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.