Gould on Generalized Diatonics

I decided to get a copy of Mark Gould's paper "Balzano and Zweifel: Another Look at Generalized Diatonic
Scales" so here's a summary.

It's starts off explaining the Balzano theory. This is roughly that you build up an MOS by alternating a
pair of intervals that add up to the generator. For the normal diatonic, these are the major and minor
third. It doesn't actually say "MOS" or "well formed" but that seems to be what we're talking about.
There's also the peculiar definition that "diatonic" means the larger size of scale step is more common and
"pentatonic" is the other way around. He decides not to use the scare quotes from then on, but I will
because I'm old fashioned enough to think that a pentatonic should have 5 notes to the octave.

The square lattice Mark mentioned on the main list, which looks like a triangular lattice rotated a bit and
reflected, is to show this pair of alternating intervals, rather than 5-limit harmony in general. All
"diatonics" mentioned in the paper are shown on similar grids.

Mark suggests a different tonic for Zweifel's 11 from 20 scale. That's so the not that changes as you move
up the chain of generators is the leading note to the tonic.

Next he gives his own criteria for "diatonics", which are like Balzano's but without worrying about there
being n(n+1) notes in the parent scale. Then there are some examples of such "diatonics".

Now the bit on ratios. He makes the startling statement "For all diatonic scales, the two intervals in the
grid approximate to a given pair of just intonation ratios" which is about as surprising as finding water
in a river. Beyond identifying traditional diatonic scales with 5:4 and 6:5, he doesn't say what ratios
were being approximated by the "diatonics" he gave before. Still, I can reverse engineer these.

There's an 11 from 27 scale which is based on a 7:6 and 10:9 adding up to a 9:7.

The normal diatonic's taken from 26, 19 and 31-equal.

11 from 31, with 6:5 and 11:9 adding up to 16:11, or 55:66:80

19 from 33 and 23 from 40 I don't know because they're not even 5-limit consistent.

11 from 41 is 9:8 plus 8:7 making 9:7, or a 7:8:9 chord.

Mark says "I believe that the diatonic scales derived from the different Cn [n-note equal temperament]
scales conform to background formations existing independently of the number of tones in the base Cn
scale." So meantone is still meantone whether it's taken from 12, 19, 26 or 31 notes. Apparently Balzano
and Zweifel failed to mention this. He goes as far as to say that 7 from 17-equal is still approximating
6:5 and 5:4.

Then he notes that the three major and four minor thirds don't add up to a 2:1 if they're tuned justly.
From which he concludes "a diatonic scale can be seen not as a reordered segment of a 'cycle of fifths' or
its microtonal equivalents, but a compact structure in a grid space of two alternating intervals." No
mention of imperfect octaves (stretching by half a syntonic comma will give just 6:5 and 5:4 thirds from a
3:2 generator) or periodicity blocks.

Then he gives a 5-limit grid using a Tenney-type notation with a 5-limit diatonic for the nominals.

After that he talks about how the concept could be extended to more dimensions, but doesn't give any
examples.

On interesting thing I noticed is that 72 is allowed by Balzano's n(n+1) criterion. So I thought I'd try
and fit this technique to the Miracle scales.

Stud loco is fulfils all the criteria for a "diatonic" I can work out. You have to generate it from 3 and
4 notes from 72-equal adding up to a secor. This is a strange primary chord, but fulfils the letter of the
specification.

Similarly, Blackjack could be generated form a quomma and secor-quomma, but it's a "pentatonic" and fails
Balzano's "coherence" test because the secor-quomma is more than twice the size of the quomma, except as a
subset of 31-equal.

Canasta is a "pentatonic" based on the chord 8:11:15 where the generator is an octave-secor instead of a
secor.

Decimal has an even number of notes, and so can't be generated by alternating intervals. It could be
thought of as a pair of septimal slendros, each generated from 12:16:21 or 16:21:28. You could also say
it's generated from 14:15:16 where the two "alternating" intervals are really the same. This kludge would
work for all even-numbered MOS.

I don't know if all odd-numbered MOS can be written this way, but I suspect so. All you need to do is fill
in an approximation to the generator, and the absolute number of generators to each internal interval have
to add up to the number of notes you want in the scale. Hopefully this will give the number you want for
either kind of generator within the octave. Magic 19 could be generated from 8:9:10.

If you take generators larger than the octave, you can always avoid Balzano's incoherence. So probably
this is cheating, and we'll have to relax that one to get Blackjack. If the rest of Balzano's criteria
were relaxed, who knows what we might find!

Graham

Graham,

Interestingly, the 4:5:6 major triad which conflates the 81/80 in
12-tone equal temperament, thereby making 9/8 and 10/9 (etc) the same
note:

10/9---5/3---5/4--15/8
\ / \ / \ / \
\ / \ / \ / \
4/3---1/1---3/2---9/8

2-----9-----4----11
\ / \ / \ / \
\ / \ / \ / \
5-----0-----7-----2

is mimicked by the 11-tone scale in 20-tet if you use the Augusto
Novarro-like identities of 13:16:19 and 17:21:25--note that 20-tet is
consistent through:

13:16:19:22:25:28:31:34:37
17:21:25:29:33:37:41:45

While each of these would conflate a slightly different comma, both
would be spelled:

4----15-----6----17-----8----19
\ / \ / \ / \ / \ / \
\ / \ / \ / \ / \ / \
9-----0----11-----2----13-----4

