scales with three sizes of steps

Hello all,

[I had some troubles with onelist earlier and asked Paul Erlich to fwd this
to the list. Now onelist has fixed the troubles and I can post myself, so if
the message also appears sent from Paul then it is my fault, sorry]

I have been investigateing into scales, that have exactly 3 different step
sizes (between successive scale members), in different equal divisions, and
also from ratio-based (n-limit) systems. Would anyone let me know what of
these scales people are using, and what properties of structure are enjoyed
about them? I would appreciate a post to the list or private email if you
prefer, or references. Such scales I find to be interesting compromise
betwen simple two-step-size and many-different-step-size from perceptual
point of view, and they can have some nice maths properties.

Also, there are scales like the ordinary (syntonic) diatonic most people use
for diatonic just intonation, who has scale steps of 9:8, 10:9 and 16:15.
This does have 3 step sizes, but because the two tone sizes are easy
equivalenced one can also say, that it is an "approximation" of a scale with
*two* step sizes (by which I mean not to say that a two-step-size version
(e.g. Pythagorean 9:8 and 256:243) is better, and the J.I. "merely"
approximates it, of cours, but hopfully you see what I mean).

So my other questions are: what J.I. scales (any cardinality) of greater
complexity might be used (or "do you use") to "approximate", in that meaning
I just explained, a scale of three-step-sizes? So I look for J.I. scales
whose step-sizes {s_1, s_2, ... s_i} can be collapsed into three equivalence
classes {S_1, S_2, S_3} such that all the s_i in S_1 are "much" closer to
each other than they are to the s_i in S_2 or S_3, if that makes sense.

And lastlly, a general question: have models been proposd, who describe, and
explain a rationale for, this kind of equivalenceing of different J.I.
intervals into classes?

Thank you very much for any answers you have for me. I was reading this list
for some time now, and I'm sorry to first post only with all this questions.
I will let you know of the result of my investigateions.

Luzius p.s. I will try to explain better if anything is not clear, sorry.

[Luzius Lanrai:]
> Also, there are scales like the ordinary (syntonic) diatonic most
people use for diatonic just intonation, who has scale steps of 9:8,
10:9 and 16:15. This does have 3 step sizes, but because the two tone
sizes are easy equivalenced one can also say, that it is an
"approximation" of a scale with *two* step sizes

Hi Luzius,

Though this ("scales with three sizes of steps") was not my motivation
for using this, this (two tetrad, scales-from-chords method) is a very
simple process that I would imagine one could also use (or alter) to
generate a variety of seven note scales with three sizes of steps...
Using 8:10:12:15 as an example, if you were to separate this tetrad by
the inversion of its (8:15) seventh (which in this case results in a
mirror inversion of the 8:10:12:15), you would generate a scale of
1/1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8, 2/1, with three distinct step
sizes @: 16/15, 75/64, and 9/8, which if it were then to be recast in
12e, where the IV permutation is the familiar Hungarian minor "exotic
scale,"

0 1 4 5 7 8 11 12
0 3 4 6 7 10 11 12
0 1 3 4 7 8 9 12
0 2 3 6 7 8 11 12
0 1 4 5 6 9 10 12
0 3 4 5 8 9 11 12
0 1 2 5 6 8 9 12
0 1 4 5 7 8 11 12

it would of course then have three sizes of steps @: 100, 200, and 300
cents.

Dan

Dan Stearns wrote,

>1/1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8, 2/1, with three distinct step
>sizes @: 16/15, 75/64, and 9/8

I was going to give the same example! Another one is my 22-tET blues scale,
which is JI is: 1/1 6/5 4/3 27/20 3/2 9/5 (2/1), where the three step sizes
are 81:80 (really small in JI), 10:9, and 6:5.

I think Luzius may have been more interested in scales with more than three
steps sizes that fall into three general size-categories. One example is the
JI harmonic minor, 1/1 9/8 6/5 4/3 3/2 8/5 15/8 2/1. There are steps of
16:15, 10:9, 9:8, and 75:64, but 10:9 and 9:8 tend to merge into a single
category.

D.Stearns wrote:

>1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8, 2/1

Thank you Dan, that is a good example. I had thought of the "gypsy minor" as
your scale sometimes is called, but not this J.I. interpreting.

There are many J.I. scales that fit my request for 3 different sizes of
step, I should not have made such wide question. But which ones are being
used by microtone-composers? I am most interested in J.I. scales (and n-tet
> 12) that are not of same cardinality as diatonic scale i.e. 7, and are
difficultly approximate by 12-tone equal.

In fact it is more for the patterning of step-sizes that I care. Dan's scale
is proceeding:

x, y, x, z, x, y, x (x y and z are the three step-sizes)

which, as he notices for us, it has inversional symmetry around the centre.

Thank you for any other examples from composers on the list.

Regards, from Luzius

[Paul H. Erlich:]
>Another one is my 22-tET blues scale, which is JI is: 1/1 6/5 4/3
27/20 3/2 9/5 (2/1), where the three step sizes are 81:80 (really
small in JI), 10:9, and 6:5.

