[tuning-math] unifying theory of interval "importance" (was: Re: First melodic spring results)

> ----- Original Message -----
> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, June 18, 2001 11:01 AM
> Subject: [tuning-math] Re: First melodic spring results
>
>
> --- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> >
> > [I wrote:]
> > > > But then how to explain the sensation of 34-tET? It _does_ have
> > > > narrow fourths (4 cents) and sixths (2 cents apiece).
> >
> > [Paul E:]
> > > Maybe the fourths and minor sixths are "less important" than the
> > > fifths and major thirds.
> >
> > Maybe, but what possible unifying theory could be behind such a
> > thing?
>
> The 4:3 less important than the 3:2 . . . the 8:5 less important than
> the 5:4 . . . higher-number ratios less important than lower-number
> ratios.

Hmmm... I see a more specific pattern. All other absolute
exponent values being equal, the absolute values of the
powers of 2 give an index of "importance".

In vector notation where the numbers are exponents of
2, 3 and 5 respectively:

3:2 = |-1 1 0|
4:3 = | 2 -1 0|

5:4 = |-2 0 1|
8:5 = | 3 0 -1|

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> Hmmm... I see a more specific pattern. All other absolute
> exponent values being equal, the absolute values of the
> powers of 2 give an index of "importance".

I don't think that's a more specific pattern, I think it's a less
specific pattern. By size of the numbers I was thinking of something
like n+d or n*d . . . by comparison, the absolute values of the
powers of 2 would be less specific, since so many intervals would
have the same value (for example, 8:3, 8:5, 8:7, 9:8, 11:8, 13:8,
15:8, 17:8, 24:11, 24:13, 24:17, 24:19, 24:23, 25:24, 29:24, 31:24,
35:24, 37:24, 41:24, 43:24, 47:24, 49:24, 5000:2401, etc.)

> ----- Original Message -----
> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, June 18, 2001 2:07 PM
> Subject: [tuning-math] unifying theory of interval "importance"
>

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > Hmmm... I see a more specific pattern. All other absolute
> > exponent values being equal, the absolute values of the
> > powers of 2 give an index of "importance".
>
> I don't think that's a more specific pattern, I think it's a less
> specific pattern. By size of the numbers I was thinking of something
> like n+d or n*d . . . by comparison, the absolute values of the
> powers of 2 would be less specific, since so many intervals would
> have the same value (for example, 8:3, 8:5, 8:7, 9:8, 11:8, 13:8,
> 15:8, 17:8, 24:11, 24:13, 24:17, 24:19, 24:23, 25:24, 29:24, 31:24,
> 35:24, 37:24, 41:24, 43:24, 47:24, 49:24, 5000:2401, etc.)

OK, got it.

Of course, I meant that powers of 2 showed the "importance"
together with the exponents of all the other primes. Guess I
just automatically think in terms of prime-factorization when
I see problems like this.

-monz
http://www.monz.org
"All roads lead to n^0"

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Hey Monz . . . at one point, you showed me a section in your book
where you equated dissonance with the area of a rectangle formed by
the numerator on one side and the denominator on the other side. This
area, of course, equals n*d. I think Dave, Graham, and I (and
Benedetti, Tenney, Pierre Lamothe, and others) have settled on n*d or
a monotonic function thereof as a measure of "complexity" -- which is
one of the factors determining dissonance, along with "tolerance"
and "span". So it seems there was something to the insight behind
your "rectangle" construction. Care to fill us in on how you came up
with that construction?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> I think Dave, Graham, and I (and
> Benedetti, Tenney, Pierre Lamothe, and others) have settled on n*d
or
> a monotonic function thereof as a measure of "complexity" -- which
is
> one of the factors determining dissonance, along with "tolerance"
> and "span".

What about "rootedness", i.e. having the smaller number be a power
of 2? Is 8:11 more consonant than 7:11?

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> What about "rootedness", i.e. having the smaller number be a power
> of 2? Is 8:11 more consonant than 7:11?

That's a tough call . . . on the other hand, it's pretty hard not to
conclude that 16:1 is more consonant than 15:1 or 17:1 . . . so
clearly octave-equivalence (and thus powers of 2) plays some role.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> >
> > What about "rootedness", i.e. having the smaller number be a power
> > of 2? Is 8:11 more consonant than 7:11?
>
> That's a tough call . . . on the other hand, it's pretty hard not to
> conclude that 16:1 is more consonant than 15:1 or 17:1 . . .

Really? I think I found such wide intervals all to be equally
consonant, or at least non-dissonant, which is the span thing.

> so
> clearly octave-equivalence (and thus powers of 2) plays some role.

Don't some folk find 8:9 to be more consonant than 7:8?

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> Really? I think I found such wide intervals all to be equally
> consonant, or at least non-dissonant, which is the span thing.

But the 16:1 sounds more "resolved" than 17:1 or 15:1, right?
>
> > so
> > clearly octave-equivalence (and thus powers of 2) plays some role.
>
> Don't some folk find 8:9 to be more consonant than 7:8?

Yeah . . . but remember, 200 cents has been a ubiquitous part of
almost everyone's musical experience in the West, and is only 4 cents
from 8:9 but 31 cents from 7:8. Even in musics where harmony is not
used per se, the 8:9 tends to show up between 4/3 and 3/2, since
those pitches are so easy to tune from a 1/1 . . . while 7:8 tends
not to show up at all.

Not saying I don't believe in rootedness, especially for larger
chords . . . although Herman's 3:4:5 version of Pachelbel's canon
somehow makes 3:4:5 sound "OK" -- one does not miss the
4:5:6's "rootedness".