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Message: 8000 Date: Thu, 06 Nov 2003 19:57:38 Subject: Re: Eponyms From: Paul Erlich i actually used log(p-q)/log(odd limit), which makes the rankings a little different for the big (in cents) intervals but does not affect the ranking of the small commas. --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
> <gdsecor@y...>
> > wrote: > >
> > > I don't have the resources to do that very readily. Wouldn't
> > putting
> > > them in order of product complexity (discarding any with
factors
> > > above some particular prime limit) accomplish this?
> > > > Not really; however we can simply take everything below a certain > > prime limit and below a limit for what I call "epipermicity",
> > i think you mean "epimericity" >
> > which > > is, for p/q>1 in reduced form, log(p-q)/log(q). It can be shown
> this
> > gives a finite list of commas if the epimermicity limit is less
> than
> > one.
> > i'll try 7-limit some other time, but since i still have my 5-limit > list in matlab's memory, here's the top rankings (for intervals <
600
> cents) by epimericity -- 1/1 shows up as best but actually its > epimericity is 0/0 so is undefined: > > numerator denominator > 1 1 > 16 15 > 6 5 > 81 80 > 4 3 > 9 8 > 10 9 > 5 4 > 25 24 > 27 25 > 128 125 > 32805 32768 > 250 243 > 135 128 > 2048 2025 > 15625 15552 > 256 243 > 648 625 > 32 27 > 3125 3072 > 75 64 > 78732 78125 > 6561 6400 > 20000 19683 > 125 108 > 27 20 > 32 25 > 25 18 > 625 576 > 1600000 1594323 > 144 125 > 393216 390625 > 256 225 > 16875 16384 > 2187 2048 > 81 64 > 2109375 2097152 > 800 729 > 6561 6250 > 1125 1024 > 3125 2916 > 100 81 > 531441 524288 > 45 32 > 20480 19683 > 2187 2000 > 729 640 > 2048 1875 > 243 200 > 16384 15625 > 125 96 > 729 625 > 1076168025 1073741824 > 3456 3125 > 6115295232 6103515625 > 1224440064 1220703125 > 1594323 1562500 > 1990656 1953125 > 274877906944 274658203125 > 262144 253125 > 625 512 > 10485760000 10460353203 > 1215 1024 > 62500 59049 > 7629394531250 7625597484987 > 78125 73728 > 273375 262144 > 4096 3645 > 67108864 66430125 > 129140163 128000000 > 2500 2187 > 162 125 > 1638400 1594323 > 531441 512000 > 4194304 4100625 > 82944 78125 > 32768 30375 > 390625000 387420489 > 244140625 241864704 > 390625 373248 > 9765625 9565938 > 1953125 1889568 > 4294967296 4271484375 > 18225 16384 > 625 486 > 34171875 33554432 > 2560 2187 > 768 625 > 140625 131072 > 31381059609 31250000000 > 9375 8192 > etc. > > considering that i had numerators and denominators well in excess
of
> 10^50 in the list, i'm inclined to believe gene that a given > epimericity cutoff will yield a finite list. and it's a good list > too -- i'm kind of pleased with this as a temperament ranking, with > meantone very near the top, augmented, schismic, pelogic, > diaschismic, blackwood, kleismic, and diminished forming a > consecutive block of interesting and eminently useful systems
(given
> their characteristic DE scales), while more unlikely choices for > human music making, like semisuper, parakleismic, and ennealimmal,
as
> well as many simpler systems with high error, fall further down -- > and of course monstrosities like atomic don't appear at all. i
wonder
> if even dave could stomach such a ranking -- the very simple > temperaments with high error are easy enough to mentally toss out
for
> the user seeking a certain goodness of approximation . . .
