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Message: 7900 Date: Sat, 01 Nov 2003 03:45:23 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> I agree with Monz. There's definitely no need to include the > 2-exponents here.

If you exlude them, you need a way of making it clear they are gone.

Message: 7901 Date: Sat, 01 Nov 2003 22:15:49 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> and as i've already been arguing with the ratios, forget it. > > there's almost always nothing valuable about retaining the > data for prime-factor 2, unless it need be considered for > (to cite two examples i can think of quickly): > > - actual orchestral scoring where the 8ve-register must > be considered, or > > - analyzing ancient Greek and Roman theory, which was > based on 4:3 "perfect-4ths" and always specified 8ves, > and gave different names to notes an 8ve apart.

if you don't retain data for prime-factor 2, how is your software able to handle torsion?

Message: 7902 Date: Sat, 01 Nov 2003 03:47:28 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> There's no need for systematic names to be so unfriendly as to call > 81/80 the [-4, 4, -1]-comma, or even the [4, -1]-comma. The name > "5-comma" can be generated and decoded systematically, as I've

shown. If 81/80 is a 5-comma, it would seem the schisma is also.

Message: 7903 Date: Sat, 01 Nov 2003 22:28:31 Subject: Re: 'neutral' intervals From: Paul Erlich Aaron, before we get into some sort of personality clash, let me just say i was apparently misinterpreting you. It sounded like you were saying that, among all musicians and music listeners, intervals of 350 cents are almost always interpreted as variants of either a minor third or a major third. You didn't say "I hear", you used the passive voice, so I thought you were generalizing about the music's affect on human listeners. I have intimate familiarity with both having the experience you describe, and then no longer having it, once the individuality of the intervals of a different culture have a chance to settle in. Therefore, I simply felt it appropriate to question what looked like an overgeneralization on your part. No offense meant, and I think we can discuss this in a sharing manner with everyone describing their own experience and point of view. Speaking of which, we should probably move this discussion to the tuning list in order to get more viewpoints in.

> Perhaps the Arabic musicians should speak for themselves? I > confess that I have not speken with any Arabic musicians on this > issue. I'm curious to know with whom you have discussed this > issue with.

Some of the finest oud players in the world (as well as middle eastern violinists . . .). If you're anywhere near boston, I'll introduce you.

> There is no need to beg in order to differ. You are free to call > these i intervals whatever you want. I offered my take on this > issue for what it's worth, and apparently it's worth little to you.

On the contrary, every viewpoint shared is very valuable to me. If it weren't for you, these lists would be that much more limited in scope. Sorry if it seemed I pounced on you but I mistook your statement as a pouncing on my experiences.

Message: 7904 Date: Sat, 01 Nov 2003 03:55:44 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> so if they're being described as monzos, just leave out > the first exponent of the vector and the first prime-factor > of the label. > > > ... looks like Gene and i support each other on this method > of description.

I think we need a way of distinguishing 2-free monzos from complete information monzos. I suggest <4, -1> vs [-4, 4, -1] to distinguish the two ways of representing 81/80; the corresponding octave class could be (4, -1). The rule would be [] represents an interval, <> represents an interval in the standard octave 1 <= q < 2, and () represents the octave class whose represetative is given by the corresponding <>.

Message: 7905 Date: Sat, 01 Nov 2003 22:34:27 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> but what i forget to emphasize here again is: even in > these cases where 8ves must be considered, it's easier > to use the monzo including 2's exponent, instead of the > actual ratio.

my "heuristics" (maybe they should be called "intuitions"), allow you to glean essential information about the commas (their complexity or distance in the lattice, and the error that tempering them out is likely to impart) directly from the numbers in the ratios. the former is particularly easy, it's just the number of digits in the numerator and denominator. for another example, you can compare two commas with about the same size numerators and denominators in them and estimate their relative sizes in cents, and errors of tempering, just by looking at the *difference* between numerator and denominator in each. The prime factorization doesn't help here at all.

Message: 7906 Date: Sat, 01 Nov 2003 01:51:11 Subject: Re: Eponyms From: Carl Lumma

>So what would your systematic names for 81/80 and 64/63 look like?

As shown.

>> With the wrong ranges, you wouldn't be able to extract the 3 exponent, >> I assumed.

> >That's true. But I don't understand what point you're making here.

No point. IIRC I was just explaining something I'd said earlier. -Carl

Message: 7907 Date: Sat, 01 Nov 2003 14:40:14 Subject: Re: Eponyms From: Carl Lumma

> the former >is particularly easy, it's just the number of digits in the numerator >and denominator.

