Tuning-Math Digests messages 8703 - 8727

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Message: 8703

Date: Sat, 06 Dec 2003 21:54:14

Subject: Re: Digest Number 863

From: Carl Lumma

>> As for transforming it into the countersubject, can you give me two
>> subjects of the same length that cannot be transformed into one
>> another with serial procedures?  I'll believe you if you say yes.
>
>No, not if you can have arbitrary operations.
//
>> What are the allowed serial procedures?
>
>Well if you're composing, you can do what you want to a row.

Aha!  And a row is just any sequence of notes then, eh, of any
length?

-Carl


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Message: 8712

Date: Sun, 07 Dec 2003 17:09:41

Subject: Re: An 11-limit linear temperament top 100 list

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
...
> The extra commas I suggested were all that was needed in the 7-limit
all had epimericity less than .46. I suggested .5 as a cutoff for
> the 7-limit and .3 for the 11-limit; I boosted this to .35, with a
50 cent cutoff for size. This gave me the following list of 51 commas,
> in order of badness of the corresponding planar temperament:
> 
> [9801/9800, 3025/3024, 3294225/3294172, 151263/151250, 441/440,
385/384, 225/224, 2401/2400, 56/55, 176/175, 4375/4374, 540/539,
64/63, 100/99, 250047/250000, 5632/5625, 36/35, 1375/1372, 126/125,
45/44, 99/98, 43923/43904, 896/891, 81/80, 49/48, 50/49, 121/120,
117649/117612, 55/54, 41503/41472, 1771561/1771470, 77/75, 4000/3993,
6250/6237, 8019/8000, 6144/6125, 1029/1024,5120/5103, 3388/3375,
3136/3125, 32805/32768, 245/242, 243/242, 128/125, 12005/11979,
245/243, 1728/1715, 19712/19683, 625/616, 1331/1323, 2200/2187]
> 
> Wedging these three at a time led to 6135 wedgies. Taking the best
100 of these by geometric badness gave me my list. 
... 

Hi Gene,

I was looking for names for linear temperaments I had found using
Graham's online finder, and I noticed this 11-limit one wasn't in your
list:

Complex aug fourths
generator mapping [[1, ?, ?, ?, ?], [0, -7, -26, -25,  3]]
minimax generators [1200., 585.14]
minimax error 4.1 c

Does this mean there is another 11-limit comma that should be added to
your list above?

I called it "complex" in deference to this one in your list:

> Tritonic
> [5, -11, -12, -3, -29, -33, -22, 3, 31, 33] [[1, 4, -3, -3, 2], [0,
-5, 11, 12, 3]]
> 
> generators [1200., 580.274408364]
> bad 6158.168745 rms 5.154394 comp 70.204409


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Message: 8714

Date: Sun, 07 Dec 2003 11:19:12

Subject: Re: Digest Number 864

From: Carl Lumma

>> Message: 5
>>    Date: Sat, 06 Dec 2003 21:54:14 -0800
>>    From: Carl Lumma <ekin@xxxxx.xxx>
>> Subject: Re: Digest Number 863
>> 
>> >> As for transforming it into the countersubject, can you give me two
>> >> subjects of the same length that cannot be transformed into one
>> >> another with serial procedures?  I'll believe you if you say yes.
>> >
>> >No, not if you can have arbitrary operations.
>> //
>> >> What are the allowed serial procedures?
>> >
>> >Well if you're composing, you can do what you want to a row.
>> 
>> Aha!  And a row is just any sequence of notes then, eh, of any
>> length?
>> 
>> -Carl
>
>I knew it was a trick question...

Nope, I just asked because I'd always heard you had to use all 12
tones before reusing any.