Message: 8001 Date: Thu, 06 Nov 2003 20:05:43 Subject: Re: Eponyms From: Paul Erlich there were actually a log of ties in the rankings -- for example log (1)=0 so all superparticulars got a zero score. here's a list again with the scores shown in the third column: 1 1 -Inf 16 15 0 6 5 0 81 80 0 4 3 0 9 8 0 10 9 0 5 4 0 25 24 0 27 25 0.210309918 128 125 0.227535398 32805 32768 0.3472592 250 243 0.35424875 135 128 0.396697481 2048 2025 0.41184295 15625 15552 0.444302056 256 243 0.466943504 648 625 0.487048023 32 27 0.488324507 3125 3072 0.493376213 75 64 0.555391285 78732 78125 0.568834688 6561 6400 0.578161697 20000 19683 0.582442033 125 108 0.586791476 27 20 0.590414583 32 25 0.604530978 25 18 0.604530978 625 576 0.604530978 1600000 1594323 0.605251545 144 125 0.6098276 393216 390625 0.610445976 256 225 0.634033151 16875 16384 0.636604282 2187 2048 0.641650248 81 64 0.644725481 2109375 2097152 0.646280585 800 729 0.646676406 6561 6250 0.653073083 1125 1024 0.65690632 3125 2916 0.663875781 100 81 0.670035965 531441 524288 0.673219541 45 32 0.673805299 20480 19683 0.675686222 2187 2000 0.680222895 729 640 0.680955483 2048 1875 0.683790172 243 200 0.684718377 16384 15625 0.686782398 125 96 0.697406178 729 625 0.704584463 1076168025 1073741824 0.706932191 3456 3125 0.721011768 6115295232 6103515625 0.72260722 1224440064 1220703125 0.723318814 1594323 1562500 0.725946912 1990656 1953125 0.727163637 2.74878E+11 2.74658E+11 0.729258578 262144 253125 0.731984665 625 512 0.7343228 10485760000 10460353203 0.739050418 1215 1024 0.739496502 62500 59049 0.741519042 7.62939E+12 7.6256E+12 0.743614519 78125 73728 0.744596936 273375 262144 0.745006093 4096 3645 0.745199871 67108864 66430125 0.745516574 129140163 128000000 0.746753932 2500 2187 0.747202793 162 125 0.747863148 1638400 1594323 0.748755323 531441 512000 0.749061613 4194304 4100625 0.75181536 82944 78125 0.752731448 32768 30375 0.753804891 390625000 387420489 0.757524855 244140625 241864704 0.757919646 390625 373248 0.758254069 9765625 9565938 0.75830862 1953125 1889568 0.763530363 4294967296 4271484375 0.765348818 18225 16384 0.766324469 625 486 0.76649026 34171875 33554432 0.768629073 2560 2187 0.770007566 768 625 0.770897186 140625 131072 0.773133533 31381059609 31250000000 0.773337708 9375 8192 0.773667414 --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> i actually used log(p-q)/log(odd limit), which makes the rankings a > little different for the big (in cents) intervals but does not
affect
> the ranking of the small commas. > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
> > <gdsecor@y...>
> > > wrote: > > >
> > > > I don't have the resources to do that very readily. Wouldn't
> > > putting
> > > > them in order of product complexity (discarding any with
> factors
> > > > above some particular prime limit) accomplish this?
> > > > > > Not really; however we can simply take everything below a
certain
> > > prime limit and below a limit for what I call "epipermicity",
> > > > i think you mean "epimericity" > >
> > > which > > > is, for p/q>1 in reduced form, log(p-q)/log(q). It can be shown
> > this
> > > gives a finite list of commas if the epimermicity limit is less
> > than
> > > one.
