Full thread (4 messages)
From: Margo Schulter (2004-05-12)
Subject: MIDI: Harmony in 17-ET/WT/ME/JOT
Hello, there, everyone.
Recently there's been some discussion of harmony in 17-tET, and I
wanted to provide some MIDI examples of one kind of four-voice
harmony that can fit nicely in 17 -- equally tempered, or more or less
unequally tempered or tuned in various ways, including George Secor's
masterful 17-tone well-temperament, known as 17-WT for short.
First, why don't I give all four versions of my composition _Cantilena
for George Secor_ to demonstrate some of the ways of dividing an
octave into 17 diatonic semitones or "thirdtones," since three of them
make a whole-tone (e.g. C-Db-C#-D), and then provide more information
on the style of harmony and on each tuning.
1. Cantilena for George Secor, 17-tET (equal temperament)
<http://www.calweb.com/~mschulter/cantigse.mid>
2. Cantilena for George Secor, 17-WT (Secor well-temperament, 1978)
<http://www.calweb.com/~mschulter/cantigsw.mid>
3. Cantilena for George Secor, 17-ME (my "modified eventone," 2000)
<http://www.calweb.com/~mschulter/cantigsm.mid>
4. Cantilena for George Secor, JOT-17 (my 17-note just system, 2002)
<http://www.calweb.com/~mschulter/cantigsj.mid>
As has often been said, 17-tET and its relatives are unlikely
candidates for styles of harmony based on regular thirds at or near
ratios of 5:4 (~386.31 cents) and 6:5 (~315.64 cents); rather, harmony
in these 17-note systems can draw upon or approximate ratios formed by
factors of 2-3-7-9-11-13.
My composition illustrates one "neo-medieval" style of harmony in 17,
with some modern touches: using narrow minor thirds and wide major
thirds as ingredients in relatively concordant but active sonorities
which resolve to stable sonorities based on fifths and fourths.
Two characteristic sonorities are a minor third, fifth, and minor
seventh above the lowest voice, with the seventh tending to contract
to a fifth; and a major third, fifth, and major sixth, with the third
typically expanding to a fifth and the sixth to an octave.
In one style of microtonal terminology, these sonorities in 17-tone
and similar systems are often called "subminor seventh chords" and
"supramajor sixth chords" respectively, to emphasize that the ratios
are different from the 5-based ones that people often associate with
major and minor thirds and sixths.
While basic progressions of this kind are inspired by a 14th-century
European style, this piece often approaches such progressions by
somewhat Renaissance-like suspensions, mixing these elements in a
texture which might also suggest the minor seventh and added sixth
chords of some 20th-century styles.
These chords and progressions illustrate a characteristic of harmony
in 17-tone and similar systems. When thirds have ratios at or near 5:4
and 6:5 -- "5-limit harmony" -- the 5:4 major third placed _below_ the
6:5 minor third, forming a sonority of 4:5:6, tends to sound more
smooth and conclusive than the converse arrangement with the minor
third below the major (10:12:15).
In contrast, with the sizes of regular thirds found in a system like
17-tET or its relatives, the smaller minor or "subminor" third is most
smoothly placed _below_ the larger major or "supramajor" third. A
common example involves the simple ratios of the 6:7 (sub)minor third
at around 266.87 cents, and the 7:9 (supra)major third at around
435.08 cents: the 6:7:9 arrangement is often notably smoother than
14:18:21 with the large major third below. The latter sonority can
bring lots of tension and excitement, although in higher registers it
can take on more smooth qualities.
It's also important to say that this "neo-medieval" style is only one
approach to harmony in 17: another approach, which can also supplement
this one, is George Secor's focus on isoharmonic chords based on
higher primes like 7:9:11, 9:11:13, or 7:9:11:13, where adjacent
partials have the same differences (here two). Indeed, Secor's 17-WT
is ingeniously designed to yield nice approximations of these and
other isoharmonic chords with the right intervals in the right
places to form the desired combinations.
Another resource of 17 is the wealth of neutral steps and intervals,
of interest for example in following or borrowing elements from a
medieval or later Near Eastern style (where such intervals abound).
Melodically, 17-tone systems feature compact and incisive diatonic
semitones, at about 70.588 cents in 17-tET and ranging from around 56
to 81 cents in the unequal systems here surveyed.
Now let's quickly consider some of the specifics of these tunings,
with links to the MIDI examples provided again for convenience.
1. 17-tone equal temperament (17-tET)
<http://www.calweb.com/~mschulter/cantigse.mid>
In 17-tET, each thirdtone has an identical size of about 70.588 cents,
and ten of these steps make the fifth at around 705.88 cents, about
3.93 cents wide. Major thirds at six scale steps are around 423.53
cents, which seems around the region of maximum complexity, richness,
or "dissonance," depending on your point of view (and the timbre).
