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Message: 9151 - Contents - Hide Contents Date: Thu, 15 Jan 2004 21:50:17 Subject: Calculating TOP From: Gene Ward Smith I've added a section on how to do this using linear programming.

Message: 9152 - Contents - Hide Contents Date: Thu, 15 Jan 2004 18:49:24 Subject: Re: The Atomischisma Scale From: Carl Lumma>This is the unique Fokker block you get by crossing a schisma with an >atom. I don't know if it is what Kirnberger got, because if it is, it >won't be in the Scala archive because the numers in the quotients are >too large to load (this provides and example of why monzos would >sometimes be useful.) I've presented it below as a Scala file, but it >isn't really. There is very little difference between this and 12- >equal, proving that 12-et is really a form of 5-limit just intonation. > >! atomschis.scl >Atom Schisma Scale >12 >! >156348578434374084375/147573952589676412928 >134217728/119574225 >1307544150375/1099511627776 >18014398509481984/14297995284350625 >10935/8192 >1709671705179880612640625/1208925819614629174706176 >16384/10935 >14297995284350625/9007199254740992 >2199023255552/1307544150375 >119574225/67108864 >295147905179352825856/156348578434374084375 >2 Rock! -C.

Message: 9153 - Contents - Hide Contents Date: Thu, 15 Jan 2004 21:56:31 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> By "unweighted" I probably mean a norm without coefficents for >>> an interval's coordinates. >> >> ? >> The norm on Tenney space... > > || |u2 u3 u5 ... up> || = log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up| > > The 'coefficients on the intervals coordinates' here are log2(2), > log2(3) etc.So 'unweighted', 9 has a length of 2 but 11 has a length of 1 . . . :(>>> This ruins the correspondence with >>> taxicab distance on the odd-limit lattice given by Paul's/Tenney's >>> norm, >>>> Huh? Which odd-limit lattice and which norm? >> It's the same norm on a triangular lattice with a dimension for each > odd number.That's not a desirable norm.> The taxicab distance on this lattice is log(odd-limit).No it isn't -- try 9:5 for example.> It's also the same distance as on the Tenney lattice, except perhaps > for the action of 2s in the latter (I forget the reasoning there).Try building up the reasoning from scratch.>>> as on that >>> lattice intervals do not have unique factorizations and thus a >>> metric based on unit lengths is likely to fail the triangle >>> inequality. >> >> Not following. >> Hmm, maybe I was wrong. I was thinking stuff like ||9|| = ||3|| = 1 > and thus ||3+3|| < ||3|| + ||3|| but that's ok. It seems bad > though, since the 3s are pointed in the same direction.What lattice/metric was this about?

Message: 9154 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:02:43 Subject: Re: TOP on the web From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> On Mon, 12 Jan 2004 18:13:59 -0000, "Gene Ward Smith" <gwsmith@s...> > wrote: >>> I've put up a TOP web page. It needs to have, at least, a discussion >> of equal and linear temperaments and Tenney complexity and badness >> added to it, but it should be valuable as a starter. Here it is: >> >> /root/tentop.htm * [with cont.] (Wayb.) >> I can plug the formulas into a program, and they seem to produce accurate > results, but I have no idea how to generalize them to higher limits. There > seems to be a pattern to them, but it'd take a long time and some guesswork > to figure it out. Could you be more explicit about how this works (in > language a non-mathematician programmer can understand)? > > Also, how would this work for more than one comma?Oh, so you were talking about one comma above? The formula is very easy then; my most recent posting of it is here: Yahoo groups: /tuning/message/51762 * [with cont.]

Message: 9155 - Contents - Hide Contents Date: Thu, 15 Jan 2004 21:57:57 Subject: Re: Two questions for Gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Hmm . . . that severely reduces the appeal of using epimericity to >> define our badness contour. In this case, we should probably use >> something that crosses zero error at some finite complexity. >> A curious remark for a physics major to make. We know the list is > compelte, we just can't prove it, and if there were one more comma we > missed it would be of no concievable musical use anyway, as Dave > would be quick to point out.Then why not define our badness contour to exclude anything of "no conceivable musical use"?

