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Message: 7775 Date: Tue, 28 Oct 2003 16:39:58 Subject: UVs for 46-ET 11-limit PB From: monz i came up with the following pseudo-periodicity-block for 46-ET, by making a 4-dimensional bingo-card lattice and selecting notes by eye. i call it a "pseudo" periodicity-block because it still includes notes at the edges of the block which are duplicates or triplicates. here it is, as a list of [3,5,7,11]-monzos, with the associated 46-ET mapping: (if viewing on the Yahoo web interface, please forward a copy to your email account to see the proper formatting) 3 5 7 11 46-ET single notes: [ 0 3 0 0] 45 [ 0 2 0 0] 30 [-2 1 0 0] 7 [-1 1 0 0] 34 [ 0 1 0 0] 15 [ 1 1 0 0] 42 [-2 0 0 0] 38 [-1 0 0 0] 19 [ 0 0 0 0] 0 [ 1 0 0 0] 27 [ 2 0 0 0] 8 [-1 -1 0 0] 4 [ 0 -1 0 0] 31 [ 1 -1 0 0] 12 [ 2 -1 0 0] 39 [ 0 -2 0 0] 16 [ 0 -3 0 0] 1 [ 0 0 1 0] 37 [ 1 0 1 0] 18 [-1 -1 1 0] 41 [ 0 -1 1 0] 22 [ 0 1 -1 0] 24 [ 1 1 -1 0] 5 [-1 0 -1 0] 28 [ 0 0 -1 0] 9 [ 0 0 0 1] 21 [ 1 0 0 1] 2 [-1 0 0 -1] 44 [ 0 0 0 -1] 25 all of the following pairs or triples of notes are the same number of steps away from the origin: the following 5 pairs are separated by a diaschisma: [-3 1 0 0] 26 [ 1 3 0 0] 26 [-3 0 0 0] 11 [ 1 2 0 0] 11 [-2 -1 0 0] 23 [ 2 1 0 0] 23 [-1 -2 0 0] 35 [ 3 0 0 0] 35 [-1 -3 0 0] 20 [ 3 -1 0 0] 20 these are the other pairs: [ 0 4 0 0] 14 [-2 -1 1 0] 14 [-1 2 0 0] 3 [ 1 -1 1 0] 3 [ 0 -4 0 0] 32 [ 2 1 -1 0] 32 [ 1 -2 0 0] 43 [-1 1 -1 0] 43 [-2 0 1 0] 29 [-1 -1 0 -1] 29 [-1 0 1 0] 10 [ 0 -1 0 -1] 10 [ 1 0 -1 0] 36 [ 0 1 0 1] 36 [ 2 0 -1 0] 17 [ 1 1 0 1] 17 the following are the triples: [ 0 1 1 0] 6 [ 0 -1 0 1] 6 [ 1 0 0 -1] 6 [ 1 1 1 0] 33 [ 1 -1 0 1] 33 [ 2 0 0 -1] 33 [-1 -1 -1 0] 13 [-2 0 0 1] 13 [-1 1 0 -1] 13 [ 0 -1 -1 0] 40 [-1 0 0 1] 40 [ 0 1 0 -1] 40 ================= i tried to derive a 46-note periodicity-block which is a subset of this list, using our software, with the following unison-vectors: [ 3 -6 1 0] [-4 -2 0 0] (diaschisma) [ 2 -3 1 0] (small septimal comma) [ 3 -4 0 -1] and the software gave me a nice 46-note periodicity-block, but it was entirely in the [3,5]-plane. as can be seen from the first list of monzos at the top of this post (the single notes), the block i want has at least 8 notes with 7 as a prime-factor, and at least 4 notes with 11 as a factor ... and there could be more, depending on how the duplicates and triplicates are filtered out. so my question is: what unison-vectors do i need, to produce the 46-tone periodicity-block in which every note is part of a subset of monzos i listed above? -monz

Message: 7776 Date: Tue, 28 Oct 2003 11:07:41 Subject: Re: comma search From: Carl Lumma

>> What I don't get is why upping the prime limit from 5 to 19 would >> make it any harder.

> >i did it this way: > >Searching Small Intervals * [with cont.] (Wayb.)