While this is fine for lots of harmony in a style for 17, and can be
pleasantly exciting, other 17-note systems often seek and obtain major
thirds at or closer to ratios such as 9:7 or 14:11 (~417.51 cents).
2. George Secor's 17-tone well-temperament (17-WT)
<http://www.calweb.com/~mschulter/cantigsw.mid>
This system, most admirably modelled by analogy to a 12-note
well-temperament of the classic late 17th-18th century type, has the
thirds in the nearer keys or transpositions yield the best
approximations of 6:7:9, with fifths in this portion of the tuning
tempered more heavily at about 707.22 cents (about 5.265 cents wide,
comparable to meantone in the opposite direction). In the more remote
portion of the circle, fifths are tempered more mildly at about 704.38
cents, yielding some just 14:11 major thirds. In this piece, Secor's
temperament brings the (sub)minor seventh sonority C-Eb-G-Bb, for
example, not too far from a simple 7-based ratio of 12:14:18:21 (with
an outer 7:4 minor seventh), about 0-267-702-969 cents in a just
version, and 0-278-707-985 cents at this position in 17-WT. The narrow
diatonic semitones in the nearer portion of the circle at around 63.90
cents also have an excellent quality for melody.
3. My 17-tone circulating "modified eventone" (17-ME)
<http://www.calweb.com/~mschulter/cantigsm.mid>
While Secor's 17-WT shows the kind of grand design that can be applied
to a 17-tone system, mine might show the flexibility of this genre.
Having started with a regular 12-note tuning (Eb-G#) with the fifths
tempered to produce pure 14:11 major thirds (an arrangement Secor also
uses, as I was later to learn, in the more remote portion of 17-WT), I
simply added five notes to complete a tuning circle, with the most
heavily tempered fifths and thirds closest to 6:7 and 7:9 in this
remote portion. For the kind of regular temperament formed by the
first 12 notes, I use the term "eventone" -- a major third, as in a
meantone, is formed by two equal or "even" whole-tones. In this
version, the fifths are more mildly tempered (about 2.42 cents wide),
and the thirds rather more complex than in Secor's 17-WT; diatonic
semitones in the nearer portion of the circle, at around 78.115 cents,
are nicely compact but not so efficient as Secor's.
4. My "Just Octachord Tuning 17" (JOT-17)
<http://www.calweb.com/~mschulter/cantigsj.mid>
This 17-note tuning -- not a "circulating" system under usual timbral
conditions, since fifths can be wide by as much as 64:63 (~27.26
cents) -- is largely based on a series of superparticular (n+1:n)
ratios for diatonic semitones or thirdtones -- 28:27, 27:26, 24:23,
23:22, and 22:21. A fourth is divided into seven thirdtone steps, or
eight notes (e.g. Bb-A#-B-C-Db-C#-D-Eb), and thus my "Just Octachord
Tuning." While all intervals are based on integer ratios, and 11 of
the 17 fifths are pure, one feature resembling a typical 18th-century
well-temperament for 12 notes is the appearance of some prominent
fifths about 5 cents impure (F-C, G-D, A-E). Here, an internal cadence
arrives at one of these "virtually tempered" fifths (A-E), while the
final cadence lands on the pure fifth Bb-F. In the nearer portion of
the circle, as in my "modified eventone," major thirds are at or near
14:11, with minor thirds at or near 13:11 (~289.21 cents).
To sum up, the elements in 17 of small minor thirds, large major
thirds, and compact diatonic semitones can all pull together nicely in
congenial styles of harmony and counterpoint. At the same time,
different 17-tone systems can render the same piece with different
shades of harmonic color and melodic proportion; as with different
shades of 16th-century meantone, or 18th-century well-temperament,
this variety has its own charm.
Most appreciatively,
Margo Schulter
mschulter@...
From: bfowol (2004-05-18)
Subject: Re: MIDI: Harmony in 17-ET/WT/ME/JOT
Hi Margo,
Could you elaborate a bit more on the structure of the Just Octachord
17 scale? From your description I gather that not all intervals are
just? Or perhaps I missed something...
Regards,
Paul
--- In MakeMicroMusic@yahoogroups.com, Margo Schulter
<mschulter@c...> wrote:
[...]