I think that this is an interesting example, because in a scale like
this I really think there is a big difference between the 1/22 and the
81/80 - as the 1/22 sounds like a scale step, or perhaps to put it
another way, the 27/20 can rather politely disappear
(phantasmagorically so to speak...) into the 4/3, whereas the 10/22
most certainly (to my ears anyway) will not.

>I think Luzius may have been more interested in scales with more than
three steps sizes that fall into three general size-categories.

Using the same (two tetrad) method you can also generate these types
of scales, say a 1/1, 19/15, 19/12, 19/10 + 20/19, 4/3, 5/3, 2/1,
which could easily be seen as a +1 +3 +1 +3 +1 +2 +1 in 12e where the
four step sizes fall into three general size-categories. Incidentally,
you could also create seven note scales with four distinct step sizes
using the same process, say the 10/9, 27/25, 21/20 & 81/70 of a
5:6:7:9, or a (rather unusual) 1/1, 17/13, 17/11, 17/9 + 18/17, 18/13,
18/11, 2/1 which could then be recast (at least as far as recognizing
four distinct step sizes is concerned) into 13, 17, 21, 23, 24, 25,
26, 28, 30, 31, 32, 34, (etc.) equal divisions of the octave.

>One example is the JI harmonic minor, 1/1 9/8 6/5 4/3 3/2 8/5 15/8
2/1. There are steps of 16:15, 10:9, 9:8, and 75:64, but 10:9 and 9:8
tend to merge into a single category.

What about something like a 12:14:18:21 (@: 1/1, 8/7, 7/6, 4/3, 3/2,
12/7, 7/4, 2/1), where in say 22e the 8/7 & the 9/8 would both be
taken, or "merge into a single category," at 4/22, whereas 26e would
differentiate, or recognize (@: 5/26 & 4/26) the two... where does one
draw the lines with these types of general size-categories? The points
that John Chalmers made (in TD 306.9) about contextual sensitivity to
commas ("Scales in the same equivalence class, defined as having the
same rank-order matrix, tend to be heard as retuning of each other.
However, this perception depends upon one's sensitivity to small pitch
deviations in context. Some listeners in some contexts might hear the
Pythagorean, 12-tet, meantone, and 5-limit JI versions of the major
scale as the same scale.") seems to me to indicate that the 'answer'
would have to take into account both the specifics of musical
contexts, and the individual abilities and limitations of a potential
listeners... I would think that with these two examples, 1/1, 8/7,
7/6, 4/3, 3/2, 12/7, 7/4, 2/1 in 22 & 26e, that it would primarily be
a choice between 'cleaning up' commas (22e), or fully recognizing them
within the scope of an equal division (26e).

Dan

[Luzius Lanrai:]
>I am most interested in J.I. scales (and n-tet > 12) that are not of
same cardinality as diatonic scale i.e. 7, and are difficultly
approximate by 12-tone equal.

One thing that quickly jumps to mind, and makes use of the same
scales-from-chords method, would be to take an overtone sequence up to
a <600� (or not, as you could also sequentially straighten everything
out after the fact, so to speak) point, and then finish the octave
starting on the overtones of the inversion of the last note... say
12:13:14:15:16 for example, this would give a nine note scale @: 1/1,
13/12, 7/6, 5/4, 4/3, 3/2, 13/8, 7/4, 15/8, 2/1, with five step sizes
falling into three general size-categories... or an eleven note scale
at 15:16:17:18:19:20 (1/1, 17/15, 6/5, 19/15, 4/3, 3/2, 8/5, 17/10,
9/5, 19/10, 2/1) with six step sizes falling into the same type of
three general size-categories... (etc.)

Dan

Luzius,

Though I'm pretty sure you'll get what I was trying to getting at, the
second example I gave (for an 11-note scale with step sizes falling
into three general size-categories) doesn't quite match up with you
wanting examples that "are difficultly approximate by 12-tone
equal..." So how about substituting the 15:16:17:18:19:20 with say a
30:36:37:38:39:40 (1/1, 6/5, 37/30, 19/15, 13/10, 4/3, 3/2, 9/5,
37/20, 19/10, 39/20, 2/1), as this will better point towards some of
the flexibility of the basic premise/method, and of course would also
be mighty "difficultly approximate by 12-tone equal."

Dan

Earlier I wrote: "1/1, 13/12, 7/6, 5/4, 4/3, 3/2, 13/8, 7/4, 15/8,
2/1, with five step sizes falling into three general size-categories."
As you would have to go all the way up to 43 before an equal division
would distinguish between the two groups of minor seconds, I believe
this constitutes an abuse of the term general - clearly this scale
would be better said to have two general size-categories as it would
be seen in 19e. However, something along the lines of 18:21:22:23:24
(1/1, 7/6, 11/9, 23/18, 4/3, 3/2, 7/4, 11/6, 23/12, 2/1) would be an
example of a 9-note scale with five step sizes falling into three
general size-categories that is also derived from this same method.

Dan