It seems to be particularly difficult to pin down... () log(d) [this list] () odd-limit(n/d) [in the tuning dictionary] () # of digits ??? [just now] -Carl

Message: 7908 Date: Sat, 01 Nov 2003 01:55:54 Subject: Re: Eponyms From: Carl Lumma

>Would you care to explain what your objection's are to the proposal,

I think I've done that.

>as opposed to your objections to my online personality?

Actually I was referring to the both of us being anal there. You mention genetic predisposition, and interestingly there's this notion of "tasters" -- that in social animals a small part of the population has a genetic factor that makes them simply must try everything exactly once. If true, I am surely one. -Carl

Message: 7909 Date: Sat, 01 Nov 2003 22:45:23 Subject: Re: ennealimmal From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Am I correct that the first ennealimmal scale with an octave > is simply 9-equal, and the next is this 17-tone one...

should be 18-tone . . . apparently you don't count 0 *or* 1200?

> > 50. > 133.3 > 183.3 > 266.7 > 316.7 > 400. > 450. > 533.3 > 583.3 > 666.7 > 716.7 > 800. > 850. > 933.3 > 983.3 > 1066.7 > 1116.7 > > ...? > > Manuel, is there a convenient way to get MOS-like scales with > non-octave periods in Scala? > > -Carl

Message: 7910 Date: Sat, 01 Nov 2003 09:57:39 Subject: The supertemperament From: Gene Ward Smith For any odd prime p, there is a finite list of superparticular ratios which belong to the p-limit. For some n, the smallest n intervals on this list will uniquely determine a val v, which will be for an equal temperament v(2) for which v is the standard val. This defines a function supertemp(p) from p to the "supertemperament" for p. If I calculated the 19-limit superparticulars correctly, we have the following: supertemp(3) = 2 supertemp(5) = 7 supertemp(7) = 72 supertemp(11) = 72 supertemp(13) = 270 supertemp(17) = 1506 supertemp(19) = 8539 For each p, there will be a range from the first n to the first n superparticulars which give the supertemperament. The ranges up through 19 are as follows: 3: 1 5: 2 7: 3 11: 5-9 13: 9-12 17: 9-13 19: 14-15

Message: 7911 Date: Sat, 01 Nov 2003 14:46:36 Subject: hey Paul From: Carl Lumma I'm interested in these scales...

>> [4, -3, 2, 13, 8, -14] [[1, 2, 2, 3], [0, 4, -3, 2]] >> complexity 14.729697 rms 12.188571 badness 2644.480844 >> generators [1200., -125.4687958]

> >25:24 chroma = -6 - 4 = -10 generators -> 10 note scale >graham complexity = 7 -> 6 tetrads

Not sure of the significance of the - in -125. I realize that might have been Gene.

>> [4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]] >> complexity 10.574200 rms 23.945252 badness 2677.407574 >> generators [600.0000000, 950.9775006]

> >// >

>> [2, 6, 6, -3, -4, 5] [[2, 0, -5, -4], [0, 1, 3, 3]] >> complexity 11.925109 rms 18.863889 badness 2682.600333 >> generators [600.0000000, 1928.512337]

> >25:24 chroma = 6 - 1 = 5 generators -> 10 note scale >graham complexity = 3*2 = 6 -> 8 tetrads

I've never noticed "generators" being expressed as larger than "periods". Why? Can't we just reduce by the periods here, getting 350.9775006 and 600., 128.512337 resp.? Again, sorry if this is more of a question for the poster of the >>'d text (Gene?).

>> [6, -2, -2, 1, 20, -17] [[2, 2, 5, 6], [0, 3, -1, -1]] >> complexity 19.126831 rms 11.798337 badness 4316.252447 >> generators [600.0000000, 231.2978354]

> >25:24 chroma = -2 - 3 = 5 generators -> 10 note scale >graham complexity = 8 -> 4 tetrads

By the way, do these temperaments have names? -Carl

Message: 7912 Date: Sat, 01 Nov 2003 09:58:58 Subject: Re: Eponyms From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >

> > There's no need for systematic names to be so unfriendly as to call > > 81/80 the [-4, 4, -1]-comma, or even the [4, -1]-comma. The name > > "5-comma" can be generated and decoded systematically, as I've

> shown. > > If 81/80 is a 5-comma, it would seem the schisma is also.

Have you actually read any of the several descriptions I've given of the proposed komma naming algorithm and its inverse? Are they all really that unclear? 81/80 is the 5-comma. 32805/32768 is the 5-schisma. And if you want, you can say that they are both 5-kommas. 64/63 is the 7-comma 59049/57344 is the 7-medium-diesis or 7-M-diesis 28/27 is the 7-large-diesis or 7-L-diesis 2048/2035 is the 25-comma 6561/6400 is the 25-small-diesis or 25-S-diesis 128/125 is the 125-small-diesis 250/243 is the 125-medium-diesis 531441/512000 is the 125-large-diesis 5120/5103 is the 5:7-kleisma 3645/3584 is the 5:7-comma

Message: 7913 Date: Sat, 01 Nov 2003 22:47:23 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> > the former > >is particularly easy, it's just the number of digits in the

numerator

> >and denominator.