> > > > i'll try 7-limit some other time, but since i still have my 5-
limit
> > list in matlab's memory, here's the top rankings (for intervals <
> 600
> > cents) by epimericity -- 1/1 shows up as best but actually its > > epimericity is 0/0 so is undefined: > > > > numerator denominator > > 1 1 > > 16 15 > > 6 5 > > 81 80 > > 4 3 > > 9 8 > > 10 9 > > 5 4 > > 25 24 > > 27 25 > > 128 125 > > 32805 32768 > > 250 243 > > 135 128 > > 2048 2025 > > 15625 15552 > > 256 243 > > 648 625 > > 32 27 > > 3125 3072 > > 75 64 > > 78732 78125 > > 6561 6400 > > 20000 19683 > > 125 108 > > 27 20 > > 32 25 > > 25 18 > > 625 576 > > 1600000 1594323 > > 144 125 > > 393216 390625 > > 256 225 > > 16875 16384 > > 2187 2048 > > 81 64 > > 2109375 2097152 > > 800 729 > > 6561 6250 > > 1125 1024 > > 3125 2916 > > 100 81 > > 531441 524288 > > 45 32 > > 20480 19683 > > 2187 2000 > > 729 640 > > 2048 1875 > > 243 200 > > 16384 15625 > > 125 96 > > 729 625 > > 1076168025 1073741824 > > 3456 3125 > > 6115295232 6103515625 > > 1224440064 1220703125 > > 1594323 1562500 > > 1990656 1953125 > > 274877906944 274658203125 > > 262144 253125 > > 625 512 > > 10485760000 10460353203 > > 1215 1024 > > 62500 59049 > > 7629394531250 7625597484987 > > 78125 73728 > > 273375 262144 > > 4096 3645 > > 67108864 66430125 > > 129140163 128000000 > > 2500 2187 > > 162 125 > > 1638400 1594323 > > 531441 512000 > > 4194304 4100625 > > 82944 78125 > > 32768 30375 > > 390625000 387420489 > > 244140625 241864704 > > 390625 373248 > > 9765625 9565938 > > 1953125 1889568 > > 4294967296 4271484375 > > 18225 16384 > > 625 486 > > 34171875 33554432 > > 2560 2187 > > 768 625 > > 140625 131072 > > 31381059609 31250000000 > > 9375 8192 > > etc. > > > > considering that i had numerators and denominators well in excess
> of
> > 10^50 in the list, i'm inclined to believe gene that a given > > epimericity cutoff will yield a finite list. and it's a good list > > too -- i'm kind of pleased with this as a temperament ranking,
with
> > meantone very near the top, augmented, schismic, pelogic, > > diaschismic, blackwood, kleismic, and diminished forming a > > consecutive block of interesting and eminently useful systems
> (given
> > their characteristic DE scales), while more unlikely choices for > > human music making, like semisuper, parakleismic, and
ennealimmal,
> as
> > well as many simpler systems with high error, fall further down --
> > and of course monstrosities like atomic don't appear at all. i
> wonder
> > if even dave could stomach such a ranking -- the very simple > > temperaments with high error are easy enough to mentally toss out
> for
> > the user seeking a certain goodness of approximation . . .
Message: 8002 Date: Thu, 06 Nov 2003 19:18:59 Subject: Re: Naming Commas From: Carl Lumma
> relative to a chain of fifths.
Sacre bleu.
>> Will it? I recently argued no; Paul seemed to argue yes.
> >I expect some overlap certainly, but also large regions that are of >interest to one and not the other. Anything with an absolute >3-exponent greater than 12 (or at most 18) is not going to be of much >interest for notational purposes. Are any of these of great interest >as vanishing in a useful linear temperament?
What I need from you/Paul is a general principle of good chromatic vectors that differs from the general principle of good commatic vectors we already have (namely "epimericity", etc.). -Carl
Message: 8003 Date: Fri, 07 Nov 2003 22:38:56 Subject: Re: Eponyms From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >
> > i.e., 3 is obviously extremely important both historically > > and theoretically, and thus deserves to be isolated by itself > > (or grouped with 2, if 2 is included).
All of the new symbols in a JI heptatonic notation will be modifying nominals in a pythagorean sequence (if we include sharped/flatted tones under a broader concept of the term nominals), so this is a natural place to make a separation.
> > the next comma appears after 11, and earlier in this thread > > i discussed the idea that 11-limit can be a kind of boundary. > > Partch thought so too. (but please, don't anyone make too > > much of this comment.)