This doesn't make a method clear to me. Could you go over it, and/or post your code? How long did it take you to do n=10^50? I've been running n=5000 for the last hour, and I'm still waiting. Here's what I do... For each d <= n, I run n up to 600 cents. That's well less than n(n+1)/2 ratios -- shouldn't be a problem for n=5000. I check if the ratio's in lowest-terms; if not, I throw it out. I then compute its prime-limit; if > p (runtime param, 5 in this case), I throw it out. I then compute the badness, using this handy-dandy formula for pi(x)... ;; Minac (unpublished, proof in Ribenboim 1995, p. 181). ;; pi(x)= (sum j=2...x) (floor ((j-1)! + 1)/j - (floor (j-1)!/j)). I believe all these tests to be quite cheap. Anywho, I then place the ratio in a running list of the best r (runtime param, in this case 10) ratios. I really don't know what other optimizations to make. Anyway, here are the results for n=1000, p=5, r=10, which takes about a minute... (badness primelimit ratio) ((0.24002696783739128 5 81/80) (0.25993019270997947 3 9/8) (0.2938674846231569 3 256/243) (0.3662040962227033 3 4/3) (0.42083442285788536 5 25/24) (0.4804530139182014 5 5/4) (0.4889023927793147 5 16/15) (0.5180580787960469 5 6/5) (0.5364217603611476 5 10/9) (0.5595027250997308 5 128/125)) ...they look roughly consistent with your results. p=7 takes no longer, but returns the same top-10 results as above. In res. 10-20 we do see some 7-limit ratios... (0.6103401603711721 3 32/27) (0.7075223009389495 7 225/224) (0.828892926073675 5 27/25) (0.8692019265111186 5 250/243) (0.9004848978857011 7 126/125) (0.9587113625980545 7 7/6) (1.0526168298843461 7 8/7) (1.1047033190174127 3 81/64) (1.1288769832608407 7 64/63) (1.2029906627249671 7 50/49)) -Carl

Message: 7777 Date: Tue, 28 Oct 2003 23:27:14 Subject: Re: Linear temperament names? From: Carl Lumma

>> >For musical types octave-equivalence is a given.

>> >> It is?

> >OK. Let me rephrase that. > >For most musical types of people who are not mathematicians, >octave-equivalence is a given.

I don't think that's true. Charles Carpenter, Brian McLaren, Jeff Scott, Wendy Carlos and many more have used non-octave scales. All these people have technical backgrounds, but the general interest on the tuning list has always seemed quite high to me. Whether they make sense on guitar is another matter, which I'm not qualified to address. Anyway, just a caveat.

>> >I was asking for help with the names. That's all. You have >> >chosen not to explain your proposed names, or point me to >> >earlier explanations, instead going on with this ridiculous >> >rant.

>> >> Didn't he say he uses names because they have already been used?

> >Wot, do you mean that if I make up a name for something, give no >explanation, and use it three times in my own posts (but no one else >does), that's enough to make it established terminology and everyone >else ought to just use it from then on? And I should just ignore any >requests for justification? :-)

That, I can't say. I just took his word that he got the names from elsewhere. There's a unidecimal (or whatever) comma, IIRC, from Wilson or earlier. As for justification, I don't think names need any. If you find something with no name, it's your right to name it whatever you like if you're so inclined, with the possible exception of naming it after yourself. I know you're one for rigorous ontology. I, on the other hand, am one for absurd, funny names. Yet another approach, and one of the more interesting approaches to naming I've come across, is that of Denny Genovese. He tends to use very obvious, descriptive multi-word names with no fancy shortening. Such as, "binary flute", "southeast just intonation center", etc. I've always been very fond of Partch's names for instruments. According to Mark Vonnegut, hippies had an unwritten rule that everything must have a name. But apparently the emphasis wasn't on naming classes of things, but on naming particular instances, often with a tendency toward anthromorphization (such as naming one's car "Charles"). Now back to your regularly-scheduled flame war. -Carl

Message: 7778 Date: Tue, 28 Oct 2003 19:07:37 Subject: Re: UVs for 46-ET 11-limit PB From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> i tried to derive a 46-note periodicity-block > which is a subset of this list, using our software, > with the following unison-vectors: > > [ 3 -6 1 0] > [-4 -2 0 0] (diaschisma) > [ 2 -3 1 0] (small septimal comma) > [ 3 -4 0 -1]

First Dave, and now you. Commas and any other interval, as opposed to an octave class, should ALWAYS be given with the correct value for 2. Anything else simply will not do. If I was referee for Dave's article, it would bounce like a rubber ball until he fixed that problem.

Message: 7779 Date: Tue, 28 Oct 2003 11:08:21 Subject: Re: heuristic and straightness From: Carl Lumma

>>> |n-d|log(d)^e / d

() Is e here supposed to be based on the prime limit of the particular ratio, or the prime limit of the group of ratios? () log-flat is supposed to give us an equal number of results in all size ranges? I tend to think this allows too much complexity in the small ratios. -Carl

Message: 7780 Date: Tue, 28 Oct 2003 19:14:20 Subject: Re: heuristic and straightness From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> What I don't get is why upping the prime limit from 5 to 19 would > >> make it any harder. The way I'd do it, is for each d < 10^50,

run

> >> n until n/d > 600 cents, kicking out any ratios where n*d has a > >> factor greater than 19. The factoring algorithm I'm using walks > >> up from 2, so aborting it after 19 or 5 wouldn't make much > >> difference.