> > It seems to be particularly difficult to pin down... > > () log(d) [this list] > () odd-limit(n/d) [in the tuning dictionary]

these are virtually identical for any small comma -- the intervals in question here.

> () # of digits ??? [just now]

yes, if the log is to the base 10, it's just the rounded number of digits.

Message: 7914 Date: Sat, 01 Nov 2003 10:00:26 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> Have you actually read any of the several descriptions I've given of > the proposed komma naming algorithm and its inverse? Are they all > really that unclear?

I don't buy kommas.

Message: 7915 Date: Sat, 01 Nov 2003 14:53:05 Subject: Re: ennealimmal From: Carl Lumma

>> Am I correct that the first ennealimmal scale with an octave >> is simply 9-equal, and the next is this 17-tone one...

> >should be 18-tone . . . apparently you don't count 0 *or* 1200?

d'oh.

>> 50. >> 133.3 >> 183.3 >> 266.7 >> 316.7 >> 400. >> 450. >> 533.3 >> 583.3 >> 666.7 >> 716.7 >> 800. >> 850. >> 933.3 >> 983.3 >> 1066.7 >> 1116.7

// 2/1

>> >> ...? >> >> Manuel, is there a convenient way to get MOS-like scales with >> non-octave periods in Scala?

-Carl

Message: 7916 Date: Sat, 01 Nov 2003 10:06:24 Subject: Re: Eponyms From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >So what would your systematic names for 81/80 and 64/63 look like?

> > As shown.

As shown where? If you mean "81/80" and "64/63", I thought we agreed that these don't qualify as names. And if we didn't, then I have to say I find, for example, "5-schisma" to be a serious improvement over "32805/32768" as a name.

Message: 7917 Date: Sat, 01 Nov 2003 14:54:33 Subject: heuristic (was Re: Re: Eponyms) From: Carl Lumma

>> It seems to be particularly difficult to pin down... >> >> () log(d) [this list] >> () odd-limit(n/d) [in the tuning dictionary]

> >these are virtually identical for any small comma -- the intervals in >question here.

Right, but why not pick a form and stick with it.

>> () # of digits ??? [just now]

> >yes, if the log is to the base 10, it's just the rounded number of >digits.

Ah yes, of course. -Carl

Message: 7918 Date: Sat, 01 Nov 2003 02:08:19 Subject: Re: The supertemperament From: Carl Lumma

>For any odd prime p, there is a finite list of superparticular ratios >which belong to the p-limit.

Here's something I can believe but which isn't immediately obvious. Can you prove it?

>For some n, the smallest n intervals on >this list will uniquely determine a val v, which will be for an equal >temperament v(2) for which v is the standard val. This defines a >function supertemp(p) from p to the "supertemperament" for p. If I >calculated the 19-limit superparticulars correctly, we have the following: > >supertemp(3) = 2 >supertemp(5) = 7 >supertemp(7) = 72 >supertemp(11) = 72 >supertemp(13) = 270 >supertemp(17) = 1506 >supertemp(19) = 8539

Cool. Howabout moving a fixed n down the list (or n's which, for each starting point in the list, uniquely define a val)? -Carl

Message: 7919 Date: Sat, 01 Nov 2003 22:57:06 Subject: heuristic (was Re: Re: Eponyms) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> It seems to be particularly difficult to pin down... > >> > >> () log(d) [this list] > >> () odd-limit(n/d) [in the tuning dictionary]

> > > >these are virtually identical for any small comma -- the intervals

in

> >question here.

> > Right, but why not pick a form and stick with it.

that's not my style :) actually, i try to use the second when supplying exact calculations . . .

> >> () # of digits ??? [just now]

> > > >yes, if the log is to the base 10, it's just the rounded number of > >digits.

> > Ah yes, of course.

obviously a different form, but a lot easier to use when you don't have a calculator handy!

Message: 7920 Date: Sat, 01 Nov 2003 02:11:00 Subject: Re: Eponyms From: Carl Lumma

>> >So what would your systematic names for 81/80 and 64/63 look like?

>> >> As shown.

> >As shown where? If you mean "81/80" and "64/63", I thought we agreed >that these don't qualify as names.

We most certainly didn't.

>And if we didn't, then I have to >say I find, for example, "5-schisma" to be a serious improvement over >"32805/32768" as a name.

