Yahoo Tuning Groups Ultimate Backup tuning Re: Fokker periodicity blocks (Joe Monzo)

>I'm interested in studying the historical conceptions of
>various shapes and sizes of periodicity blocks in music
>all over the world.

I'm with you on that one!

>The most important aspect, it seems to me, is to determine
>the proper lattice metric to portray all the different tunings.

I'm not sure why that would be important. As far as I can tell the most
important aspect is getting reliable information on how musicians actually
tune their instruments. Music theory in other countries is no more (and
probably less) reliable than music theory in the West.

>There is no clear consensus on this yet, altho I recall Pauls
>Erlich and Hahn being the most outspoken in favor of the one(s?)
>they like, and doesn't Carl Lumma agree, at least partially?

First, you have to decide what you want to measure. Psychoacoustic
consonance? If so, the lattice really isn't of much use, since consonance
has little or nothing to do with factoring. However, Paul E. does seem to
consider his metric a lattice metric. He only recognizes factoring by 2's,
so he's got a lattice dimension for every odd factor, to odd-limit
infinity. Because he uses a triangular lattice, every ratio has an
octave-equivalent representation that spans one rung. So his metric is
simply the log of the odd limit of the ratio. Why log, I don't know. The
only advantage of using logs that I ever understood was that it unifies the
results of odd and prime limit factorizations, which shouldn't matter to
Paul E, given his stance on prime numbers in consonance measures. Last I
heard, Paul E. was considering switching to an octave-specific rectangular
lattice; perhaps he can update us on the pro's and con's of such a thing.

Or, you could measure the modulation distance (Paul H, if you can think of
a better way to explain this...), so to speak, of an interval. In this
approach, factoring is important. First, you declare which intervals you
consider consonant, and you give lattice dimensions only to the identities
needed. Then you count the number of rungs (each of which are defined as
consonant) along the shortest route to the target interval. In effect, you
are counting the number of common-tone modulations (by consonances) you
need to get to the interval. Paul Hahn uses this metric in his diameter
and consistency measurements.

-C.

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

>Because he uses a triangular lattice, every ratio has an
>octave-equivalent representation that spans one rung. So his metric is
>simply the log of the odd limit of the ratio. Why log, I don't know.

So that, for example, 9 is the same distance whether considered on its own
or as two steps of 3.

>The
>only advantage of using logs that I ever understood was that it unifies the
>results of odd and prime limit factorizations, which shouldn't matter to
>Paul E, given his stance on prime numbers in consonance measures.

Using logs doesn't in any way unify this with any of the prime-based
approaches such as Euler's, Barlow's, or Wilson's complexity measures. I
disagree with all of those.

>Last I
>heard, Paul E. was considering switching to an octave-specific rectangular
>lattice; perhaps he can update us on the pro's and con's of such a thing.

It's not a matter of switching, it's just that octave-equivalence is a
convenient assumption for many purposes when you don't want to deal with the
complexity of considering all octave-specific versions of intervals, chords,
and tunings. For working with Fokker periodicity blocks, octave-equivalence
is essential. If you're doing something in an octave-specific context,
though, I see nothing wrong with n*d as a complexity measure, therefore
Tenney's rectangular lattice, where the distance between rungs is the log of
the prime number along whose axis you're moving, could work well for
octave-specific complexity (the convenience of only having prime axes is
much appreciated here).

>Or, you could measure the modulation distance (Paul H, if you can think of
>a better way to explain this...), so to speak, of an interval. In this
>approach, factoring is important. First, you declare which intervals you
>consider consonant, and you give lattice dimensions only to the identities
>needed. Then you count the number of rungs (each of which are defined as
>consonant) along the shortest route to the target interval. In effect, you
>are counting the number of common-tone modulations (by consonances) you
>need to get to the interval. Paul Hahn uses this metric in his diameter
>and consistency measurements.

This is what I would use if the musical context operates under a given
overall odd-limit of consonance (as Paul Hahn assumes). I would weight the
consonances by the log of their odd-limits (note the two different meanings
of odd-limit here). When the overall odd-limit goes to infinity, you get the
thing you were talking about at the top of this message.

I should point out that by "factoring" above you don't mean anything like
prime factorization.

The 3-d lattices I've been drawing can be said to conform with this last
option, with 7 as the overall odd limit. Since you are projecting the 3-d
lattice onto 2-d anyway, there's no point in worrying about the relative
lengths of the rungs.

🔗Joe Monzo <monz@xxxx.xxxx>

> [Paul Erlich, TD 329.12]
>
>> Because he uses a triangular lattice, every ratio has an
>> octave-equivalent representation that spans one rung.
>> So his metric is simply the log of the odd limit of the
>> ratio. Why log, I don't know.
>
> So that, for example, 9 is the same distance whether
> considered on its own or as two steps of 3.
>
> <etc. - snip>

Paul,

I've been reading your postings on this thread with much
interest, but since my name is part of the subject line,
I just wanted it to be clear to everyone that it was
Carl Lumma who wrote all the things (in TD 328.10) you
quoted in your posting here.

- monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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🔗Carl Lumma <clumma@nni.com>

>Using logs doesn't in any way unify this with any of the prime-based
>approaches such as Euler's, Barlow's, or Wilson's complexity measures. I
>disagree with all of those.

As do I. However, I was referring simply to this:

>So that, for example, 9 is the same distance whether considered on its own
>or as two steps of 3.

That is, that prime limit folks measuring 9 as two 3's will agree with odd
limit people measuring 9 as 9.

So my question is, why do you care?

>>Or, you could measure the modulation distance (Paul H, if you can think of
>>a better way to explain this...), so to speak, of an interval. In this
>>approach, factoring is important. First, you declare which intervals you
>>consider consonant, and you give lattice dimensions only to the identities
>>needed. Then you count the number of rungs (each of which are defined as
>>consonant) along the shortest route to the target interval. In effect, you
>>are counting the number of common-tone modulations (by consonances) you
>>need to get to the interval. Paul Hahn uses this metric in his diameter
>>and consistency measurements.

>When the overall odd-limit goes to infinity, you get the
>thing you were talking about at the top of this message.

As I posted months ago, and as is appropriate when looking at
psychoacoustic consonance. This is inappropriate for modulation distance,
however, because we wish to know how certain intervals function within a
given system of consonances. As you are so fond of pointing out, sensory
consonance and functional consonance are not the same --- the latter
evolves with music. 225/224 might be a basic consonance in another 2000
years, but we wish to know how it functions in the 9-limit systems of
today. By making the odd limit infinity, we destroy this information.

>I should point out that by "factoring" above you don't mean anything like
>prime factorization.

How not?

-C.

🔗PERLICH@xxxxxxxxxxxxx.xxx

Carl Lumma wrote,

>That is, that prime limit folks measuring 9 as two 3's will agree with odd
>limit people measuring 9 as 9.

>So my question is, why do you care?

Because it's convenient to have only one lattice point for both.

>>>Or, you could measure the modulation distance (Paul H, if you can think of
>>>a better way to explain this...), so to speak, of an interval. In this
>>>approach, factoring is important. First, you declare which intervals you
>>>consider consonant, and you give lattice dimensions only to the identities
>>>needed. Then you count the number of rungs (each of which are defined as
>>>consonant) along the shortest route to the target interval. In effect, you
>>>are counting the number of common-tone modulations (by consonances) you
>>>need to get to the interval. Paul Hahn uses this metric in his diameter
>>>and consistency measurements.

>>I should point out that by "factoring" above you don't mean anything like
>>prime factorization.

>How not?

Using your own example, 225/224 on a 9-limit lattice could be factored as 9/7 *
5 * 5, or 9 * 5/7 * 5, showing it to be three rungs in distance. Paul Hahn went
through this (was it this very example?) when we were refining the algorithm.
On a lattice such as Monzo's, the prime factorization, 3 * 3 * 5 * 5 * 7^(-1),
shows the interval to traverse five rungs.

🔗Carl Lumma <clumma@xxx.xxxx>

>>>I should point out that by "factoring" above you don't mean anything like
>>>prime factorization.
>>
>>How not?
>
>Using your own example, 225/224 on a 9-limit lattice could be factored as
>9/7 * 5 * 5, or 9 * 5/7 * 5, showing it to be three rungs in distance. Paul
>Hahn went through this (was it this very example?) when we were refining the
>algorithm. On a lattice such as Monzo's, the prime factorization, 3 * 3 * 5
>* 5 * 7^(-1), shows the interval to traverse five rungs.

Of course the results can be different on different lattices, why point
this out? The city-block process is _something_ like prime factorization
(and on a rectangular prime-limit lattice, it is everything like it). In
particular, I was explaining that the city-block concept does little for
the measurement of psychoacoustic consonance; despite weak hints from my
own experience, I have come to accept your idea that the ear has no
facility for factoring bare intervals.

-C.

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

🔗Joe Monzo <monz@xxxx.xxxx>

> [Paul Erlich, TD 334.12]
>
> Carl wrote,
>
>> In particular, I was explaining that the city-block concept
>> does little for the measurement of psychoacoustic consonance;
>> despite weak hints from my own experience, I have come to
>> accept your idea that the ear has no facility for factoring
>> bare intervals.
>
> OK, I'm glad we agree.

You'll be glad to know that I pretty much agree with all
of this too, including the 'weak hints'. I don't think the
ear/brain system does much prime-factoring in the case of
hearing dyads, but in understanding more complex chunks of
musical processing, it probably does.

-monz

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Joe Monzo wrote,

>You'll be glad to know that I pretty much agree with all
>of this too, including the 'weak hints'. I don't think the
>ear/brain system does much prime-factoring in the case of
>hearing dyads, but in understanding more complex chunks of
>musical processing, it probably does.

I have a different interpretation for the latter: in more complex chunks of
musical material, the prime-factorization is, at best, spelled out by the
pitch materials themselves. I feel I can successfully defend this
interpretation against any example you could care to point out that purports
to support your theory.