Message: 9156 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:08:41 Subject: Re: The Four Mistyschism Scales From: Paul Erlich Is Kirberger's approximation to 12-equal in Scala? It would be interesting if you actually had to go all the way out to atomic to maximize the number of Scala matches in this big 12-tone exercise of yours. --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> A hoary eccelesiastical joke has it that mysticism should really be > spelled "misty schism", so this is how I am spelling the four Fokker > blocks which come from misty and the schisma. The two most notable > things about these scales is that Scala classifies all four of them > as well-temperments, and that this concludes the classification, > though I'll check the latter claim. Mistyschism2 is the same scale as > Scala's duoden12.scl, about which we read "Almost equal 12-tone > subset of Duodenarium". > > ! mistyschism1.scl > Mistyschism scale 2048/2025 67108864/66430125 > 12 > ! > 524288/492075 > 9/8 > 1215/1024 > 512/405 > 4/3 > 64/45 > 3/2 > 262144/164025 > 2048/1215 > 3645/2048 > 256/135 > 2 > > ! mistyschism2.scl > Mistyschism scale 2048/2025 67108864/66430125 = duoden12.scl > 12 > ! > 135/128 > 9/8 > 1215/1024 > 512/405 > 4/3 > 64/45 > 3/2 > 405/256 > 2048/1215 > 3645/2048 > 256/135 > 2 > > ! mistyschism3.scl > Mistyschism scale 2048/2025 67108864/66430125 > 12 > ! > 135/128 > 9/8 > 1215/1024 > 512/405 > 4/3 > 1476225/1048576 > 3/2 > 405/256 > 2048/1215 > 3645/2048 > 256/135 > 2 > > ! mistyschism4.scl > Mistyschism scale 2048/2025 67108864/66430125 > 12 > ! > 524288/492075 > 9/8 > 1215/1024 > 512/405 > 4/3 > 64/45 > 3/2 > 405/256 > 2048/1215 > 3645/2048 > 256/135 > 2

Message: 9157 - Contents - Hide Contents Date: Thu, 15 Jan 2004 02:53:22 Subject: Re: TOP on the web From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> I can plug the formulas into a program, and they seem to produce accurate > results, but I have no idea how to generalize them to higher limits. There > seems to be a pattern to them, but it'd take a long time and some guesswork > to figure it out. Could you be more explicit about how this works (in > language a non-mathematician programmer can understand)?I can get formulas like this for linear temperaments directly from the wedgie, but I found out today that this approach is out of hand from a practical point of view. It's also out of hand for higher prime limits anyway. The solution to all of this is to add to the page a discussion of how to set the thing up as a linear programing problem, which I plan on doing, including dealing with the special cases which arise when you are in a subgroup space and dual space, one which uses only some of the primes in the prime limit.> (I've noticed there's a new notation for monzos that involves different > symbols like angle brackets and vertical bars, but I'm continuing to use > the old notation with square brackets since I don't know the appropriate > usage of the new notation.)It distinguishes monzos, |...>, fron vals, <...|. It worked for physics.

Message: 9158 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:20:14 Subject: Re: Two questions for Gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Then why not define our badness contour to exclude anything of "no > conceivable musical use"?Fine, but then it would be worth mentioning that you really didn't need to.

Message: 9159 - Contents - Hide Contents Date: Thu, 15 Jan 2004 07:53:37 Subject: The Four Mistyschism Scales From: Gene Ward Smith A hoary eccelesiastical joke has it that mysticism should really be spelled "misty schism", so this is how I am spelling the four Fokker blocks which come from misty and the schisma. The two most notable things about these scales is that Scala classifies all four of them as well-temperments, and that this concludes the classification, though I'll check the latter claim. Mistyschism2 is the same scale as Scala's duoden12.scl, about which we read "Almost equal 12-tone subset of Duodenarium". ! mistyschism1.scl Mistyschism scale 2048/2025 67108864/66430125 12 ! 524288/492075 9/8 1215/1024 512/405 4/3 64/45 3/2 262144/164025 2048/1215 3645/2048 256/135 2 ! mistyschism2.scl Mistyschism scale 2048/2025 67108864/66430125 = duoden12.scl 12 ! 135/128 9/8 1215/1024 512/405 4/3 64/45 3/2 405/256 2048/1215 3645/2048 256/135 2 ! mistyschism3.scl Mistyschism scale 2048/2025 67108864/66430125 12 ! 135/128 9/8 1215/1024 512/405 4/3 1476225/1048576 3/2 405/256 2048/1215 3645/2048 256/135 2 ! mistyschism4.scl Mistyschism scale 2048/2025 67108864/66430125 12 ! 524288/492075 9/8 1215/1024 512/405 4/3 64/45 3/2 405/256 2048/1215 3645/2048 256/135 2