> > > >ok, so why don't you do it? (seriously -- my factoring algorithm > >refuses numbers higher than 2^32). see if you can reproduce my 5- > >limit results first.

> > Ok, maybe later tonight/this morning. But how'd you do 10^50 if > you can't factor above 2^32?

i started with the factors!

Message: 7781 Date: Tue, 28 Oct 2003 19:26:17 Subject: Re: comma search From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> What I don't get is why upping the prime limit from 5 to 19 would > >> make it any harder.

> > > >i did it this way: > > > >Searching Small Intervals * [with cont.] (Wayb.)

> > This doesn't make a method clear to me. Could you go over it, > and/or post your code?

i did it with command-line instructions. first create your 2-d array of 5-limit ratios (the 5-limit lattice) and then calculate the numerator and denominator of each ratio . . .

> How long did it take you to do n=10^50?

the computer time was negligible.

> I then > compute its prime-limit; if > p (runtime param, 5 in this case), > I throw it out. I then compute the badness, using this handy-dandy > formula for pi(x)... > > ;; Minac (unpublished, proof in Ribenboim 1995, p. 181). > ;; pi(x)= (sum j=2...x) (floor ((j-1)! + 1)/j - (floor (j-1)!/j)).

you're misunderstanding something. the highest prime in the ratio is not necessarily the prime limit in which it will be used. witness the pythagorean limma and pythagorean comma in the 5-limit case, for example.

Message: 7782 Date: Tue, 28 Oct 2003 19:26:43 Subject: Re: UVs for 46-ET 11-limit PB From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >

> > i tried to derive a 46-note periodicity-block > > which is a subset of this list, using our software, > > with the following unison-vectors: > > > > [ 3 -6 1 0] > > [-4 -2 0 0] (diaschisma) > > [ 2 -3 1 0] (small septimal comma) > > [ 3 -4 0 -1]

> > First Dave, and now you. Commas and any other interval, as opposed

to

> an octave class, should ALWAYS be given with the correct value for

2.

> Anything else simply will not do. If I was referee for Dave's > article, it would bounce like a rubber ball until he fixed that > problem.

good think you weren't a referree for fokker's original papers.

Message: 7783 Date: Tue, 28 Oct 2003 19:27:55 Subject: Re: heuristic and straightness From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >>> |n-d|log(d)^e / d

> > () Is e here supposed to be based on the prime limit of the > particular ratio, or the prime limit of the group of ratios?

the prime limit of the lattice you're tempering.

> () log-flat is supposed to give us an equal number of results > in all size ranges?

when they're defined logarithmically, yes.

> I tend to think this allows too much > complexity in the small ratios.

so you want to penalize complexity even more? go for it.

Message: 7784 Date: Tue, 28 Oct 2003 11:35:28 Subject: Re: UVs for 46-ET 11-limit PB From: Carl Lumma

>> First Dave, and now you. Commas and any other interval, as opposed >> to an octave class, should ALWAYS be given with the correct value for >> 2. Anything else simply will not do. If I was referee for Dave's >> article, it would bounce like a rubber ball until he fixed that >> problem.

> >good think you weren't a referree for fokker's original papers.

Why? Sounds like Fokker could have used the feedback too. -Carl

Message: 7785 Date: Tue, 28 Oct 2003 19:50:06 Subject: Re: UVs for 46-ET 11-limit PB From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> good think you weren't a referree for fokker's original papers.

Why? Referees are always insisting on one change or another, generally for the better.

Message: 7786 Date: Tue, 28 Oct 2003 19:52:33 Subject: Re: UVs for 46-ET 11-limit PB From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> i tried to derive a 46-note periodicity-block > which is a subset of this list,

of course, *any* 46-note subset which has each of the 46-equal degrees exactly once is already a periodicity block . . . (by the way, i don't agree with your reckoning of "equally close" . . .)

> using our software, > with the following unison-vectors: > > [ 3 -6 1 0] > [-4 -2 0 0] (diaschisma) > [ 2 -3 1 0] (small septimal comma) > [ 3 -4 0 -1] > > > and the software gave me a nice 46-note > periodicity-block, but it was entirely > in the [3,5]-plane.

the determinant of this matrix is 14, so i'm not sure how you're getting a 46-note periodicity block out of it!