> 4. My "Just Octachord Tuning 17" (JOT-17)
> <http://www.calweb.com/~mschulter/cantigsj.mid>
>
> This 17-note tuning -- not a "circulating" system under usual
> timbral conditions, since fifths can be wide by as much as 64:63
> (~27.26 cents) -- is largely based on a series of superparticular
> (n+1:n) ratios for diatonic semitones or thirdtones -- 28:27,
> 27:26, 24:23, 23:22, and 22:21. A fourth is divided into seven
> thirdtone steps, or eight notes (e.g. Bb-A#-B-C-Db-C#-D-Eb), and
> thus my "Just Octachord Tuning." While all intervals are based on
> integer ratios, and 11 of the 17 fifths are pure, one feature
> resembling a typical 18th-century well-temperament for 12 notes
> is the appearance of some prominent fifths about 5 cents impure
> (F-C, G-D, A-E). Here, an internal cadence arrives at one of
> these "virtually tempered" fifths (A-E), while the final cadence
> lands on the pure fifth Bb-F. In the nearer portion of
> the circle, as in my "modified eventone," major thirds are at or
> near 14:11, with minor thirds at or near 13:11 (~289.21 cents).
>
From: Margo Schulter (2004-05-19)
Subject: Re: MIDI: Harmony in 17-ET/WT/ME/JOT
Dear Paul (who asked about JOT-17),
Please let me clarify that JOT-17 is a just tuning under the
definition that all intervals are derived from integer ratios,
although some of the fifths are impure, that is, at integer ratios
other than 3:2 (~701.955 cents).
There's an article which I wrote, "JOT-17: A Just Thirdtone System for
Neomedieval Music," in _1/1: The Journal of the Just Intonation
Network_, Vol. 11, No. 3 (Autumn 2003), pp. 10-25, with some musical
examples from articles in the current issue available on the Web in
MIDI format:
<http://www.justintonation.net/soundfiles.html#jot-17>
Why don't I show a diagram of the octachord structure, recognizing
that the results for people viewing this on Yahoo could be
unpredictable, and so also providing a URL for an ASCII version of
this diagram:
<www.calweb.com/~mschulter/jot17asc.txt>
First 4:3 octachord (498.04)
|-------------------------------------------------|
0 62.96 128.30 208.84 266.87 340.55 417.51 498.04
1/1 28/27 14/13 44/39 7/6 28/23 14/11 4/3
|-------|------|------|------|------|------|------|
3696 3564 3432 3276 3168 3036 2904 2772
Bb A# B C Db C# D Eb
28:27 27:26 22:21 91:88 24:23 23:22 22:21
62.96 65.34 80.54 58.04 73.68 76.96 80.54
Middle 9:8 tone (203.91)
|------------------------|
498.04 561.01 626.34 701.96
4/3 112/81 56/39 3/2
|-------|-------|-------|
2772 2673 2574 2464
Eb D# E F
28:27 27:26 117:112
62.96 65.34 75.61
Second 4:3 octachord (498.04)
|-------------------------------------------------------|
701.96 764.92 830.25 910.79 968.83 1042.51 1119.46 1200
3/2 14/9 21/13 22/13 7/4 42/23 21/11 2/1
|-------|-------|-------|-------|-------|-------|-------|
2464 2376 2288 2184 2112 2024 1936 1848
F Gb F# G Ab G# A Bb
28:27 27:26 22:21 91:88 24:23 23:22 22:21
62.96 65.34 80.54 58.04 73.68 76.96 80.54
Here's a Scala file for an octave of Bb-Bb:
----------------- Scala file starts on next line of text ------------
! jot17a.scl
!
Just octachord tuning -- 4:3-9:8-4:3 division, 17 steps (7 + 3 + 7), Bb-Bb
17
!
28/27
14/13
44/39
7/6
28/23
28/22
4/3
112/81
56/39
3/2
14/9
21/13
22/13
7/4
42/23
21/11
2/1
----------- Scala file concluded on newline following '2/1' ------------
Here's another Scala file with a C-C arrangement (same system, but with
intervals with respect to C rather than Bb):
----------------- Scala file starts on next line of text ------------
! jot17.scl
!
Just octachord tuning -- 9:8-4:3-9:8 division, 17 steps (7 + 3 + 7), C-C
17
!
91/88
273/253
273/242
13/11
364/297
14/11
117/88
91/66
63/44
3/2
273/176
819/506
819/484
39/22
182/99
21/11
2/1
----------- Scala file concluded on newline following '2/1' ------------
Most appreciatively,
Margo
mschulter@...
From: Margo Schulter (2004-05-19)
Subject: Re: MIDI: Harmony in 17-ET/WT/ME/JOT
On Tue, 18 May 2004, Margo Schulter wrote:
> Why don't I show a diagram of the octachord structure, recognizing
> that the results for people viewing this on Yahoo could be
> unpredictable, and so also providing a URL for an ASCII version of
> this diagram:
>
> <www.calweb.com/~mschulter/jot17asc.txt>
This should have been written:
<http://www.calweb.com/~mschulter/jot17asc.txt>
Most appreciatively,
Margo
mschulter@...