Partch thought that 7 was implied in a 12-tone octave in that 5:7 or 7:10 suggest consonant tritones, so 11 is the first prime that is completely foreign to conventional harmony. Since ratios of 13 are similar in effect to ratios of 11 (and since 11 harmonically bridges to 13 rather easily), this makes a good case for stopping with 11. Also, those who advocate achieving new harmonic resources by extending the meantone temperament until it forms a closed system of 31 tones find that they have ended up with an 11-limit (tempered) system. So I think that there are many that would agree with 11 as a boundary.
> > the next comma appears after 19, which i myself used as > > a limit from about 1988-98.
Mapping a harmonic series consistently into a 12-tone octave (without skipping over any primes to reach other primes, and without skipping over any odd harmonics to reach other odd harmonics) yields a 19- limit set that's a favorite of mine: 16:17:18:19:20:21:22:24:25:26:28:30:32 So I also like 19 as a boundary.
> > the next comma appears after 31, which is the highest limit > > Ben Johnston has used in his music. > > > > interesting.
This complete 5 octaves of harmonics, and there's a big gap between 31 and 37, the next prime.
> so the primes are arranged as > 2 3 , 5 7 11 , 13 17 19 , 23 29 31 , 37 41 43 , 47 53 59 , 61 67 71 > > looks like the next comma after 31 makes sense too -- isn't 43 the > highest limit used by george secor at least in some context?
If you're referring to something that I posted some time ago, I think that was 41. (But I did happen to come across a use for 43 when I was subsequently rummaging through some of my old papers.) Trouble with 43 is that 32:43 readily invites confusion with 3:4. --George
Message: 8004 Date: Fri, 07 Nov 2003 22:46:41 Subject: Re: Eponyms From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> no reply, george? > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
gdsecor@y...>
> > wrote: > >
> > > BTW, have you ever tried collapsing an 11-limit lattice into 2 > > > dimensions by mapping 11/8 to <10, 5>?
> > > > you're probably referring to the 3-5-11 lattice? > > > > the full 11-limit lattice is at least 3 dimensional if you use > > one "xenharmonic bridge" as above. for the 3-5-11 case, this
choice
> > (184528125:184549376) is probably a very good one. for the full
11-
> > limit case, 9800:9801 is probably better for most purposes, since > > it's both a little smaller (in cents) and much simpler (i.e.,
shorter
> > in the lattice).
Sorry, I started something that sudden time constraints didn't permit me to finish. Also, in my haste I also skipped over 7 -- 7/4 would also be mapped to the [7, -5] position. --George
Message: 8005 Date: Fri, 07 Nov 2003 22:47:52 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
>>
> considering that i had numerators and denominators well in excess
of
> 10^50 in the list, i'm inclined to believe gene that a given > epimericity cutoff will yield a finite list.
I've shown that it does, for cutoffs less than 1, using Baker's theorem. i wonder
> if even dave could stomach such a ranking -- the very simple > temperaments with high error are easy enough to mentally toss out
for
> the user seeking a certain goodness of approximation . . .
Tossing out powers such as 6561/6400 is what I'd recommend also, though Dave might not go for it.
Message: 8006 Date: Fri, 07 Nov 2003 22:57:17 Subject: Re: Linear Temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" > <paul.hjelmstad@u...> wrote:
> > A few questions - > > > > I know how to derive generators-to-primes using commas in
matrices.
> > How is it done using values? > > > > Second question - how do you go the other way? That is, derive
> commas
> > from generators.
> > you need the mapping; then it's straightforward.
I derive commas from the wedgie in my software.
Message: 8007 Date: Fri, 07 Nov 2003 23:07:51 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> So I think that there are many that would agree with 11 as a
boundary. I've recently done some pieces with full 13-limit chords as harmony; the results may convince people that stopping earlier is a good plan. So far as complete otonal and utonal chords go, however, 5-limit has a natural 3-et triadic nature, 7-limit 4-et tetradic, 9-limit 5-et quintadic, and 13-limit 7-et septadic. For 11-limit complete harmony, we are stuck with a 6 val, which is a little ungainly.
