Balzano's generalized diatonic scales

I just read "The Group-theoretic description of 12-fold and microtonal
pitch systems", by Gerald Balzano. It was interesting. Here's an
inventory of the properties he uses to define generalized-diatonic scales...

(1) propriety; "coherence" of scale degrees across modes.
Search list archives for "Rothenberg".

(2) tuning coverage; rank order matrix and interval matrix are the same.
I don't think this has much to do with anything. It does mean that the
scale's intervals will be tuned proportionally to their rank, and one can
speculate if this adds anything to their perception...

(3) linear connectivity; transposing by generator changes one note.
But not less than one --- closed chains are not allowed. Makes for
symmetry at the generator. The interval of equivalence must be 2/1.

(4) ...this is the hard one. All of a sudden, the generators of the linear
representation must be fifths, and the scale must contain exactly two kinds
of thirds, which sum to the generator (it is allowed that the two thirds be
the same --- spliting the generator into two equal parts). This is the
most arbitrary of his properties.

-C.

I wrote...

>(3) linear connectivity; transposing by generator changes one note.
> But not less than one --- closed chains are not allowed.

The reason for this is to prevent the (Rothenberg) efficiency from becoming
very low (see TD 262.14).

-C.

>(4) ...this is the hard one. All of a sudden, the generators of the linear
>representation must be fifths

Carl, I don't know what you mean. Balzano doesn't say this. In fact, the
generator for 72-tET in Balzano-world would be a flat minor third.

>(it is allowed that the two thirds be
>the same --- spliting the generator into two equal parts)

That could never happen in Balzano-world.

I find Balzano's paper to be beautiful for the very mathematical structures
you seem to have misunderstood (those not covered in your (1) - (3)).
However, I feel strongly that these structures have little to do with music.

[Paul H. Erlich:]
>the generator for 72-tET in Balzano-world would be a flat minor
third.

I guess that these scales would also be contained in the group of (mod
n) symmetric scales that I recently posted on (where x*y=n and y=x+1)?
If so, I'm curious as to what form these scales are given in - i.e.,
is the Balzano 9-out-of-20 +2+2+3+2+2+2+3+2+2 (0, 2, 4, 7, 9, 11, 13,
16, 18, 20 )?

Dan

Dan --

I'm not sure I understand your x and y terminolgy, but Balzano's "major
mode" -- which he does not cling to very strongly -- is
(0,2,5,7,9,11,14,16,18).

[Paul H. Erlich:]
> I'm not sure I understand your x and y terminolgy,

Just any two intervals of n (or I guess in a practical sense, any
whole number >0 & <n).

>but Balzano's "major mode" -- which he does not cling to very
strongly -- is (0,2,5,7,9,11,14,16,18).

Thanks. Does he give any others, or were the 17-out-of-72 type
examples (x=8 & y=9, where x*y=n and y=x+1, in the ad hock terminology
that I used in the mod n symmetric scale post) you mentioned
extrapolations of the basic premises behind this 9-out-of-20?

Dan

>>(4) ...this is the hard one. All of a sudden, the generators of the linear
>>representation must be fifths
>
>Carl, I don't know what you mean. Balzano doesn't say this. In fact, the
>generator for 72-tET in Balzano-world would be a flat minor third.

I meant scale-step fifths. He says that the scale must be covered with
triads, and that they must fall on every other note of the scale. And, he
says that the generator in the linear representation must be the outside
note of the triad. So, he's saying there must be exactly two kinds of
third and two kinds of fifth, with the restriction that the second kind of
fifth appears only once. Shoot me if this isn't sufficient, with my first
three properties, to get everything he says.

>>(it is allowed that the two thirds be the same --- spliting the generator
>>into two equal parts)
>
>That could never happen in Balzano-world.

I see now that this is true.

-C.

[Carl Lumma:]
> I meant scale-step fifths. He says that the scale must be covered
with triads, and that they must fall on every other note of the scale.
And, he says that the generator in the linear representation must be
the outside note of the triad. So, he's saying there must be exactly
two kinds of third and two kinds of fifth, with the restriction that
the second kind of fifth appears only once.

Hmm... now I'm confused. If fifths are fifths, and triads are triads
so to speak, then wouldn't this only apply to the (3+4, 3*4)
7-out-of-12? It would seem that 'fifths' and 'triads' must be only
referring to some sort of sequential consistency here (i.e., a
9-out-of-20 would then have an 0-5-9 triad and a 9/20, and one 8/20,
fifth etc.)... but (as I haven't actually read anything by Balzano)
there's a pretty good chance that I've got something or the other here
horribly garbled... What I'm mostly trying to understand right now is
to what extent these (m-out-of-n) scales (where x*y=n & y=x+1) are a
subset of the mod n symmetrical scales that I recently posted on - are
Balzano's papers/theories anywhere on the net?

Thanks,
Dan

Dan wrote,

>Thanks. Does he give any others, or were the 17-out-of-72 type
>examples (x=8 & y=9, where x*y=n and y=x+1, in the ad hock terminology
>that I used in the mod n symmetric scale post) you mentioned
>extrapolations of the basic premises behind this 9-out-of-20?

He does give the 11-out-of-30 type, but the extrapolation to 17-out-of-72 is
quite unambiguous.

Carl Lumma wrote,

>I meant scale-step fifths. He says that the scale must be covered with
>triads, and that they must fall on every other note of the scale. And, he
>says that the generator in the linear representation must be the outside
>note of the triad. So, he's saying there must be exactly two kinds of
>third and two kinds of fifth, with the restriction that the second kind of
>fifth appears only once. Shoot me if this isn't sufficient, with my first
>three properties, to get everything he says.