I didn't say I'd use 32805/32768 as a name. -Carl

Message: 7921 Date: Sat, 01 Nov 2003 10:20:11 Subject: Re: Eponyms From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: >

> > Have you actually read any of the several descriptions I've given of > > the proposed komma naming algorithm and its inverse? Are they all > > really that unclear?

> > I don't buy kommas.

Do you mean you don't like spelling it with a "k" when it's being used as a generic term. That's fine. That's not part of the naming algorithm. That's just me. There are no "kommas" in the automatically generated names. Are you seriously saying you haven't read any of them because of "kommas"?

Message: 7922 Date: Sat, 01 Nov 2003 00:21:38 Subject: Re: Eponyms From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >The 5, 7 and 11 are all on the same side of the ratio, or there would > >have been a colon ":" in there.

> > How do you pronounce that?

Good question. I haven't been pronouncing the colon at all. 385/384 is the first time I've felt any desire to indicate multiplication (as 5.7.11-kleisma). I don't want to pronounce the dot either, so that's a bit of a problem. If we call it 385-kleisma that problem doesn't occur, but I tend to think the systematic name should give the factorisation when it gets bigger than most people can easily factorise mentally. I'm guessing that's around 125, but we could simply declare it to be 385. :-) And then the need for dots would be very rare. Another vague idea: The order of mention of primes could be different depending whether they are being multiplied (dot) or divided (colon).

> >They are all only to the power given, namely 1.

> > How do you do it with higher powers?

5, 25, 125, 5^4, 5^5, ... where the latter are pronounced "five to the four" etc. 7, 49, 343 (or 7^3), 7^4, 7^5, ... 11, 121, 11^3, 11^4, ... I suppose 11^3 should be pronounced "eleven cubed" rather than "eleven to the three". Tanaka's kleisma (_the_ kleisma) has the systematic name of 5^6-kleisma (five-to-the-six-kleisma)

> >It's a kleisma so it's in the range 4.5 c (a bit arbitrary at present) > >to 11.7 c (actually, exactly half a pythagorean comma).

> > 385:383 is in that range.

Yes, but the system says that the only factors omitted from the first part of the name are factors of 2 and 3. 383 contains other primes (in fact _is_ a rather large prime) which would therefore have to be upfront in the name.

> >That's how a dumb algorithm would have to do it, but you or I > >(assuming we knew something about the system) would say: Its got 385 > >as a factor along with some powers of 2 and 3. I know roughly how big > >it is so I wonder if it's 386/385 or 385/384.

> > Oh, I thought you always gave the numerator.

No. To convert a comma ratio to its systematic name: 1. Remove all factors of 2 and 3. 2. Replace slash with colon. 3. Swap the two sides of the ratio if necessary to put the smallest number first. 4. If it now starts with "1:", eliminate the "1:". 5. If any side of the (2,3-reduced) ratio is bigger than 125 (or maybe 385) then give its prime factorisation in some form (details yet to be decided). 6. Calculate the comma size in cents and use it to look up and append the category name, preceded by a hyphen. This is not guaranteed to give a unique name (although clashes will be exceedingly rare). To be certain that your comma actually deserves the name, you have to run the process in reverse (as I've described already) trying 3-exponents in the series 0, 1, -1, 2, -2, 3, -3, ... and octave reducing, until you get a hit on the correct size-category. Then see if you've got your original comma ratio back again.

> >Oh 384 has prime factors of only 2's and 3's.

> > How do we know it's only got 2's and 3's if we're only given > "385-kleisma"?

Because that's the system. Even if you didn't know the system to start with, you should soon notice, when looking at any kind of list of systematic comma names, that there isn't an explicit power of 2 or 3 mentioned anywhere.

Message: 7923 Date: Sat, 01 Nov 2003 02:24:08 Subject: Re: Eponyms From: Carl Lumma

>Are you seriously saying you haven't read any of them >because of "kommas"?

I must admit that the first time this went around, I stopped reading when I saw it. -Karl

Message: 7924 Date: Sat, 01 Nov 2003 00:27:43 Subject: Re: Eponyms From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> > >>5.7.11-kleisma has no advantages over 385/384 that I can see.

> > > > The latter must be factored to see what it's good for, and > > log'ed to give an exact size. The former gives a size range, > > and with the addition of the 3 exponent tells you what it's > > good for (otherwise how'reyou going to say what pythagorean > > commas are good for?). But with the addition of the 3 exponent, > > we loose the ability to draft size ranges. What say you to > > this, Dave?

> > If you want to make this systematic, why not simply monzo-size range?

I agree with this for the _really_ complex commas, but I want a reasonably non-mathematician friendly system where for example the systematic names for 81/80 and 64/63 are 5-comma and 7-comma respectively. You can even pronounce the "7" as "septimal" if you want, and then it's the same as its common name.

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