Message: 9160 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:22:57 Subject: Re: Question for Dave Keenan From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> If a timbre has >> >> 2nd partial off by < 10.4 cents >> 3rd partial off by < 16.5 cents >> 4th partial off by < 20.8 cents >> 5th partial off by < 24.1 cents >> 6th partial off by < 26.9 cents >> >> does it 'hold together' as a single pitch, or does it fall apart into >> multiple pitches? >> Or is it experienced as somewhere in between, having a single but > poorly-defined pitch. I don't know. I trust your ears more than mine > on such questions, but I'd like to hear it.OK, I'll put something together when I have a chance . . .> Note that my objection to your claimed 5-limit temperaments with very > large errors was not a disagreement between yours and my perception, > but more of an epistemological problem. I sense you are setting up > something to try to catch me in a similar logical mistake. This is > fun. :-) >>> (I'll try to prepare some examples, playing random scales . . .) >> >> If yes: >> What does "yes" mean here?the sound holds together as a single pitch.>> If I take any inharmonic timbre with one loud partial and some quiet, >> unimportant ones (very many fall into this category), and use a >> tuning system where >> >> 2:1 off by < 10.4 cents >> 3:1 off by < 16.5 cents >> 4:1 off by < 20.8 cents >> 5:1 off by < 24.1 cents >> 6:1 off by < 26.9 cents >> >> and play a piece with full triadic harmony, doesn't it follow that >> the harmony should 'hold together' the way 5-limit triads should? >> I don't know. What has the single loud partial got to do with it? Is > this partial one of those mentioned above?No, it essentially determines the pitch of the timbre.> We know that with quiet sine waves nothing special happens with any > dyad except a unison, and that loud sine waves work like harmonic > timbres presumably due to harmonicsand combinational tones . . .> being generated in the > nonlinearities of the ear-brain system.quiet harmonic timbres don't generate combinational tones, so they won't "work like" loud sine waves.> Don't we?That also ignores virtual pitch. A set of quiet sine waves can evoke a single pitch which does not agree with any combinational tone . . . at certain intervals, the pitch evoked will be least ambiguous, which is certainly 'something special happening' . . . The fact is that, when using inharmonic timbres of the sort I described, Western music seems to retain all it meaning: certain (dissonant) chords resolving to other (consonant) chords, etc., all sounds quite logical. My sense (and the opinion expressed in Parncutt's book, for example) is that *harmony* is in fact very closely related to the virtual pitch phenomenon. We already know, from our listening tests on the harmonic entropy list, that the sensory dissonance of a chord isn't a function of the sensory dissonances of its constituent dyads. Furthermore, you seem to be defining "something special" in a local sense as a function of interval size, but in real music you don't get to evaluate each sonority by detuning various intervals various amounts, which this "specialness" would seem to require for its detection. The question I'm asking is, with what other tonal systems, besides the Western one, is this going to be possible in.

Message: 9161 - Contents - Hide Contents Date: Thu, 15 Jan 2004 11:36:29 Subject: Re: Question for Dave Keenan From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> If a timbre has > > 2nd partial off by < 10.4 cents > 3rd partial off by < 16.5 cents > 4th partial off by < 20.8 cents > 5th partial off by < 24.1 cents > 6th partial off by < 26.9 cents > > does it 'hold together' as a single pitch, or does it fall apart into > multiple pitches?Or is it experienced as somewhere in between, having a single but poorly-defined pitch. I don't know. I trust your ears more than mine on such questions, but I'd like to hear it. Note that my objection to your claimed 5-limit temperaments with very large errors was not a disagreement between yours and my perception, but more of an epistemological problem. I sense you are setting up something to try to catch me in a similar logical mistake. This is fun. :-)> (I'll try to prepare some examples, playing random scales . . .) > > If yes:What does "yes" mean here?> If I take any inharmonic timbre with one loud partial and some quiet, > unimportant ones (very many fall into this category), and use a > tuning system where > > 2:1 off by < 10.4 cents > 3:1 off by < 16.5 cents > 4:1 off by < 20.8 cents > 5:1 off by < 24.1 cents > 6:1 off by < 26.9 cents > > and play a piece with full triadic harmony, doesn't it follow that > the harmony should 'hold together' the way 5-limit triads should?I don't know. What has the single loud partial got to do with it? Is this partial one of those mentioned above? We know that with quiet sine waves nothing special happens with any dyad except a unison, and that loud sine waves work like harmonic timbres presumably due to harmonics being generated in the nonlinearities of the ear-brain system. Don't we?