Message: 8008 Date: Fri, 07 Nov 2003 23:10:38 Subject: Re: Eponyms From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
<gdsecor@y...> wrote:
>
> > I don't have the resources to do that very readily. Wouldn't
putting
> > them in order of product complexity (discarding any with factors > > above some particular prime limit) accomplish this?
> > Not really; however we can simply take everything below a certain > prime limit and below a limit for what I call "epipermicity", which > is, for p/q>1 in reduced form, log(p-q)/log(q). It can be shown
this
> gives a finite list of commas if the epimermicity limit is less
than
> one. >
> > One would only > > need to determine at what point two commas were forced to share
the
> > same name and then decide if they were both "important".
> > NEVER! Two commas with the same name makes no sense at all.
I wasn't suggesting giving them the same name, as I said in msg. #7420: << As for the answer to your second question: I don't know if there will be unique names for *all* of them (probably not), but I believe that there are enough names to cover all of those that would be of use to most theorists. Those with very specialized purposes (such as yourself) could devise a modification to Dave's naming system (such as appending letters a, b, c, etc. in order of ratio complexity) to distinguish equivocal names -- I think you're creative enough to come up with something that would be meaningful (or perhaps Dave might have some ideas). >> I really haven't had very much time lately to participate in this sort of discussion, but when it seemed that Dave had dropped out I saw that there were things that hadn't been resolved, I felt that I had to jump in and say something. If you want a couple of specific examples to pursue this further, I can mention a couple of instances from which we were trying to set the kleisma-comma boundary: 152:153 (~11.4c) is definitely a 17:19-kleisma, but 1114112:1121931 (~12.1c) will either be either a (subordinate) 17:19-kleisma or the 17:19-comma, depending on where the boundary is set. 135:136 (~12.8c) will either be the 5:17-kleisma or the 5:17-comma, but 327680:334611 (~36.2c) also claims the name 5:17-comma with our present comma/S-diesis boundary. These are issues that we still need to work out. --George
Message: 8009 Date: Fri, 07 Nov 2003 00:30:14 Subject: Re: Eponyms From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: ...
> > considering that i had numerators and denominators well in excess of > 10^50 in the list, i'm inclined to believe gene that a given > epimericity cutoff will yield a finite list. and it's a good list > too -- i'm kind of pleased with this as a temperament ranking, with > meantone very near the top, augmented, schismic, pelogic, > diaschismic, blackwood, kleismic, and diminished forming a > consecutive block of interesting and eminently useful systems (given > their characteristic DE scales), while more unlikely choices for > human music making, like semisuper, parakleismic, and ennealimmal, as > well as many simpler systems with high error, fall further down -- > and of course monstrosities like atomic don't appear at all. i wonder > if even dave could stomach such a ranking -- the very simple > temperaments with high error are easy enough to mentally toss out for > the user seeking a certain goodness of approximation . . .
It's not bad. It seems to sufficiently penalise excessive complexity, but as you observe, it does not sufficiently penalise excessive error.
Message: 8010 Date: Fri, 07 Nov 2003 01:25:03 Subject: Naming Commas From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > hi George, > > > > --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
> <gdsecor@y...> wrote:
> >
> > > Loocs like you kaught something from Dave Ceenan. ;-)
> > > > i usually try to follow suggestions for standardization > > ... unless i very strongly disagree, as i did with Sims > > 72edo notation.
Thanks Monz, but I actually never proposed it as any kind of standard. I first just did it as an expedient in one particular post, where I thought there was potential for confusion, and I explained the usage at the start of that post. Yahoo groups: /tuning-math/message/6875 * [with cont.] But apparently a few people failed to interpret this sentence correctly, and instead assumed I was inventing a new term, and they didn't like it so they didn't read any further. But I didn't learn this until much later, so I carried on using it. It seemed to me to be a convenient way to keep the two meanings distinct, at least in writing. Had I known that this was actually a _barrier_ to understanding the proposed naming system, I would have stopped using it sooner.
