Yahoo Tuning Groups Ultimate Backup tuning Dave Beardsley's "centric modulation"

[Monzo, TD 99:]
> the way we perceive harmony is usually a combination
> of these two models, the chain of exponents and the
> prime[/odd]-factoring, each one being employed
> to varying degrees for different people and circumstances.
> The chain model tends to have a lateral orientation,
> while the prime model is overwhelmingly centric.

[Beardsley, TD 100:]
> In my work "Sonic Bloom" (CD's ready soon!),
> I play chords that use 13 - 11 - 9 without a 1/1,
> I feel that this is a sort of vertical modulation
> that doesn't sound as abrupt as 12tet modulation.
>
> Maybe there's a JI theorist who as a term for
> this type of modulation - as if there's none on this
> list?

How about "centric modulation"?
Where the old-fashioned kind would be "lateral modulation".

I thought this idea worth examining.

First let's look at the differences in interval structure;
I've arranged them the best I can in ASCII to show
relative interval sizes:

6 \
13 \ \ |
\ | 316c |
289c | / |
/ | 5 |
11 637c 702c
\ | \ |
347c | 386c |
/ | / |
9 / 4 /

Proportions of chords are generally compared by
relating all higher tones to the bottom tone,
and using the same number for this tone in the
two different chords. Is is calculated by finding
the least common multiple.

If we do this first for the individual intervals
we can compare their sizes rationally:

the "major 3rds":
The 5/4 [= 45:36] is 45/44 [= 39 cents] wider
than 11/9 [44:36].

the "minor 3rds":
The 6:5 [= 66:55] is 66/65 wider [= 26 cents]
than 13/11 [= 65:55].

the "5th":
The 3:2 [= 27:18] is 27/26 [= 65 cents] wider
than 13:9 [= 26:18].

So finding the l.c.m. for all six tones,
the 4:5:6 chord can be called 36:45:54,
and 9:11:13 can be called 36:44:52.

If we factor out all powers of 2, we can easily
put this on a prime-factor lattice:

27
/
45 _ /
'-._ /
11 9
| /
| / _ 13
| / _.-'
| ( ) _.-'
| / _.-'
| / _.-'
|/_.-'
(1)

Assuming Dave was using chords that gave an unambiguous
center on 1 to start with, this "centric modulation"
goes up the prime axes 3, 9, and 11, still staying
centered on 1, rather than sliding laterally along
the old 3x5 matrix the way 9:45:27 (the 4:5:6 "major
chord" rooted on 9) would have.

This is exactly one of the new types of harmonic
procedure which I discussed in my posting, and
what Schoenberg was hinting at with his new use
of the 12-Eq system.

Dave does it in JI, and he is introducing the a
similar kind of ambiguity as Schoenberg by employing
the 26-, 39-, and 65-cent xenharmonic bridges described
above to make the 9:11:13 function like a 4:5:6.

To a certain degree, it's difficult to get away from
the centric feeling on the local 1/1 imposed by this
type of harmony. Thus the old story about how it's easy
to recognize Partch's music because it's all in "G".

But an important point contra this, recognized by
both Schoenberg and Partch, is that the higher the
odd/prime factors become, the less strongly they
imply the local 1/1. The ambiguity is introduced
by the very largeness, or complexity, of the numbers.

By imagining very large lattices filled with small-interval
xenharmonic bridges to the higher primes, JI can
be used just as ambiguously as the 12-tone serialists
used that tuning.

- Monzo

BTW Dave, "Sonic Bloom" is cool.
Thanks for the advance copy. :)

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🔗Daniel Wolf <DJWOLF_MATERIAL@xxxxxxxxxx.xxxx>

Message text written by Joe Monzo

>But an important point contra this, recognized by
both Schoenberg and Partch, is that the higher the
odd/prime factors become, the less strongly they
imply the local 1/1. The ambiguity is introduced
by the very largeness, or complexity, of the numbers.
<

I have to disagree with this. For the time being, let's accept the premise
that all of the primary tonalities of the diamond are 'in G' (which is,
itself, a non-traditional definition of a tonality). Ambiguity in the
Partch diamond occurs only with the 9 and /9 tonalites in that they also
form (via "wormholes" periodicities) 3 relationships to the 3 and /3
tonalities.

If we go further and consider the resources of the extended Partchian
gamut, it is far richer in non-G tonalities at lower limits than at higher
limits. The resources involving 7, 9, and 11 identities are increasingly
restricted to the G tonalities and have a net effect of reifying the
centrality of G, not of introducing ambiguity.

On the other hand, the ambiguity with which Schoenberg was dealing is
essentially a property of temperament.