Message: 9162 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:22:36 Subject: Re: The Four Mistyschism Scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Is Kirberger's approximation to 12-equal in Scala? It would be > interesting if you actually had to go all the way out to atomic to > maximize the number of Scala matches in this big 12-tone exercise of > yours.Is atomic a comma of 12-et?

Message: 9164 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:24:50 Subject: Re: A temperament naming convention? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> It seems to be that if a temperament in one prime limit has the same > TOP tunings of the primes it includes as the tunings in the next > higher prime limit, it ought to get the same name (I'm not suggesting > the converse.) From that point of view, I've already screwed up once, > with the meantone names. > > 5-limit meantone: [1201.698522, 1899.262910, 2790.257558] > > septimal meantone: [1201.698521, 1899.262909, 2790.257556, 3370.548328] > > Septimal meantone and 5-limit meantone both are simply "meantone" > under this naming convention. > > The 11-limit 31&43, what I called "meantone", has a TOP tuning of > [1201.611156, 1899.198965, 2790.351234, 3371.044615, 4145.302457]. > This would not necessarily get the name "meantone". > > The 11-limit 31&50, which I called "meanpop", has a top tuning > [1201.698521, 1899.262909, 2790.257556, 3370.548328, 4150.346670]. > Under the proposed convention, it should be called "meantone". > > Should 31&50, with wedgie [1, 4, 10, -13, 4, 13, -24, 12, -44, -71], > be given the name "meantone" and some other name be found for 31&43, > with wedgie [1, 4, 10, 18, 4, 13, 25, 12, 28, 16]?Sounds somewhat reasonable to me . . .

Message: 9165 - Contents - Hide Contents Date: Thu, 15 Jan 2004 18:41:41 Subject: A temperament naming convention? From: Gene Ward Smith It seems to be that if a temperament in one prime limit has the same TOP tunings of the primes it includes as the tunings in the next higher prime limit, it ought to get the same name (I'm not suggesting the converse.) From that point of view, I've already screwed up once, with the meantone names. 5-limit meantone: [1201.698522, 1899.262910, 2790.257558] septimal meantone: [1201.698521, 1899.262909, 2790.257556, 3370.548328] Septimal meantone and 5-limit meantone both are simply "meantone" under this naming convention. The 11-limit 31&43, what I called "meantone", has a TOP tuning of [1201.611156, 1899.198965, 2790.351234, 3371.044615, 4145.302457]. This would not necessarily get the name "meantone". The 11-limit 31&50, which I called "meanpop", has a top tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328, 4150.346670]. Under the proposed convention, it should be called "meantone". Should 31&50, with wedgie [1, 4, 10, -13, 4, 13, -24, 12, -44, -71], be given the name "meantone" and some other name be found for 31&43, with wedgie [1, 4, 10, 18, 4, 13, 25, 12, 28, 16]?

Message: 9166 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:27:35 Subject: Re: More on the naming convention From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> The 5-limit temperament with Ampersand's comma, 7-limit miracle, and > 11-limit miracle all have identical TOP tunings of the primes they > cover.Wow. I'm assuming you gave this tuning in your message giving a few 7- limit linear TOP tunings.> Should the Ampersand temperament simply be called miracle?If it's all the same pitches, I suppose that makes some sense . . .> One might extend this convention to closest points among reasonable > temperament choices. For instance, the 225/224-planar temperament has > TOP tuning [1200.493660, 1901.172569, 2785.167472, 3370.211784]. The > nearest reasonable TOP point in val space to this seems to be 11- limit > marvel, with TOP tuning [1200.508704, 1901.148724, 2785.132538, > 3370.019002, 4149.558115]. The projection of this point into 7-limit > val space is at a distance of 0.06867 cents from 225/224-planar TOP, > which seems to me to justify my proposal to give the 7 and 11 limit > planar temperaments the same name of "marvel".But they have different pitches, right?