Message: 7787 Date: Tue, 28 Oct 2003 12:04:19 Subject: Re: heuristic and straightness From: Carl Lumma

>> I then >> compute its prime-limit; if > p (runtime param, 5 in this case), >> I throw it out. I then compute the badness, using this handy-dandy >> formula for pi(x)... >> >> ;; Minac (unpublished, proof in Ribenboim 1995, p. 181). >> ;; pi(x)= (sum j=2...x) (floor ((j-1)! + 1)/j - (floor (j-1)!/j)).

> >you're misunderstanding something. the highest prime in the ratio is >not necessarily the prime limit in which it will be used. witness the >pythagorean limma and pythagorean comma in the 5-limit case, for >example.

//

>> >>> |n-d|log(d)^e / d

>> >> () Is e here supposed to be based on the prime limit of the >> particular ratio, or the prime limit of the group of ratios?

> >the prime limit of the lattice you're tempering.

Aha! Well then, I certainly don't need to compute pi(x) more than once. There's no apparent speedup, but at least these should be correct... (primelimit=5 max-d=1000, res=10) (badness primelimit ratio) ((0.24002696783739128 5 81/80) (0.4023163202708607 3 4/3) (0.42083442285788536 5 25/24) (0.4804530139182014 5 5/4) (0.4889023927793147 5 16/15) (0.5180580787960469 5 6/5) (0.5364217603611476 5 10/9) (0.5405096406579765 3 9/8) (0.5595027250997308 5 128/125) (0.828892926073675 5 27/25)) (primelimit=7 max-d=1000, res=20) (badness primelimit ratio) ((0.4419896533813025 3 4/3) (0.6660493039778589 5 5/4) (0.7075223009389495 7 225/224) (0.8337823128571303 5 6/5) (0.9004848978857011 7 126/125) (0.9587113625980545 7 7/6) (1.0518045661034596 5 81/80) (1.0526168298843461 7 8/7) (1.1239582004626365 3 9/8) (1.1288769832608407 7 64/63) (1.1786390756834733 5 10/9) (1.2029906627249671 7 50/49) (1.20863712518982 7 49/48) (1.284039331328201 7 36/35) (1.312860066467948 7 15/14) (1.3239722230853748 5 16/15) (1.3259689601439075 7 28/27) (1.3374344495057697 5 25/24) (1.3442467614814364 7 21/20) (1.3641655968558717 7 245/243))

>> () log-flat is supposed to give us an equal number of results >> in all size ranges?

> >when they're defined logarithmically, yes.

Does this also give an equal number of results in all complexity ranges? 'Cause that seems more intuitive to me.

>> I tend to think this allows too much >> complexity in the small ratios.

> >so you want to penalize complexity even more? go for it.

Prosibly. If I can get this working. Looks like the start-from- factors approach is the only workable solution. There's no native array in scheme, but I'm sure you can make them with vectors of vectors. Weirdly, though, my method is apparently O(n^2) no matter what the limit, whereas your lattice-based method is O(n^2) at the 5-limit, O(n^3) at the 7-limit, etc. No wait, my n is max-d, and your n is radius on the lattice. Not the same. Did you just up the radius until max-d was exceeded somewhere? -Carl

Message: 7788 Date: Tue, 28 Oct 2003 20:13:07 Subject: Re: heuristic and straightness From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> () log-flat is supposed to give us an equal number of results > >> in all size ranges?

> > > >when they're defined logarithmically, yes.

> > Does this also give an equal number of results in all > complexity ranges? 'Cause that seems more intuitive to me.

by 'size', i meant complexity, or size of the numerator and denominator. i didn't mean cents size of the comma in JI!

> Weirdly, though, my method is apparently O(n^2) no matter what the > limit, whereas your lattice-based method is O(n^2) at the 5-limit, > O(n^3) at the 7-limit, etc. No wait, my n is max-d, and your n is > radius on the lattice. Not the same. Did you just up the radius > until max-d was exceeded somewhere?

i made sure it was large enough to enclose kees' figure. note the relationship with his work mentioned here: Definitions of tuning terms: heuristic complex... * [with cont.] (Wayb.)

Message: 7789 Date: Tue, 28 Oct 2003 20:35:42 Subject: Re: UVs for 46-ET 11-limit PB From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >

> > i tried to derive a 46-note periodicity-block > > which is a subset of this list,

> > of course, *any* 46-note subset which has each of the 46-equal > degrees exactly once is already a periodicity block . . . (by the > way, i don't agree with your reckoning of "equally close" . . .) >

> > using our software, > > with the following unison-vectors: > > > > [ 3 -6 1 0] > > [-4 -2 0 0] (diaschisma) > > [ 2 -3 1 0] (small septimal comma) > > [ 3 -4 0 -1] > > > > > > and the software gave me a nice 46-note > > periodicity-block, but it was entirely > > in the [3,5]-plane.

