Can of worms?

Does anyone think an attempt* at organizing (or at least bandying about) a
systematized organization and classification index of tuning systems (and
their accompanying terminology) is a good idea? (Or perhaps more of a
hopeless bother that works itself out over time?)

Respectfully,
Dan

*Perhaps there has already been a substantive attempt at this that I�m
unaware of?

On Mon, 11 Jan 1999 08:43:28 -0500, "Dan Stearns" <stearns@capecod.net>
wrote:

>Does anyone think an attempt* at organizing (or at least bandying about) a
>systematized organization and classification index of tuning systems (and
>their accompanying terminology) is a good idea? (Or perhaps more of a
>hopeless bother that works itself out over time?)

I think that would be useful, especially for newcomers like me. There must
be many different ways to classify scales, though. On the one hand, for
instance, we could divide scales into equal and non-equal, but then the
similarity between 31-tet and quarter-comma meantone would be obscured.

There seem to be four general categories that include most of the equal
divisions of the octave. Each of the meantone scales can be divided into
five whole steps and two half steps, which can be arranged in a diatonic
scale. Another category (which includes 15-tet and 22-tet) has diatonic
scales with 3 large whole steps and 2 small whole steps instead of 5 whole
steps of the same size. The remaining scales either have relatively small
fifths (such as 16-tet and multiples of 7-tet) or relatively large fifths
(such as 18-tet and multiples of 5-tet).

group I (meantone): 12, 19, 24, 26, 31, 36, 38, 43, 45, 50, ...
group II: 15, 22, 27, 29, 34, 39, 41, 46, 48, 53, ...
group III: 7, 9, 11, 14, 16, 21, 23, 28, 33, 35, 40, 47, 52, ...
group IV: 5, 6, 8, 10, 13, 17, 18, 20, 25, 30, 32, 37, 42, 44, 49, 51, ...

(although 17-tet seems to me more similar to the first two groups than to
any of the other scales in the last group).

Well-tempered scales could be classified by the number of notes in the
scale and the amount of tempering of each fifth in the scale away from 3/2.
Traditional well-tempered scales have 12 notes, but others such as 19-note
well-tempered scales might also be useful.

>I think that would be useful, especially for newcomers like me. There must
be many different ways to classify scales, though

As far as defining terminology goes, I found a perfect example of what I had
in mind @ :

http://www.ixpres.com/interval/dict/index.htm

Dan

Herman Miller wrote:

>
> There seem to be four general categories that include most of the equal
> divisions of the octave. Each of the meantone scales can be divided into
> five whole steps and two half steps, which can be arranged in a diatonic
> scale. Another category (which includes 15-tet and 22-tet) has diatonic
> scales with 3 large whole steps and 2 small whole steps instead of 5 whole
> steps of the same size. The remaining scales either have relatively small
> fifths (such as 16-tet and multiples of 7-tet) or relatively large fifths
> (such as 18-tet and multiples of 5-tet).
>
> group I (meantone): 12, 19, 24, 26, 31, 36, 38, 43, 45, 50, ...
> group II: 15, 22, 27, 29, 34, 39, 41, 46, 48, 53, ...
> group III: 7, 9, 11, 14, 16, 21, 23, 28, 33, 35, 40, 47, 52, ...
> group IV: 5, 6, 8, 10, 13, 17, 18, 20, 25, 30, 32, 37, 42, 44, 49, 51, ...
>
> (although 17-tet seems to me more similar to the first two groups than to
> any of the other scales in the last group).
>
> Well-tempered scales could be classified by the number of notes in the
> scale and the amount of tempering of each fifth in the scale away from 3/2.
> Traditional well-tempered scales have 12 notes, but others such as 19-note
> well-tempered scales might also be useful.

There are many of us who use just intonation as well as higher harmonics. Also
some of us use scale without a 1/1 or a single point of reference. I could
come up with quite a few categories of just Just scales. Hey was that an
anaphoria!
-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

>>Does anyone think an attempt* at organizing (or at least bandying
about) a
>>systematized organization and classification index of tuning systems
(and
>>their accompanying terminology) is a good idea? (Or perhaps more of a
>>hopeless bother that works itself out over time?)

Herman Miller wrote,

>I think that would be useful, especially for newcomers like me. There
must
>be many different ways to classify scales, though. On the one hand, for
>instance, we could divide scales into equal and non-equal, but then the
>similarity between 31-tet and quarter-comma meantone would be obscured.