>The fact is, there *is* a consistent serial practice, especially in the
>2nd Viennese School, that doesn't include arbitrary operations on rows.  
>And what would be the point of using a row, if your serial procedures can
>turn it into any other row you happened to feel like writing? Just write
>the notes you feel like, in that case (and afterwards you could even
>pretend there *was* a row, and that everything was derived from it via
>your secret procedures). But theorists' progresively more sophisticated
>modelling of serial procedures isn't some sort of numerological claptrap,
>and the musical structures described really are fundamental to that
>repertoire. I mean, Schoenberg actually believed it when he said he had
>"discovered a method of composing that would ensure the supremacy of 
>German music for the next 100 years".

But in fact Schoenberg was the end of 200 years of German musical
supremacy.  Actually Mahler had already come to America, IIRC.

-Carl


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Message: 8717

Date: Mon, 08 Dec 2003 16:50:21

Subject: (unknown)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> then a set of such 
> classes is a set of pitch classes, and so should be a set-pitch-
>class.

Not in English -- in English, a set of pitch classes would be a pitch-
class-set, or PC set for short.

> This is not simple, this is a crazy-quilt of needless complexity.

The pot calling the kettle black if I ever saw it.


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Message: 8719

Date: Mon, 08 Dec 2003 17:00:27

Subject: Re: Digest Number 862

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:

> This is also an excellent point. One issue I have grappled with is 
> the mathematical versus the musical definition of harmonics. 
> The mathematical 'harmonic series' as I understand it always 
> represents harmonics as 1/n, whereas in music we often talk 
> about harmonics as whole number multiples, or what would be 
> called in math the 'arithmetic series'. What is your take on this?
> 
> Aaron

The reason for this is historical. We say whole number multiples 
today because everyone since Fourier talks about frequency 
measurements. In the old days, it was string length measurements (or 
still today, period or wavelength) where the numbers are *inversely 
proportional* to the frequency numbers. So the harmonic series in the 
old days *was* 1/1, 1/2, 1/3, 1/4, etc, and yet it's the same 
harmonic series that today goes 1, 2, 3, 4 . . .


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Message: 8720

Date: Mon, 08 Dec 2003 17:16:18

Subject: Question for Manuel, Gene, Kees, or whomever . . .

From: Paul Erlich

What is the Kees van Prooijen expressibility-reduced (aka odd-limit 
reduced) 72-tone 11-limit periodicity block? In other words, each 
interval of 72-equal expressed as the simplest (in odd limit) 11-
limit ratio with which it is epimorphic, or whatever the right way of 
saying that is.

George Secor's paper includes a big 72-equal keyboard diagram. It's 
marked with ratios, and I don't like them :)


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Message: 8722

Date: Mon, 08 Dec 2003 19:10:15

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > What is the Kees van Prooijen expressibility-reduced (aka odd-
limit 
> > reduced) 72-tone 11-limit periodicity block? In other words, each 
> > interval of 72-equal expressed as the simplest (in odd limit) 11-
> > limit ratio with which it is epimorphic, or whatever the right 
way 
> of 
> > saying that is.
> 
> I was doing this sort of thing using MT reduction. What is the 
> criterion for van Prooijen reduction?

each ratio is a "ratio of" the smallest possible odd number. see

Definitions of tuning terms: ratio of, (c) 2003 by Joe Monzo *

Searching Small Intervals *


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Message: 8724

Date: Mon, 08 Dec 2003 20:52:56

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > > I was doing this sort of thing using MT reduction. What is the 
> > > criterion for van Prooijen reduction?
> > 
> > each ratio is a "ratio of" the smallest possible odd number. see
> > 
> > Definitions of tuning terms: ratio of, (c) 2003 by Joe Monzo *
> > 
> > Searching Small Intervals *
> 
> Why is this preferable to removing any factors of 2 and taking the 
> product of numerator and denominator?

It's *way* preferable. The latter is based on a false view of octave-
reducing the tenney lattice, at best. Do you think 5:3 and 15:8 
should count as equally 'distant' octave-equivalence classes from 
1:1? What I was asking about is supported by Partch, octave-
equivalent harmonic entropy, and pretty straighforward explanations I 
posted for Maximiliano on the tuning list . .


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