> > the determinant of this matrix is 14, so i'm not sure how you're > getting a 46-note periodicity block out of it!

i grouped a few quartets of 11-limit commas together at random, and after a couple of 72s, found that 9801:9800, 3025:3024, 441:440 and 176:175 together give 46. then i tried multiplying and dividing pairs of these to get simpler ratios (being sure to keep 4 linearly independent ones at each stage); one possibility is 896:891, 385:384, 125:126, and 176:175. the matrix of these: -4 0 1 -1 -1 1 1 1 -2 3 -1 0 0 -2 -1 1 now i looked at the periodicity block defined by the unit hypercube lying between 0 and 1 (instead of the usual -.5 and .5) along each of the four transformed coordinate axes using the matrix above: numerator denominator 55 54 33 32 25 24 16 15 275 256 11 10 10 9 9 8 55 48 7 6 33 28 6 5 175 144 99 80 5 4 32 25 165 128 33 25 4 3 27 20 11 8 7 5 99 70 275 192 35 24 165 112 3 2 55 36 99 64 25 16 8 5 825 512 33 20 5 3 297 175 55 32 7 4 99 56 9 5 11 6 297 160 15 8 48 25 495 256 99 50 2 1 this has plenty of ratios with factors of 7 and 11 -- hopefully it's close to what you need!

Message: 7790 Date: Tue, 28 Oct 2003 12:35:57 Subject: Re: heuristic and straightness From: Carl Lumma

>> >> () log-flat is supposed to give us an equal number of results >> >> in all size ranges?

>> > >> >when they're defined logarithmically, yes.

>> >> Does this also give an equal number of results in all >> complexity ranges? 'Cause that seems more intuitive to me.

> >by 'size', i meant complexity, or size of the numerator and >denominator. i didn't mean cents size of the comma in JI!

Oh. I guess I'm satisfied, then.

>> Weirdly, though, my method is apparently O(n^2) no matter what the >> limit, whereas your lattice-based method is O(n^2) at the 5-limit, >> O(n^3) at the 7-limit, etc. No wait, my n is max-d, and your n is >> radius on the lattice. Not the same. Did you just up the radius >> until max-d was exceeded somewhere?

> >i made sure it was large enough to enclose kees' figure. note the >relationship with his work mentioned here: > >Definitions of tuning terms: heuristic complex... * [with cont.] (Wayb.)

...

>heuristic complexity > >for a unison vector n/d, which is a ratio of R, the heuristic for >complexity is log(R). > >the complexity heuristic is identical to kees van prooijen's >'expressibility' measure

Huh? For n/d, I thought it was just log(d). What's R here, odd- or prime-limit? Further, isn't there something on kees' page about your meassure and and his being subtly different? -Carl

Message: 7791 Date: Tue, 28 Oct 2003 20:40:02 Subject: Re: heuristic and straightness From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >heuristic complexity > > > >for a unison vector n/d, which is a ratio of R, the heuristic for > >complexity is log(R). > > > >the complexity heuristic is identical to kees van prooijen's > >'expressibility' measure

> > Huh? For n/d, I thought it was just log(d). What's R here, odd- > or prime-limit?

ratio of R, related to odd limit (click on "ratio of" for the definition). for a small (in cents) comma, d is a very good approximation of R.

> Further, isn't there something on kees' page about your meassure > and and his being subtly different?

that was a different measure of mine, namely taxicab distance on the isosceles triangular lattice.

Message: 7792 Date: Tue, 28 Oct 2003 12:44:11 Subject: Re: heuristic and straightness From: Carl Lumma

>> >heuristic complexity >> > >> >for a unison vector n/d, which is a ratio of R, the heuristic for >> >complexity is log(R). >> > >> >the complexity heuristic is identical to kees van prooijen's >> >'expressibility' measure

>> >> Huh? For n/d, I thought it was just log(d). What's R here, odd- >> or prime-limit?

> >ratio of R, related to odd limit (click on "ratio of" for the >definition). for a small (in cents) comma, d is a very good >approximation of R. >

>> Further, isn't there something on kees' page about your meassure >> and and his being subtly different?

> >that was a different measure of mine, namely taxicab distance on the >isosceles triangular lattice.

