Tuning-Math Archive Section 9: 8700

Message: 8700 - Contents - Hide Contents Date: Sat, 06 Dec 2003 19:11:01 Subject: Re: Digest Number 862 From: Gene Ward Smith --- 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?

I don't see it as a big problem; in math a series is summed. The sequence 1, 1/2, 1/3 ... is the harmonic sequence, or even worse, the harmonic progression, but that isn't likely to cause confusion.

Message: 8701 - Contents - Hide Contents Date: Sat, 06 Dec 2003 19:14:12 Subject: Re: Transitive groups of degree 12 and low order containing a 12-cycle From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"

> I've been thinking about this. If you map the 12 pitches of 12-et > in a 4 X 3 grid thus: > > 0 3 6 9 > 4 7 10 1 > 8 11 2 5 > > Selecting any subset of 12-et (coloring the squares for example) > If you flip the whole thing along a vertical axis in the middle, > you will get the P1<->P7 transform. Flipping along a horizontal > axis in the middle will give the mirror image and P1<->P7 transform. > (rotating the colors 180 degrees gives the mirror image only) > So it makes sense that this is D(4) X S(3). Gene, is this > illustration any good? I'm still kind of figuring out the distinction > between Dihedral and Cyclic groups.

I like it, but for the S(3) part you need to be willing to permute the three rows in any order.

Message: 8703 - Contents - Hide Contents Date: Sat, 06 Dec 2003 21:54:14 Subject: Re: Digest Number 863 From: Carl Lumma

>> >s 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

Message: 8706 - Contents - Hide Contents Date: Sat, 06 Dec 2003 03:52:41 Subject: Re: Digest Number 862 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> What I really wonder about is this - is there any math (number theory > algebra or the like) that underlies BOTH the study of tuning (commas, > generators, vals) and permutations (counting sets, etc)? I would > be excited to see if there are any connections between the two > sides of music theory - or are they mutually exclusive?

I'd read up on groups.

Message: 8707 - Contents - Hide Contents Date: Sat, 06 Dec 2003 04:15:39 Subject: (unknown) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx jon wild <wild@f...> wrote:

> Gene writes: > No, a set class is a class of sets; specifically it's an equivalence class > of pc sets (sets of pcs). Is this really so confusing?

Mathematicians are always using sets of sets of sets, etc, but no one calls them set-set-sets--how would you tell one sort of set of sets of sets of sets from another? The definitions involved here are way, way easier than trying to define schemes or vertex operator algebras or suchlike complicated things, but the nomenclature is a mess. If you tried to define rings in terms of sets of sets, its spectrum in terms of sets of sets of sets of sets, its locally ringed space in terms of sets of sets of sets of sets of sets, and schemes in terms of sets of sets of sets of sets of sets of sets, and called schemes set-set-set-set-set-sets you would drive people even more nuts than they already are trying to understand it all now.

>> {... -20, -8, 4, 16, 30...} isn't a residue mod 12, it is an >> equivalence class mod 12; >

> sorry, I always screw that up. I think of the residue "4" as representing > the equivalence class, I know it's not quite right.

"4" is a class representive and can be and often is used to represent the equivalence class in question.

>> Are you sure music theorists don't know we are talking about the >> dihedral group of degree 12? There's a certain amount of group >> awareness percolating through academia--e.g., Lewin. >

> Lewin and many others certainly know it! Do they carry music theory > journals at your library? Or the three books I mentioned above? Rahn's is > for beginners but is solid, Lewin's has some quite advanced stuff in the > appendices. But the most advanced treatment I know of, which I'm sure > you'd appreciate a lot, is Guerino Mazzola's new book "The Topos of > Music", which recasts everything in category theory.

Years ago I read something by someone trying to apply category theory to music. I felt like taking a gun and shooting him, because the categories he was so pround of were abelian groups--not the *category* of abelian groups, the groups themselves. I hope Mazzola does better. I've looked at Lewin's book, and the man is sound, but he is using groups, not categories, which makes a great deal of sense.

> By the way, I don't know if anyone else here knew Lewin, but you might not > have heard the very sad news that he died in May at the age of 69. He was > a really lovely guy and certainly the giant of music theory in the second > half of the 20th-century.

Sorry to hear that. He made some definitions that music theory was in sore need of.

>> Is the definition of "pitch" an integer associated to standard >> musical notation in such a way that C corresponds to 0, sharp to the >> addition of 1, and flat to the subtraction of 1? If so, at some point >> it ought to be said; if not, whatever else it means ought to be given. >> I presume someone actually has done this? >

> Yes. It's the first chapter in Rahn's book "Basic Atonal Theory". The > chapter is called "The Integer Model of Pitch". Sorry, I thought you knew > about that.

Why would I know a definition in a book I havn't read? Defining things by means of a map from the integers to a system of notation is not the most obvious approach, at least to me. "Integer model of pitch", by the way, doesn't exactly sound like this is what Rahn does.

Message: 8710 - Contents - Hide Contents Date: Sat, 06 Dec 2003 07:45:49 Subject: (unknown) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx jon wild <wild@f...> wrote:

> Well I'm not sure, either, that he "defines" pitches as integers--that > sounds wrong. He doesn't attempt an ontological examination of what > pitches actually *are*, he assumes you know what pitches are, and starts > by *associating* integers to pitches, then labels pitches by the integer > associated to them, choosing middle C arbitrarily as zero.

You know what things are unless you are doing pure math, where you define what they are. This is applied math, but I think that works better if you give clean definitions here also. What happens if I make assumptions about what you mean by "pitch" and they turn out to be different assumptions that yours? I'm still trying to digest the idea that "pitch" is a feature of a score, not of any actual sound.

> And it's not dependent on notation--sing me a song, I'll tell you what the > pcs are, no notation involved.

What if I'm singing along with Paul and he's playing a 22-et guitar?

Message: 8712 - Contents - Hide Contents 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

Message: 8714 - Contents - Hide Contents Date: Sun, 07 Dec 2003 11:19:12 Subject: Re: Digest Number 864 From: Carl Lumma

>> >essage: 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

Message: 8715 - Contents - Hide Contents Date: Sun, 07 Dec 2003 20:35:39 Subject: Re: Digest Number 864 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx jon wild <wild@f...> wrote:

>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".

After which, of course, he moved to the Big Orange.

Message: 8717 - Contents - Hide Contents 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.

Message: 8718 - Contents - Hide Contents Date: Mon, 08 Dec 2003 16:52:50 Subject: (unknown) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

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

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

"Val" contains exactly three letters.

Message: 8719 - Contents - Hide Contents 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:

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 . . .

Message: 8720 - Contents - Hide Contents 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 :)

Message: 8721 - Contents - Hide Contents Date: Mon, 08 Dec 2003 18:27:34 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Gene Ward Smith --- 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?

Message: 8722 - Contents - Hide Contents 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) 200... * [with cont.] (Wayb.) Searching Small Intervals * [with cont.] (Wayb.)

Message: 8723 - Contents - Hide Contents Date: Mon, 08 Dec 2003 20:33:04 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Gene Ward Smith --- 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) 200... * [with cont.] (Wayb.) > > Searching Small Intervals * [with cont.] (Wayb.)

Why is this preferable to removing any factors of 2 and taking the product of numerator and denominator?

Message: 8724 - Contents - Hide Contents 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) 200... * [with cont.] (Wayb.) >> >> Searching Small Intervals * [with cont.] (Wayb.) >

> 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 . .