It's certainly interesting, but probably irreconcilably alien to the
(5-limit diatonic) model on which it's ostensibly based.

take care,

--Dan Stearns

----- Original Message -----
From: <graham@microtonal.co.uk>
To: <tuning-math@yahoogroups.com>
Sent: Saturday, March 16, 2002 12:06 PM
Subject: [tuning-math] Gould on Generalized Diatonics

> I decided to get a copy of Mark Gould's paper "Balzano and Zweifel:
Another Look at Generalized Diatonic
> Scales" so here's a summary.
>
> It's starts off explaining the Balzano theory. This is roughly that
you build up an MOS by alternating a
> pair of intervals that add up to the generator. For the normal
diatonic, these are the major and minor
> third. It doesn't actually say "MOS" or "well formed" but that
seems to be what we're talking about.
> There's also the peculiar definition that "diatonic" means the
larger size of scale step is more common and
> "pentatonic" is the other way around. He decides not to use the
scare quotes from then on, but I will
> because I'm old fashioned enough to think that a pentatonic should
have 5 notes to the octave.
>
> The square lattice Mark mentioned on the main list, which looks like
a triangular lattice rotated a bit and
> reflected, is to show this pair of alternating intervals, rather
than 5-limit harmony in general. All
> "diatonics" mentioned in the paper are shown on similar grids.
>
> Mark suggests a different tonic for Zweifel's 11 from 20 scale.
That's so the not that changes as you move
> up the chain of generators is the leading note to the tonic.
>
> Next he gives his own criteria for "diatonics", which are like
Balzano's but without worrying about there
> being n(n+1) notes in the parent scale. Then there are some
examples of such "diatonics".
>
> Now the bit on ratios. He makes the startling statement "For all
diatonic scales, the two intervals in the
> grid approximate to a given pair of just intonation ratios" which is
about as surprising as finding water
> in a river. Beyond identifying traditional diatonic scales with 5:4
and 6:5, he doesn't say what ratios
> were being approximated by the "diatonics" he gave before. Still, I
can reverse engineer these.
>
> There's an 11 from 27 scale which is based on a 7:6 and 10:9 adding
up to a 9:7.
>
> The normal diatonic's taken from 26, 19 and 31-equal.
>
> 11 from 31, with 6:5 and 11:9 adding up to 16:11, or 55:66:80
>
> 19 from 33 and 23 from 40 I don't know because they're not even
5-limit consistent.
>
> 11 from 41 is 9:8 plus 8:7 making 9:7, or a 7:8:9 chord.
>
> Mark says "I believe that the diatonic scales derived from the
different Cn [n-note equal temperament]
> scales conform to background formations existing independently of
the number of tones in the base Cn
> scale." So meantone is still meantone whether it's taken from 12,
19, 26 or 31 notes. Apparently Balzano
> and Zweifel failed to mention this. He goes as far as to say that 7
from 17-equal is still approximating
> 6:5 and 5:4.
>
> Then he notes that the three major and four minor thirds don't add
up to a 2:1 if they're tuned justly.
> From which he concludes "a diatonic scale can be seen not as a
reordered segment of a 'cycle of fifths' or
> its microtonal equivalents, but a compact structure in a grid space
of two alternating intervals." No
> mention of imperfect octaves (stretching by half a syntonic comma
will give just 6:5 and 5:4 thirds from a
> 3:2 generator) or periodicity blocks.
>
> Then he gives a 5-limit grid using a Tenney-type notation with a
5-limit diatonic for the nominals.
>
> After that he talks about how the concept could be extended to more
dimensions, but doesn't give any
> examples.
>
>
> On interesting thing I noticed is that 72 is allowed by Balzano's
n(n+1) criterion. So I thought I'd try
> and fit this technique to the Miracle scales.
>
> Stud loco is fulfils all the criteria for a "diatonic" I can work
out. You have to generate it from 3 and
> 4 notes from 72-equal adding up to a secor. This is a strange
primary chord, but fulfils the letter of the
> specification.
>
> Similarly, Blackjack could be generated form a quomma and
secor-quomma, but it's a "pentatonic" and fails
> Balzano's "coherence" test because the secor-quomma is more than
twice the size of the quomma, except as a
> subset of 31-equal.
>
> Canasta is a "pentatonic" based on the chord 8:11:15 where the
generator is an octave-secor instead of a
> secor.
>
> Decimal has an even number of notes, and so can't be generated by
alternating intervals. It could be
> thought of as a pair of septimal slendros, each generated from
12:16:21 or 16:21:28. You could also say
> it's generated from 14:15:16 where the two "alternating" intervals
are really the same. This kludge would
> work for all even-numbered MOS.
>
> I don't know if all odd-numbered MOS can be written this way, but I
suspect so. All you need to do is fill
> in an approximation to the generator, and the absolute number of
generators to each internal interval have
> to add up to the number of notes you want in the scale. Hopefully
this will give the number you want for
> either kind of generator within the octave. Magic 19 could be
generated from 8:9:10.
>
> If you take generators larger than the octave, you can always avoid
Balzano's incoherence. So probably
> this is cheating, and we'll have to relax that one to get Blackjack.
If the rest of Balzano's criteria
> were relaxed, who knows what we might find!
>
>
> Graham
>
>
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