Sweet. Tx. -Carl

Message: 7793 Date: Tue, 28 Oct 2003 02:16:37 Subject: Linear temperament names? From: Dave Keenan Have names been proposed for any of the linear microtemperaments below? The generators and errors are for minimax. The mappings given are octave equivalent: [gens_per_3 gens_per_5 gens_per_7 gens_per_11 gens_per_13; periods_per_3 periods_per_5 etc...] Limit Period Gen Max gens Max err Prime mapping Rep ET 7-limit 1 oct 193.87 c 16 1.4 c [16 2 5] 99 11-limit 1/2 oct 216.74 c 30 3.1 c [-6 -1 10 -3; 1 1 0 0] 72 11-limit 1/2 oct 183.21 c 30 2.4 c [-6 -11 2 3; 1 0 1 0] 72 15-limit 1/3 oct 83.02 c 48 2.8 c [-6 -5 2 -3 -14; 0 2 2 2 2] 72 15-lm-wo-13 1 oct 193.24 c 35 2.8 c [-15 2 5 -22] 118 These linear temps will appear in the microtempered guitar article I'm preparing for Xenharmonikon 18. If you do propose a name, please also say why you think it is appropriate. If I don't think the name makes much sense I may just include the temperament without a name. Others being included, whose names I have accepted, are: miracle magic kleismic schismic diaschismic neutral thirds My preference is for names that are descriptive of the generator (and also the period, when it isn't a whole octave). And when the period isn't an octave then the name might describe any period-equivalent inversion or extension of the generator. Graham, have you ever thought of spelling it "majic" since it's generated by MAJor thirds? Then the second one above might be called "twin majic" since its period is a half-octave, and its period minus its gen is a major third. The third one might be called "twin minor tones". Perhaps the first one is "semi major thirds" or "semi-majic", except how will we distinguish it from the last one which also has a gen which is half a major third, but has a different mapping for the prime 3. Perhaps the fourth one is "triple minor thirds" or "triple kleismic". Your thoughts on this will be appreciated. -- Dave Keenan

Message: 7794 Date: Tue, 28 Oct 2003 21:48:27 Subject: Re: Linear temperament names? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> The article deals only with linear temperaments of octave-repeating > octave-equivalent scales, so the reader is only interested in how

many

> periods there are modulo the number in the octave.

Someone reading about temperaments presumably wants to know what they are. Your sloppy method means they must work at it even to get the mapping of primes in terms of period and generator. How can you simultaneously maintain you are dumbing down *and* increase the number of mathematical hoops you expect your readers to jump through? If you want to make things easy, you are going about it in a very, very bad way. So when the period

> _is_ the octave these are all zero and I prefer to omit them. It's > easier to omit them without confusion if they come _last_. I do not > want to use any vector or matrix math in the article.

It's pitched at

> an audience with more basic math skills.

So that is why you insist on making the math difficult??

> The article deals only with octave-repeating octave-equivalent

scales

> so why should I bother saying that there are zero generators in the > 1:2 every time.

You should "bother" to give your poor readers a break by explicitly giving them a mapping to primes. What is this--pledge week for microtonalists? If they understand your article without the secret decoder ring they are in? In any case sloppy is sloppy.

> Commas and wedgies are utterly irrelevant to my article.

Commas are irrelevant to explaining temperaments? I don't think so. Once again, you propose leaving your readers clueless.

Message: 7795 Date: Tue, 28 Oct 2003 23:16:23 Subject: Re: Linear temperament names? 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: >

> > The article deals only with linear temperaments of octave-repeating > > octave-equivalent scales, so the reader is only interested in how

> many

> > periods there are modulo the number in the octave.

> > Someone reading about temperaments presumably wants to know what they > are.

Yes, but there are many ways to answer that. Those that are best for mathematicians are not necessarily best for musical types. For musical types octave-equivalence is a given. It is difficult to consider its absence.

> Your sloppy method means they must work at it even to get the > mapping of primes in terms of period and generator.

What sloppy method? You don't even know what my method of explanation is. Take another look at Graham's temperament catalog. Catalogue of linear temperaments * [with cont.] (Wayb.) He doesn't bother saying that there are zero generators in the octave every time either, and he doesn't give period mappings at all. I haven't heard you berating _him_ lately. Sorry Graham :-)

> How can you > simultaneously maintain you are dumbing down *and* increase the > number of mathematical hoops you expect your readers to jump through? > If you want to make things easy, you are going about it in a very, > very bad way.

You have absolutely no idea how I am going about it. All you know is the format in which I am tabulating the mappings. I have to say, you would be the last person I would ask for advice on explaining things to non-mathematicians. I was asking for help with the names. That's all. You have chosen not to explain your proposed names, or point me to earlier explanations, instead going on with this ridiculous rant. So I may well use my own names if I think they are sufficiently obvious.

