Non-Octave ETs: Spectrum Scales

Non-Octave ETs: Spectrum Scales

Here are two different Non-Octave Equal Temperaments, which are
constructed to take into account the spectra of the intended
instruments they will be played on.

First, let's look at a Non-Octave ET which is formed to give some
good approximations of 5-7 Limit JI. Obviously, this would be
intended for use with timbres which possess Harmonic Spectra. The
below scale is built by taking the ratio 15/8 divided into 45 parts.
Column one shows us the approximated ratio, two - the scales degrees
of the ET, cents values and the consecutive scales degrees (revealing
inversional symmetry).

19 Tone Non-Octave Scale based on division of 15/8 into 45 parts.

Aprox. Degree Cents Consecutive.
Ratio
0.000
5 120.919 120.919
8 193.470 72.551
10 241.837 48.367
7/6 11 266.021 24.184
6/5 13 314.389 48.367
5/4 16 386.940 72.551
9/7 18 435.307 48.367
21 507.859 72.551
7/5 24 580.410 72.551
26 628.777 48.367
3/2 29 701.329 72.551
32 773.880 72.551
34 822.247 48.367
37 894.799 72.551
39 943.166 48.367
7/4 40 967.350 24.184
42 1015.717 48.367
15/8 45 1088.269 72.551
50 1209.187 120.919

This kind of scale has the best of three worlds; 1. The evenness of
ET scale degrees, 2. The warmth of JI ratios, and 3. A stretched
octave of 9.187 cents. The Scala show/data function reveals 16
fifths at an average of 701.329 cents, and 16 energetic "wide"
fourths at 507.859.

Now the next obvious question in the reader's mind, is: "Well, what
about instruments with Inharmonic Spectra?"

I thought you'd ask that!

So imagining you've got that old idiophone laying around that you
wish to sample and tune to a very sweet scale which matches the
spectra of the timbre, and the results of your FFT analysis reveal
the prominent spectra in the first octave to be (frequencies
converted to cents values):

45.647
257.405
430.673
936.429
1196.692
1239.199

So we find an optimal scale by taking the near-octave found in the
spectra of a Gong, and dividing this by 29, to obtain the following
scale:

15 Tone Non-Octave Scale from division of the Near-Octave of Gong
Spectra into 29 Parts

Degree Cents Consecutive
0.000
1 42.724 42.724
3 128.172 85.448
6 256.345 128.172
7 299.069 42.724
10 427.241 128.172
12 512.690 85.448
14 598.138 85.448
15 640.862 42.724
17 726.310 85.448
19 811.759 85.448
22 939.931 128.172
23 982.655 42.724
26 1110.828 128.172
28 1196.276 85.448
29 1239.000 42.724

Here I have filled in the scale with intervals of choice, which were
not found in the spectra of the Gong, but one can easily see where
the spectra match to the scale degrees of this Non-Octave ET.

This can be a good strategy for tuning a single instrument with
inharmonic spectra, and in every case I've used this method, I've
found it to produce a much more resonant tuning for timbre being used.

Worth noting too, is that this inversionally symmetrical scale has
three step sizes. Scala data reveals: Pseudo Myhill's Property (I
guess this is the 3 step sizes), is a Winograd Deep Scale (someone
can tell me what that is and I'll be grateful), and "0" recognizable
fifths (to be expected in Gong spectra - and I'd still like to
understand what is the range that Scala considers a fifth). Anyway,
one must trust little of their biases toward a ratio-centric approach
here, because what we may find true in the area of harmonic timbres,
can have little to do with what sounds best on inharmonic timbres.

Notes:

After a bit of deep contemplation of recent discussions with Jeff
Scott on the topic of Non-Octave Scales, and much real-time playing
on my own, to internalize and appreciate the aesthetic qualities of a
wide variety of them, I've came to realize that the magic of the Non-
Octave species of microtonal tuning, is that a "stretched"
or "compressed" octave possesses a unique "identity" and sonic
fingerprint that is entirely different than the effect of tunings
with a perfect 2/1 (even with harmonic timbres). That there is not
the "blending effect" of a "JI 2/1", but is a subtle or not so subtle
beating at the near "stretched" or "compressed" octave, makes this
kind of tuning have a rare energetic sound all its own, which invites
exciting new kinds of compositional and stylistic thought. The "near-
octaves" are each unique and stand out of the musical texture in a
manner that is alien to JI, but not to Gamelan. For instance, the
tension between an ostinato bass, and a melodic line being played 2
near-octaves above, has a sound of energetic motion because of the
near-octave displacement over the musical range. Interestingly too,
is that these kinds of tunings are rich with illusions, where
sometimes an interval that is quite wide or narrow of 2/1, will
function aurally as the octave. New harmonies and melodies abound in
this kind of tuning.

Worth noting too is that there is extensive psychoacoustical evidence
that wide intervals are prefered and even played intuitively in many
cultures around the world (including trained Western orchestral
performers). Discussions with Jeff Scott about his experience in
playing Gamelan music, has shined even more personal light on why the
musical phenomenon of the stretched octave sounds so good. It is the
controlled tuning of beats into the scales that makes the intervals
have this unique identity, which actually helps them to be heard as
distinct in large ensemble settings (a point widely known and
discussed with regard to Gamelan tunings). This is a wonderful
property to have in a tuning. Hearing the beauty of it, is believing.

All this is in keeping with my continued quest to use scales which
are complimentary to the spectra of the instruments being used. Even
though there can be great challenges to one's compositional style,
when on occasion one must make the necessary great leap of using
tuning systems which contain no exact 2/1 relationship, the rewards
of musical beauty are well worth the quest.

Thanks,

Jacky Ligon

Cents Tables for Scala and Li'l Miss' Scale Oven (LMSO), for quick
tuning:

19 Tone Non-Octave Scale based on division of 15/8 into 45 parts.

120.919
193.470
241.837
266.021
314.389
386.940
435.307
507.859
580.410
628.777
701.329
773.880
822.247
894.799
943.166
967.350
1015.717
1088.269
1209.187

15 Tone Non-Octave Scale from division of the Near-Octave of Gong
Spectra into 29 Parts

42.724
128.172
256.345
299.069
427.241
512.690
598.138
640.862
726.310
811.759
939.931
982.655
1110.828
1196.276
1239.000