39
138.573 150.637 208.835 203.910 138.573 150.637 208.835

Pepmnt: E F* G A B C* D E
0 138.202 287.713 495.904 701.955 842.297 991.809 1200
0 -0.371 -1.497 -2.141 +2.141 +1.770 +0.644 0
138.202 149.512 208.191 208.191 138.202 149.512 208.191
-0.371 -1.126 -0.644 +4.281 -0.371 -1.126 -0.644

Here is a score of the three-voice piece, which opens with a brief
two-voice introduction, and follows a 6/8 meter:

123 456 123 456 123 4 5 6 123 456
1 2 | 1 2 | 1 2 + + | 1 2 |
E5 C*5 B4 A4 B4 C*5 D5 E5
E4 F*4 G4 A4 G4 F*4 E4

5
123 456 123 456 123 4 5 6 123 456
1 2 | 1 2 | 1 2 + + | 1 2 |
E5 C*5 B4 A4 B4 C*5 D5 E5
B4 C*5 D5 E5 D5 C*5 A4 B4
E4 F*4 G4 A4 G4 F*4 E4

10
123 4 5 6 1 2 3 456 1 2 3 456 123 456
1 2 | 1 2 | 1 2 | 1 2 |
E5 D5 C*5 D5 C*5 D5 E5 F*5
B4 A4 G4 A4 A4 G4 A4 B4 C*5
E4 D4 E4 F*4 G4 F*4

15
123 456 123 4 5 6 123 456 123 456
1 2 | 1 2 | 1 2 | 1 2 |
F*5 D5 E5 E5 F*5 G5 F*5 E5
C*5 B4 A4 B4 C*5 D5 E5
F*4 G4 A4 A4 G4 A4
20
123 4 5 6 123 4 5 6 123 4 5 6 123 456
1 2 | 1 2 | 1 2 | 1 2 |
E5 F*5 E5 G5 D4 E4 F*5 D5 C*5 E5
D5 C*5 D4 C*4 B4 C*5 A4 B4
A4 G4 F*4 E4

123 456 123 456 123 456 123 456
1 2 | 1 2 | 1 2 | 1 2 ||
F*5 D5 E5
C*5 B4
F*4 E4

A few cadences might invite comment, illustrating some of the uses of
neutral intervals (vertical or melodic) in what is generally a
13th-century European style of polyphony. At measures 15-16, we have:

| 1 2 | 1 2 |
G5 F*5 E5
D5 E5
G4 A4

Here F*5 forms a cadential sonority G4-D5-F*5 approximating 12:18:22
(about 0-702-1049 cents in JI, and 0-704-1050 in Peppermint), with an
outer 11:6 neutral seventh contracting to a fifth and an upper 11:9
neutral third contracting to a unison. The relative simplicity of 11:6
might give it some degree of "compatibility" or consonance.

At measures 19, there is a tribute to Kyle Gann:

123 4 5 6 123 456
1 2 | 1 2 |
F*5 D5 C*5 E5
C*5 A4 B4
F*4 E4

This is a variation on the Landini cadence alluded to in Kyle's recent
13-limit piece. The Landini idiom involves two voices at a sixth which
is expected to expand stepwise to a stable octave (here the outer
voices in the three-voice progression F*4-A4-D5 to E4-B4-E4), with the
upper voice momentarily descending to the fifth before resolving to
the expected octave by the upward leap of a third (here D5-C*5-E5).
Here the Landini figure in the upper voice of D5-C*5-E5 includes a
melodic neutral third, C*5-E5, at about 16:13.

The conclusion also highlights some neutral intervals:

123 456 123 456 123 456 123 456
1 2 | 1 2 | 1 2 | 1 2 ||
F*5 D5 E5
C*5 B4
F*4 E4

The cadential sonority F*4-C*5-D5 is approximately 22:33:36, with a
fifth F*4-C*5 plus an 18:11 neutral F*4-D5 sixth above the lowest
voice, and a neutral second C*5-D5 of about 12:11 between the upper
voices (around 0-702-853 cents JI, and 0-704-854 cents in Peppermint).
The outer neutral sixth expands to an octave E4-E5, while the middle
voice holds its C*5 to produce a sonority of E4-C*5-E5, near 8:13:16
(about 0-841-1200 cents in JI, and 0-842-1200 in Peppermint) before
resolving obliquely to its expected note of arrival B4. This is a kind
of variation on a typical 13th-century cadence where a major or minor
sixth moves obliquely to a fifth.

Anyway, thanks for the opportunity and encouragement to write this
piece and learn a bit more about 13-based voice-leading.

Peace and love,

Margo

Thank you margo for reminding us how easy it might have been to have music develop along this path.
I t shows how an interval like the 13 can be incorporated into this idiom without any problem at all. in fact it hard to imagine that some one might have tried this at some point and just couldn't get anywhere which is not uncommon.
i like the second and third best, but maybe i just like the 2nd and third repeat of it more than the timbres it is always hard to pin down ones subjectivity

Margo Schulter wrote:

>Hello, everyone, and thanks to everyone for that thread on "13-limit
>counterpoint," which inspired me to write a short composition using
>tempered approximations of 13-based intervals in Peppermint. >
> <http://www.bestii.com/~mschulter/huscond3.mid>
> <http://www.bestii.com/~mschulter/huscond6.mid>
> <http://www.bestii.com/~mschulter/huscond7.mid>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

> Date: Fri, 25 Nov 2005 16:22:59 -0800
> From: Kraig Grady <kraiggrady@...>
> Subject: Re: 13-based voice-leading (MIDI files)
>
> Thank you margo for reminding us how easy it might have been to have
> music develop along this path.
> I t shows how an interval like the 13 can be incorporated into this
> idiom without any problem at all. in fact it hard to imagine that some
> one might have tried this at some point and just couldn't get anywhere
> which is not uncommon.

Thank you for your encouragement, and following your impressions, I'd say
that this is the kind of music which might have evolved had the Near
Eastern influence on Western European culture during the 12th and 13th
centuries included some musical styles using neutral intervals: just
ratios like 12:11, 13:12, and 14:13 are part of some Near Eastern theory
during this general era.

As you say, it fits together nicely.

> i like the second and third best, but maybe i just like the 2nd and
> third repeat of it more than the timbres it is always hard to pin down
> ones subjectivity

This is an interesting point: is a repeat preferred because of different
instrumentation, or simply because it is a repeat and one can listen to
something already known from a different "point of view"? One possibly
connecting idea: a given timbre might at once provide general novelty and
draw you in to concentrate on a particular melodic line in a contrapuntal
texture, for example.

Love and peace,

Margo