> So when the period

> > _is_ the octave these are all zero and I prefer to omit them. It's > > easier to omit them without confusion if they come _last_. I do not > > want to use any vector or matrix math in the article.

> > It's pitched at

> > an audience with more basic math skills.

> > So that is why you insist on making the math difficult??

Again you have no idea what math I'm using, so I don't know where you get off with this sort of arrogant nonsense.

> > The article deals only with octave-repeating octave-equivalent

> scales

> > so why should I bother saying that there are zero generators in the > > 1:2 every time.

> > You should "bother" to give your poor readers a break by explicitly > giving them a mapping to primes.

I know you have trouble grasping this, but they don't actually care about "primes" per se. Only octave-reduced primes: 3/2, 5/4, 7/4, 11/8, 13/8, etc.

> What is this--pledge week for > microtonalists? If they understand your article without the secret > decoder ring they are in? In any case sloppy is sloppy.

How did you get to be such a prick?

> > Commas and wedgies are utterly irrelevant to my article.

> > Commas are irrelevant to explaining temperaments? I don't think so. > Once again, you propose leaving your readers clueless.

My article is not intended as a treatise on temperaments in general. It use temperaments with very specific properties to achieve a very specific task - guitar fretboard optimisation. Due to limited space, the use of temperaments will have to be partly in the nature of a recipe. The understanding of exactly _why_ one is doing certain things may have to come later. However I do mention commas in general and the fact that temperaments distribute them. It's just that the knowledge of which _particular_ commas any given microtemperament distributes has absolutely no application in the fretboard optimisation method, so their inclusion would just be clutter. -- Dave Keenan

Message: 7796 Date: Tue, 28 Oct 2003 03:55:59 Subject: Re: Linear temperament names? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >Graham, have you ever thought of spelling it "majic" since it's > >generated by MAJor thirds?

> > I thought it was an acronym. > > -Carl

Yes, but no less contrived than MIRACLE. Multiple Approximations Generated Iteratively and Consistently. Catalogue of linear temperaments * [with cont.] (Wayb.) and it has a [5 1 12] mapping, so I realise now that none of the LTs I gave qualify as semi-magic since they would have to start out [10 2 ...] to qualify. Instead the ones that are generated by a half major-third have [16 2 ...] and [-15 2 ...]. I just found that the latter has been called "semithirds", and since [8 1] is called "wuerschmidt" the former could be called "semiwuerschmidt". I'm guessing that it's ok to call this one "triple-kleismic" 15-limit 1/3 oct 83.02 c [-6 -5 2 -3 -14; 0 2 2 2 2] although there may well be other candidates for that name. That just leaves two: 11-limit 1/2 oct 183.21 c [-6 -11 2 3; 1 0 1 0] maybe "twin minortones", and 11-limit 1/2 oct 216.74 c [-6 -1 10 -3; 1 1 0 0] maybe "twin thirds", although these don't agree with the thirds of "semithirds".

Message: 7797 Date: Tue, 28 Oct 2003 07:34:07 Subject: Re: Linear temperament names? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> Have names been proposed for any of the linear microtemperaments below? > > The generators and errors are for minimax. The mappings given are > octave equivalent: > [gens_per_3 gens_per_5 gens_per_7 gens_per_11 gens_per_13; > periods_per_3 periods_per_5 etc...]

Graham convinced me to switch to his convention. Do you have a reason for preferring [generator, period] over [period, generator]? I think we should try for some degree of standardization. Noreover, you are ignoring 2, and to me this is simply not acceptable.

> Limit Period Gen Max gens Max err Prime mapping Rep ET > 7-limit 1 oct 193.87 c 16 1.4 c [16 2 5]

Hemiwuerschmidt. You should give all of the mapping and give it in a canonical reduced form, or a give a reduced comma basis, or a wedgie--or best of all, all three. 99

> 11-limit 1/2 oct 216.74 c 30 3.1 c [-6 -1 10 -3; 1 1 0 0] > 11-limit 1/2 oct 183.21 c 30 2.4 c [-6 -11 2 3; 1 0 1 0]

Unidec.

> 15-limit 1/3 oct 83.02 c 48 2.8 c [-6 -5 2 -3 -14; 0 2 2 2 2]

Trikleismic.

> 15-lm-wo-13 1 oct 193.24 c 35 2.8 c [-15 2 5 -22]

For the 7-limit temperament, I have it listed as Hemithird.

> If you do propose a name, please also say why you think it is > appropriate.

The names I give are ones which have already been used; they are not new proposals. If I don't think the name makes much sense I may just

> include the temperament without a name.