> It's very nice that you brought this up, because that's the next > thing I was going to suggest. Why not modify my suggestion above by > dropping the commas entirely, then changing the semicolons that > remain back to commas, so that the above example (133:72) would be > done this way: > [-3 -2, 0 1 0, 0 0 1] or > [-2, 0 1 0, 0 0 1]. > This makes the grouping by threes more obvious (and the higher primes > much easier to locate), and angle brackets would no longer be > necessary.
I like this very much. It is also MATLAB/Octave compatible. Since commas are optional, but semicolons indicate the end of a row (and so would make a matrix, not a vector). Isn't that right Paul? The angle-bracket thing was OK too. But I was going to suggest that the octave-specific vectors should use the angle-brackets, since I think octave-equivalent ones using square brackets have been around a lot longer. But with this scheme they can all use square brackets. And as pointed out by Monz, it will be easy to remember that the commas come after the exponents of 3, 11, 19 and 31 since these have indeed seemed to be natural stopping (resting?) places. So a Pythagorean comma(generic sense) could be given in octave-specific form as [x y] or [x y,] without ambiguity, but in octave equivalent form it would have to be [y,] although I don't recall ever having seen a vextor with only octaves in it, so if you saw [y] you'd be pretty sure it was meant to be [y,]. Then there's the possibility of 2-and-3-free monzos being used to name very complex commas in George's and my system. These would have to _start_ with a comma. For example, the atom of Kirnberger is also the [,12]-schismina. Although I don't know how you pronounce that so it's clear you're giving a prime exponent, and not a factor. "five-to-the-twelve-schismina" works just as well for me in this case. And indeed, how should we pronounce the commas(punctuation sense) so they don't get confused with commas(generic sense) or commas(specific size range sense)? I'm guessing these won't be very comma-n in spoken conversation between musicians, so we can ignore this problem. But I think we've comma long way.
Message: 8011 Date: Fri, 07 Nov 2003 01:31:25 Subject: Naming Commas From: Dave Keenan George made a good point about who "the rest of us" might be. I then realised that mathematical types searching for temperaments want names for the commas(generic sense) that vanish, while musicians using those temperaments will need names for the commas(generic sense) that _don't_ vanish. Why would they need a name for something that isn't there? This dichotomy will also lead to quite different rankings of "the most important commas(generic sense)".
Message: 8013 Date: Fri, 07 Nov 2003 02:14:16 Subject: Re: Naming Commas From: Carl Lumma
>I then realised that mathematical types searching for temperaments >want names for the commas(generic sense) that vanish, while musicians >using those temperaments will need names for the commas(generic
sense)
>that _don't_ vanish. Why would they need a name for something that >isn't there?
The names can be the same.
>This dichotomy will also lead to quite different rankings of "the
most
>important commas(generic sense)".
Will it? I recently argued no; Paul seemed to argue yes. -Carl
Message: 8014 Date: Fri, 07 Nov 2003 02:38:08 Subject: Re: Naming Commas From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:
> >I then realised that mathematical types searching for temperaments > >want names for the commas(generic sense) that vanish, while musicians > >using those temperaments will need names for the commas(generic
> sense)
> >that _don't_ vanish. Why would they need a name for something that > >isn't there?
> > The names can be the same.
Certainly. But these two different purposes may lead to diffferent ideas about what would constitute a good name. George and I think that a good name will indicate the simplest ratios that can be notated with that comma, relative to a chain of fifths.
>
> >This dichotomy will also lead to quite different rankings of "the
> most
> >important commas(generic sense)".
> > Will it? I recently argued no; Paul seemed to argue yes.
I expect some overlap certainly, but also large regions that are of interest to one and not the other. Anything with an absolute 3-exponent greater than 12 (or at most 18) is not going to be of much interest for notational purposes. Are any of these of great interest as vanishing in a useful linear temperament?