Well, there are scales that satisfy these properties but don't fall into his
schema, since he also requires the product of the two sizes of thirds to
equal the number of notes in the tuning.

Dan -

As you can see I was confused by Carl too, but Carl really just meant thirds
and fifths in the Yasser sense -- the number of scale degrees (plus one)
that the intervals subtend. So no, you didn't get anything garbled.

-Paul

>>I meant scale-step fifths. He says that the scale must be covered with
>>triads, and that they must fall on every other note of the scale. And, he
>>says that the generator in the linear representation must be the outside
>>note of the triad. So, he's saying there must be exactly two kinds of
>>third and two kinds of fifth, with the restriction that the second kind of
>>fifth appears only once. Shoot me if this isn't sufficient, with my first
>>three properties, to get everything he says.
>
>Well, there are scales that satisfy these properties but don't fall into his
>schema, since he also requires the product of the two sizes of thirds to
>equal the number of notes in the tuning.

Yeah, but he requires that because he thinks it is required to get all the
things he says he wants. But it isn't, which is my point.

-C.

>>I meant scale-step fifths. He says that the scale must be covered
>>with triads, and that they must fall on every other note of the scale.
>>And, he says that the generator in the linear representation must be
>>the outside note of the triad. So, he's saying there must be exactly
>>two kinds of third and two kinds of fifth, with the restriction that
>>the second kind of fifth appears only once.
>
>Hmm... now I'm confused.

Sorry Dan, there was some poor wording here:

>>the outside note of the triad

I meant the outside interval.

>If fifths are fifths, and triads are triads so to speak, then wouldn't
this >only apply to the (3+4, 3*4) 7-out-of-12?

Which 7-out-of-12 is that?

>It would seem that 'fifths' and 'triads' must be only referring to some
sort >of sequential consistency here (i.e., a 9-out-of-20 would then have
an 0-5-9 >triad and a 9/20, and one 8/20, fifth etc.)... but (as I haven't
actually >read anything by Balzano) there's a pretty good chance that I've
got >something or the other here horribly garbled...

All Balzano wants is the scale to be covered with three-note chords that
fall on every-other degree of the scale. Which means they'll be made of
thirds and fifths. It is actually a huge mistake to consider them chords,
tho, since Balzano hasn't given any property that defines "chords" (he's
deliberately thrown out the usual one: harmony). Which means that his
whole idea amounts to a lot of nothing.

Well, not quite. He does require that the same interval appears as a fifth
in exactly n-1 modes of the scale, when the scale has n notes per 2:1.
Which is no more and no less than MOS when the generator turns out to be a
fifth and the interval of equivalence a 2:1. So the final list of stuff is
now...

(1) propriety; "coherence" of scale degrees across modes.
Search list archives for "Rothenberg".

(2) tuning coverage; rank order matrix and interval matrix are the same.
I don't think this has much to do with anything. It does mean that the
scale's intervals will be tuned proportionally to their rank, and one can
speculate if this adds anything to their perception...

(3) linear connectivity; transposing by generator changes one note.
But not less than one --- closed chains are not allowed. Makes for
symmetry at the generator, and prevents the (Rothenberg) efficiency from
becoming very low (see TD 262.14).

(4) fifths; the generator is a fifth in exactly n-1 modes of the scale.
Means that scale degrees will be related in an interesting way to the
acoustic intervals of the scale. Choice of fifths is completely arbitrary.

-C.

[Carl Lumma:]
> Which 7-out-of-12 is that?

Just the standard 12e diatonic (in its mirror inversion rotation) as a
sequence of (+3, +4, +3, etc.) thirds... I've also used whole (odd
numbered) groups of these types of symmetrical arrays in an attempt to
increase and decrease a compositional sense of (what I suppose you
could call) structural chromaticism... For example, using the 9/20 &
11/20 of 20e, you could create the following 19, 17, 15, 13, 11, 9, 7,
5, & 3-out-of-20 sets (which also has the mirror inversion rotation of
the Balzano 9-out-of-20)

0 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 20
0 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 20
0 2 3 4 5 6 7 9 11 13 14 15 16 17 18 20
0 2 4 5 6 7 9 11 13 14 15 16 18 20
0 2 4 5 7 9 11 13 15 16 18 20
0 2 4 7 9 11 13 16 18 20
0 2 7 9 11 13 18 20
0 2 9 11 18 20
0 9 11 20

where x=1, 8, 3, 6, 5, 4, 7, 2, & 9, and y=10, 1, 8, 3, 6, 5, 4, 7, &
2 respectively.

Dan

I wrote,

>>Well, there are scales that satisfy these properties but don't fall into
his
>>schema, since he also requires the product of the two sizes of thirds to
>>equal the number of notes in the tuning.

Carl wrote,

>Yeah, but he requires that because he thinks it is required to get all the
>things he says he wants. But it isn't, which is my point.

Balzano is a much better mathematician than you seem to be giving him credit
for. The product property _is_ required to get his three group-theoretic
representations of the pitch set -- melodic, key, and harmonic. Since the
title of his paper is _The Group-theoretic Description of 12-fold and
Microtonal Pitch Systems_, and he gives these representations a huge amount
of emphasis in his paper, there's no way you can exclude them from the list
of "all the things he says he wants".

Carl wrote,

>All Balzano wants is the scale to be covered with three-note chords that
>fall on every-other degree of the scale. Which means they'll be made of
>thirds and fifths. It is actually a huge mistake to consider them chords,
>tho, since Balzano hasn't given any property that defines "chords" (he's
>deliberately thrown out the usual one: harmony).

Totally wrong. Read the paper.

>Which means that his
>whole idea amounts to a lot of nothing.

Well, maybe, but it's an awfully pretty (and dangerously convincing)
nothing.