My preference is for you not to sow confusion by introducing new names for already named and cataloged temperaments; not naming if you object to a name seems like a plan.

Message: 7798 Date: Wed, 29 Oct 2003 03:23:01 Subject: Re: Linear temperament names? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:

> >For musical types octave-equivalence is a given.

> > It is?

OK. Let me rephrase that. For most musical types of people who are not mathematicians, octave-equivalence is a given.

> >I was asking for help with the names. That's all. You have chosen > >not to explain your proposed names, or point me to earlier > >explanations, instead going on with this ridiculous rant.

> > Didn't he say he uses names because they have already been used?

Wot, do you mean that if I make up a name for something, give no explanation, and use it three times in my own posts (but no one else does), that's enough to make it established terminology and everyone else ought to just use it from then on? And I should just ignore any requests for justification? :-)

Message: 7799 Date: Wed, 29 Oct 2003 20:16:18 Subject: Re: UVs for 46-ET 11-limit PB From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> hi paul, > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >

> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > > wrote:

> > > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > > >

> > > > i tried to derive a 46-note periodicity-block > > > > which is a subset of this list,

> > > > > > of course, *any* 46-note subset which has each of > > > the 46-equal degrees exactly once is already a > > > periodicity block . . .

> > > right, i know that ... but i deliberately left in > the duplicates and triplicates to see what you, Gene, > et al would come up with. > > >

> > > (by the way, i don't agree with your reckoning of > > > "equally close" . . .)

> > > i know ... you use the hexagonal reckoning rather than > the rectangular one i used. i considered doing that > from the start, but it was just easier for me to do > the one i did since i'm still using Excel for this > kind of stuff. > > >

> > >

> > > > using our software, > > > > with the following unison-vectors: > > > > > > > > [ 3 -6 1 0] > > > > [-4 -2 0 0] (diaschisma) > > > > [ 2 -3 1 0] (small septimal comma) > > > > [ 3 -4 0 -1] > > > > > > > > > > > > and the software gave me a nice 46-note > > > > periodicity-block, but it was entirely > > > > in the [3,5]-plane.

> > > > > > the determinant of this matrix is 14, so i'm not sure how

you're

> > > getting a 46-note periodicity block out of it!

> > > > a sign in the first comma was reversed. its complete > [2,3,5,7,11]-monzo should be [12 3 -6 -1 0] . > see my post to Gene. > > > >

> > i grouped a few quartets of 11-limit commas together > > at random, and after a couple of 72s, found that > > 9801:9800, 3025:3024, 441:440 and 176:175 together > > give 46. then i tried multiplying and dividing pairs > > of these to get simpler ratios (being sure to keep > > 4 linearly independent ones at each stage); one > > possibility is 896:891, 385:384, 125:126, and 176:175. > > the matrix of these: > > > > -4 0 1 -1 > > -1 1 1 1 > > -2 3 -1 0 > > 0 -2 -1 1 > > > > now i looked at the periodicity block defined by the > > unit hypercube lying between 0 and 1 (instead of the > > usual -.5 and .5) along each of the four transformed > > coordinate axes using the matrix above: > > > > numerator denominator > > 55 54 > > 33 32 > > 25 24 > > 16 15 > > 275 256 > > <etc., snip> > > > > this has plenty of ratios with factors of 7 and 11 -- > > hopefully it's close to what you need!

> > > > thanks ... but it would be easier for me to tell if > instead of ratios the notes had already been factored > into monzos. if anyone else cares to do it for me, > that would be nice! ;-) > > > > > -monz

here are the factorizations: first column is 3, second column is 5, third column is 7, fourth column is 11: -3 1 0 1 1 0 0 1 -1 2 0 0 -1 -1 0 0 0 2 0 1 0 -1 0 1 -2 1 0 0 2 0 0 0 -1 1 0 1 -1 0 1 0 1 0 -1 1 1 -1 0 0 -2 2 1 0 2 -1 0 1 0 1 0 0 0 -2 0 0 1 1 0 1 1 -2 0 1 -1 0 0 0 3 -1 0 0 0 0 0 1 0 -1 1 0 2 -1 -1 1 -1 2 0 1 -1 1 1 0 1 1 -1 1 1 0 0 0 -2 1 0 1 2 0 0 1 0 2 0 0 0 -1 0 0 1 2 0 1 1 -1 0 1 -1 1 0 0 3 -2 -1 1 0 1 0 1 0 0 1 0 2 0 -1 1 2 -1 0 0 -1 0 0 1 3 -1 0 1 1 1 0 0 1 -2 0 0 2 1 0 1 2 -2 0 1 0 0 0 0

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