Message: 8015 Date: Fri, 07 Nov 2003 03:00:29 Subject: Re: Eponyms From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> > wrote:
> > One would only > > need to determine at what point two commas were forced to share the > > same name and then decide if they were both "important".
> > NEVER! Two commas with the same name makes no sense at all.
Gene, perhaps before you assume you are the last remaining defender of good sense, and leap, with battle-cry, down the throat of the imagined enemy, you might check to see if they really meant what you assumed they meant. In the very next sentence George wrote:
> > Dave already discussed this: > > > > Yahoo groups: /tuning-math/messages/7320 * [with cont.]
In that, you will find: "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." So only the comma with the lowest absolute 3-exponent gets the simple systematic name. I think what George intended was, if only one is "important" then it gets the name and the other little piggy has none. But if both are important we have to figure a way to give them different names. I suggest simply adding the adjective "complex" to the start of the one with the second lowest 3-exponent, then if you need to go beyond that, which seems very unliklely, then "hypercomplex" or some such. For example, we have [12 19] as the Pythagorean-comma 23.5 c and so [41 65] might be called the complex-Pythagorean-comma 19.8 c.
Message: 8016 Date: Fri, 07 Nov 2003 03:51:16 Subject: Re: Naming Commas From: Dave Keenan Oops! I serously screwed up the monzos for those commas(specific size-range sense). I should have written: For example, we have [-19 12] as the Pythagorean-comma 23.5 c and so [65 -41] might be called the complex-Pythagorean-comma 19.8 c.
Message: 8017 Date: Fri, 07 Nov 2003 03:58:08 Subject: Re: Naming Commas From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> What I need from you/Paul is a general principle of good chromatic > vectors that differs from the general principle of good commatic > vectors we already have (namely "epimericity", etc.).
I'll leave that to Paul, since that is slightly different again from good commas for a general-purpose notation system.
Message: 8018 Date: Fri, 07 Nov 2003 05:23:51 Subject: Re: Naming Commas From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> > [-3 -2, 0 1 0, 0 0 1] or > > [-2, 0 1 0, 0 0 1]. > > This makes the grouping by threes more obvious (and the higher
primes
> > much easier to locate), and angle brackets would no longer be > > necessary.
> > I like this very much. It is also MATLAB/Octave compatible. Since > commas are optional, but semicolons indicate the end of a row (and
so
> would make a matrix, not a vector). Isn't that right Paul?
i checked and you're right!
Message: 8019 Date: Fri, 07 Nov 2003 05:25:09 Subject: Re: Naming Commas From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> George made a good point about who "the rest of us" might be. > > I then realised that mathematical types searching for temperaments > want names for the commas(generic sense) that vanish, while
musicians
> using those temperaments will need names for the commas(generic
sense)
> that _don't_ vanish. Why would they need a name for something that > isn't there?
because it rules harmonic scale construction.
Message: 8020 Date: Fri, 07 Nov 2003 05:26:48 Subject: Re: Linear Temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:
> A few questions - > > I know how to derive generators-to-primes using commas in matrices. > How is it done using values? > > Second question - how do you go the other way? That is, derive
commas
> from generators.
you need the mapping; then it's straightforward.
> Are Linear Temperaments always based on commas?
sure, it always can be, as long as it has a mapping.
> Any information, even partial is appreciated.
the mapping represents the primes in terms of the generators.
Message: 8021 Date: Fri, 07 Nov 2003 05:27:58 Subject: Re: Naming Commas From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:
> > >I then realised that mathematical types searching for
temperaments
> > >want names for the commas(generic sense) that vanish, while
musicians
> > >using those temperaments will need names for the commas(generic
> > sense)
> > >that _don't_ vanish. Why would they need a name for something
that
> > >isn't there?
> > > > The names can be the same.
> > Certainly. But these two different purposes may lead to diffferent > ideas about what would constitute a good name. George and I think
that
> a good name will indicate the simplest ratios that can be notated
with
> that comma, relative to a chain of fifths. >
> >
> > >This dichotomy will also lead to quite different rankings
of "the
> > most
> > >important commas(generic sense)".