Agreed!

>There seem to be four general categories that include most of the equal
>divisions of the octave. Each of the meantone scales can be divided
into
>five whole steps and two half steps, which can be arranged in a
diatonic
>scale. Another category (which includes 15-tet and 22-tet) has diatonic
>scales with 3 large whole steps and 2 small whole steps instead of 5
whole
>steps of the same size.

The relative positions of the large whole steps and small whole steps
depend on which triads you want consonant and which you are willing to
give up. Only in the meantone tunings can a diatonic scale have six
consonant triads. Also, the unequal whole steps of 15 and 22 sound poor
when used in succession melodically.

>The remaining scales either have relatively small
>fifths (such as 16-tet and multiples of 7-tet) or relatively large
fifths
>(such as 18-tet and multiples of 5-tet).

>group I (meantone): 12, 19, 24, 26, 31, 36, 38, 43, 45, 50, ...
>group II: 15, 22, 27, 29, 34, 39, 41, 46, 48, 53, ...
>group III: 7, 9, 11, 14, 16, 21, 23, 28, 33, 35, 40, 47, 52, ...
>group IV: 5, 6, 8, 10, 13, 17, 18, 20, 25, 30, 32, 37, 42, 44, 49, 51,
...

>(although 17-tet seems to me more similar to the first two groups than
to
>any of the other scales in the last group).

There are many criteria for classifying equal-tempered scales, such as
the consistency limits which I have discussed before and the diameter
measures proposed by Paul Hahn. Putting those aside, Miller's system is
good; I would prefer instead a more hierarchical classification. I stop
at 34-tET because beyond that the distinctions begin to blur (e.g., one
can use two different sizes of fifth to construct pentatonic scales in
35-tET). First, I would isolate the scales with fifths worse than those
of 5-tet (which I think are just usable with most timbres):

group I: 1, 2, 3, 4, 6, 8, 9, 11, 13, 16, 18, 23

Group II would have all other ETs within it; they all have acceptable
pentatonic scales, although the multiples of 5 have equipentatonic
scales which are quite special due to their symmetry (and consequent
propensity for atonality):

group IIA: 5, 10, 15, 20, 25, 30

Within that group the only multiples of 15 have good triads and contain
symmetrical 6-tone and 10-tone scales rich in those triads:

group IIA1: 5, 10, 20, 25
group IIA2: 15, 30

Now all other ETs have good pentatonic scales of two step sizes:

group IIB: 7, 12, 14, 17, 19, 21, 24, 26, 27, 28, 29, 31, 32, 33, 34

Of these, the multiples of 7 have equiheptatonic scales:

group IIB1: 7, 14, 21, 28

of which only 28 has consonant triads and symmetrical octatonic scales
rich in those triads:

group IIB1a: 7, 14, 21
group IIB1b: 28

The rest have heptatonic scales (defined Pythagoreically) with two step
sizes:

group IIB2: 12, 17, 19, 22, 24, 26, 27, 29, 31, 32, 33

which can be subdivided into those where the "Pythagorean" heptatonic
scales do not have consonant triads:

group IIB2a: 17, 22, 27, 29, 32, 33

and those where they do:

group IIB2b: 12, 19, 24, 26, 31 (this is Miller's meantone group again
-- I hesitate in calling 24 meantone, though; perhaps "double meantone"
is more appropriate for 24.)

One can divide group IIB2a into ETs without good 5-limit triads:

group IIB2a(1): 17, 32, 33

and those with good triads:

group IIB2a(2): 22, 27, 29

While all three in this group have passable 7-limit tetrads, only 22 has
10-note scales with two step sizes rich in consonant 7-limit tetrads:

group IIB2a(2)(a): 27, 29
group IIB2a(2)(b): 22

Once can divide group IIB2b into ETs without consonant 7-limit tetrads:

group IIB2b(1): 12, 19, 24

and those with good tetrads:

group IIB2b(2): 26, 31

In this group only 26 had 14-note scales with two step sizes rich in
consonant 7-limit tetrads:

group IIB2b(2)(a): 31
group IIB2b(2)(b): 26

That completes the classification of ETs under 35. While most other
scales have analogues within those ETs, some do not, such as Keenan's
scale, or 17-limit JI, or non-octave scales.