Message: 9167 - Contents - Hide Contents Date: Thu, 15 Jan 2004 19:36:05 Subject: More on the naming convention From: Gene Ward Smith The 5-limit temperament with Ampersand's comma, 7-limit miracle, and 11-limit miracle all have identical TOP tunings of the primes they cover. Should the Ampersand temperament simply be called miracle? One might extend this convention to closest points among reasonable temperament choices. For instance, the 225/224-planar temperament has TOP tuning [1200.493660, 1901.172569, 2785.167472, 3370.211784]. The nearest reasonable TOP point in val space to this seems to be 11-limit marvel, with TOP tuning [1200.508704, 1901.148724, 2785.132538, 3370.019002, 4149.558115]. The projection of this point into 7-limit val space is at a distance of 0.06867 cents from 225/224-planar TOP, which seems to me to justify my proposal to give the 7 and 11 limit planar temperaments the same name of "marvel". A little farther out, at a distance of 0.08519 cents is the {225/224, 14700/14641}-planar temperament, which is both a worse temperament and farther away. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 9168 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:29:03 Subject: Re: Two questions for Gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Then why not define our badness contour to exclude anything of "no >> conceivable musical use"? >> Fine, but then it would be worth mentioning that you really didn't > need to.If there is some other comma out there satisfying the epimericity requirement, then you *did* really need to.

Message: 9169 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:35:23 Subject: Re: More on the naming convention From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> One might extend this convention to closest points among reasonable >> temperament choices. For instance, the 225/224-planar temperament > has>> TOP tuning [1200.493660, 1901.172569, 2785.167472, 3370.211784]. The >> nearest reasonable TOP point in val space to this seems to be 11- > limit>> marvel, with TOP tuning [1200.508704, 1901.148724, 2785.132538, >> 3370.019002, 4149.558115]. The projection of this point into 7- limit >> val space is at a distance of 0.06867 cents from 225/224-planar TOP, >> which seems to me to justify my proposal to give the 7 and 11 limit >> planar temperaments the same name of "marvel". >> But they have different pitches, right?Yes, but that is a situation which often arises for names already in use among planar temperaments. I'm suggesting we systematize things a little better by using TOP and the val space to sort things out for us.

Message: 9170 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:37:36 Subject: Re: Two questions for Gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> If there is some other comma out there satisfying the epimericity > requirement, then you *did* really need to.Yes, but there won't be. We all knew Catalan's conjecture was true even before it was proven. There are tons of number theoretic conjectures about which there is essentially no doubt about their truth but which have not been proven.

Message: 9171 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:31:30 Subject: Re: The Four Mistyschism Scales From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Is Kirberger's approximation to 12-equal in Scala? It would be >> interesting if you actually had to go all the way out to atomic to >> maximize the number of Scala matches in this big 12-tone exercise > of >> yours. >> Is atomic a comma of 12-et?Yes. <12 19 28|161 -84 -12> = 0 (Did I notate this incorrectly?)

Message: 9172 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:45:40 Subject: Re: The Four Mistyschism Scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Yes. <12 19 28|161 -84 -12> = 0 > (Did I notate this incorrectly?)Wow. Is the tuning list ready for this kind of stuff? I could go up to the epimericity of the atom.

Message: 9173 - Contents - Hide Contents Date: Thu, 15 Jan 2004 22:58:12 Subject: Re: The Four Mistyschism Scales From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> Yes. <12 19 28|161 -84 -12> = 0 >> (Did I notate this incorrectly?) >> Wow. Is the tuning list ready for this kind of stuff?Well, this is probably the only way you're going to get Kirnberger's tuning (the one that's a JI approximation of 12-equal: a chain of schisma-flattened fifths), and that's the sort of tuning you might find in Scala.

Message: 9174 - Contents - Hide Contents Date: Thu, 15 Jan 2004 23:34:20 Subject: TOP history From: Gene Ward Smith Paul, could you tell us something about when and how you discovered TOP? I'd like to add some history to my top page.

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