> > > > Will it? I recently argued no; Paul seemed to argue yes.
> > I expect some overlap certainly, but also large regions that are of > interest to one and not the other. Anything with an absolute > 3-exponent greater than 12 (or at most 18) is not going to be of
much
> interest for notational purposes. Are any of these of great interest > as vanishing in a useful linear temperament?
depends what you mean by useful, but i'd say no.
Message: 8022 Date: Fri, 07 Nov 2003 00:47:24 Subject: Re: Naming Commas From: Carl Lumma
>What I need from you/Paul is a general principle of good chromatic >vectors that differs from the general principle of good commatic >vectors we already have (namely "epimericity", etc.).
I could see just 1/complexity, or somehow weakening the size term, as size doesn't seem important beyond preserving propriety. -Carl
Message: 8023
Date: Fri, 07 Nov 2003 10:15:07
Subject: Comma size categories extended
From: Dave Keenan
I'm sorry, but I can't stop playing with the idea of systematic comma
names. But please note that these are intended to be _in_addition_to_
(not replacing) common or historical names in the case of well known
commas - just like in the naming of chemical compounds or biological
organisms.
Here's a proposal to extend the systematic comma size categories out
to 115 cents. And again I would like to acknowledge Joe Monzo's
pioneering work in this area.
The following tables will be much easier to read in email. If you're
reading on Yahoo's web interface, you can "Forward" it to yourself.
Size category name Boundary Alternative name
----------------------------------------------------------
0 c
schismina
[-84 53]/2 ~= 1.81 c
schisma
[317 -200]/2 ~= 4.50 c
kleisma
[-19 12]/2 ~= 11.73 c
comma
[-57 36]/2 ~= 35.19 c
minor-diesis or small-diesis
[8 -5]/2 ~= 45.11 c
diesis or medium-diesis
[-11 7]/2 ~= 56.84 c
major-diesis or large-diesis
[-30 19]/2 ~= 68.57 c
chromatic-semitone or small-semitone
[35 -22]/2 ~= 78.49 c
limma or medium-semitone
[-3 2]/2 ~= 101.96 c
diatonic-semitone or large-semitone
[62 -39]/2 ~= 111.88 c
apotome
[-106 67]/2 ~= 115.49 c
These will give many systematic names that are the same as the
historical or common names used in Scala (with the substitution of
"Pythagorean" for 3, "classic" for 5, "septimal" for 7, etc.)
They also have the useful property that each category has an exact
apotome-complement category, except for schisma and kleisma which must
be combined to give a complement to the diatonic-semitone category.
But that's all right because the distinction between schisma and
kleisma isn't necessary for making names unique, but only for matching
historical usage.
Category Complementary category
---------------------------------------
schismina apotome
schisma/kleisma diatonic-semitone
comma limma
minor-diesis chromatic-semitone
diesis major-diesis
In case anyone has just joined the discussion: The purpose of these
precise boundaries is to make it possible to uniquely and
unambiguously name many commas without having to refer to the power of
3 (or 2) contained in the comma, since this is effectively encoded in
the size category.
George,
Please remind me why we didn't use the terms minor-diesis, diesis,
major-diesis?
Message: 8024 Date: Sat, 08 Nov 2003 10:50:05 Subject: Re: Eponyms From: Carl Lumma
>To give a simple example, in the 5-limit, (5+1)/2 = 3, and we may >start from the 3-chord [5/4, 3/2, 2]. If we solve for a val [a, b, c] >such that 5/4, or [-2, 0, 1] is mapped to 1, 3/2 is mapped to 2, and >2 is mapped to 3 we get the equations a5 - 2 a2 = 1, a3 - a2 = 2, and >a2 = 3, the solution of which is a2 = 3, a3 = 5, and a5 = 7, so the >val in question is uniquely determined to be [3, 5, 7], the standard >3-val for the 5-limit.
What's the definition of "standard val"? -Carl
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