As I have posted several times before, the essential point of contact
between Schoenberg and Partch is the fact that a diamond and a row box are
operationally identical. It is extremely interesting, as a question in the
history of musical style, that both figures would have developed tonal
systems predicated on inversional symmetry and further, that both tonal
styles develop vertically by the addition of identities. Tonal progression
around the diamond could be reduced to a kind of harmonic "planing" on
overtone series whose fundamentals are related as undertone series (and
vice versa) while 12-tone technique progresses by series whose origins are
related by inversion.

One deficit of both systems, from opposite causes, is in the quality of
progressions. The movement from X-tonality to X-tonality in the diamond
would seem to be directed inevitably at the tonic major. Partch responded
by favoring interesting harmonic ostinati (in 'tonal flux' relations), and
used the tonic Otonality with relatively low frequency. Twelve-tone
technique, in the progression from row to row, lacks any directional
priority.

🔗Joseph L Monzo <monz@xxxx.xxxx>

>> [Monzo, TD 102:]
>> But an important point contra this, recognized by
>> both Schoenberg and Partch, is that the higher the
>> odd/prime factors become, the less strongly they
>> imply the local 1/1. The ambiguity is introduced
>> by the very largeness, or complexity, of the numbers.
>
[Wolf, TD 102:]
> I have to disagree with this. <snip> Ambiguity in the
> Partch diamond occurs only with the 9 and /9 tonalites
> in that they also form (via "wormholes" periodicities)
> 3 relationships to the 3 and /3 tonalities.

Your point about Partch's music is well taken.
I should have made it clear that I was referring more
to his theories as expounded in _Genesis_ than in
his actual compositional practice, which was sometimes
a bit different. Partch speaks clearly of this
amgbiguity (of higher-limit resources) in his book.

> If we go further and consider the resources of the
> extended Partchian gamut, it is far richer in non-G
> tonalities at lower limits than at higher
> limits. The resources involving 7, 9, and 11 identities
> are increasingly restricted to the G tonalities and
> have a net effect of reifying the centrality of G,
> not of introducing ambiguity.

Again, I can agree with you here regarding Partch's
compositional practice. Two things that were just
posted here recently reinforce what you're saying:

[Monzo, TD 99:]
> Even along the higher-prime axes, we still tend to
> think in terms of a chain of powers of 3. For example,
> while avoiding all ratios having 5 as a factor, La Monte
> Young, in the lattice of his tunings, still deploys
> chains of powers of 3 among the 7-limit ratios.

[Lumma, TD 102:]
> Wilson has shown that Partch (unconsciously) selected
> his secondary ratios based on a linear series
> (chain of fifths).

It's true that in Partch's system, lower-limit resources
reinforced "lateral modulation" while his higher-limit
resources produced centricity.

My main point was that as the limit increases, the
size of the numbers themselves introduces an element
of ambiguity into the system. While Partch's 7-9-11
resources may have emphasized the 1/1 "G", they
certainly did so with much less strength than the 1-3-5
resources.

Perhaps if his tonal system had been constructed
with more higher-limit resources on other "local 1/1s"
the type of ambiguity I'm discussing would have
been more pronounced in his music.

[Wolf:]
> On the other hand, the ambiguity with which Schoenberg
> was dealing is essentially a property of temperament.

Also agreed. My main reason for bringing Schoenberg
into this was to show why he decided to keep the 12-Eq
resources, when he was aware of more subtle tuning
possibilites. One would think that someone interested
in expanded tonal resources would make use of these
alternative tunings, but he didn't, and it was because
he wanted to preserve and *make use of the ambiguity*
built into the 12-Eq system.

Schoenberg realized how vast the possibilities of rational
implications in music are, and from a practical point
of view, he didn't feel it was necessary *at that point
in musical history* to grapple with the expanded resources.
He made clear his belief that by early in the next century
(the one approaching now), musicians would indeed begin
to use more finely-divided scales than the 12-Eq system.

{Wolf:]
> It is extremely interesting, as a question in the
> history of musical style, that both figures would have
> developed tonal systems predicated on inversional symmetry
> and further, that both tonal styles develop vertically
> by the addition of identities.

I find it very interesting indeed that Partch and
Schoenberg made use of similar formal structures in
their tonal systems while, in a sense, going about
it in exactly opposite ways.

[Wolf:]
> The movement from X-tonality to X-tonality in the diamond
> would seem to be directed inevitably at the tonic major.
> Partch responded by favoring interesting harmonic ostinati
> (in 'tonal flux' relations)

Please - give some examples of this from his music.
I'd be interested in studying it.

- Monzo
http://www.ixpres.com/interval/monzo/homepage.html