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Message: 7950 Date: Mon, 03 Nov 2003 21:05:11 Subject: Some 5-limit TM reduced et bases From: Gene Ward Smith 12: [81/80, 128/125] [[-4, 4, -1], [7, 0, -3]] 15: [128/125, 250/243] [[7, 0, -3], [1, -5, 3]] 19: [81/80, 3125/3072] [[-4, 4, -1], [-10, -1, 5]] 31: [81/80, 393216/390625] [[-4, 4, -1], [17, 1, -8]] 34: [2048/2025, 15625/15552] [[11, -4, -2], [-6, -5, 6]] 53: [15625/15552, 32805/32768] [[-6, -5, 6], [-15, 8, 1]] 65: [32805/32768, 78732/78125] [[-15, 8, 1], [2, 9, -7]] 118: [32805/32768, 1224440064/1220703125] [[-15, 8, 1], [8, 14, -13]] 171: [32805/32768, 7629394531250/7625597484987] [[-15, 8, 1], [1, -27, 18]]

Message: 7951 Date: Mon, 03 Nov 2003 16:54:14 Subject: Re: Some 11-limit TM reduced et bases From: Carl Lumma

>This gives the basis for the corresponding standard val, and then the >characteristic temperament (the linear temperament obtained from the >first three of the four basis commas.)

It seems you order these largest-to-smallest. Why do we want to leave out the smallest comma in the basis -- my guess was we'd want to omit the largest. -Carl

Message: 7952 Date: Mon, 03 Nov 2003 19:25:58 Subject: Re: Some 11-limit TM reduced et bases From: Carl Lumma

>> It seems you order these largest-to-smallest. Why do we want to >>leave out the smallest comma in the basis -- my guess was we'd want >>to omit the largest.

> >No, I ordered them by Tenney height, but why in the world would we >ditch the largest comma?

Why in the world would we ditch the highest comma? Once again, isn't it a combination of these two factors that decides a comma's 'goodness'? Then again, I'm not sure of the relative benefit of ditching vs. keeping the best commas. -Carl

Message: 7953 Date: Tue, 04 Nov 2003 15:35:00 Subject: Re: Some 11-limit TM reduced et bases From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >This gives the basis for the corresponding standard val, and then

the

> >characteristic temperament (the linear temperament obtained from

the

> >first three of the four basis commas.)

> > It seems you order these largest-to-smallest.

no, simplest to most complex.

> Why do we want to leave > out the smallest comma in the basis -- my guess was we'd want to

omit

> the largest.

he leaves out the most complex, which is intuitive. the simplest will have the most effect on harmonic progressions in the tuning.

Message: 7954 Date: Tue, 04 Nov 2003 15:55:14 Subject: Re: reduced basis for 24-ET?? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> hi Gene, > > > i tried deriving a periodicity-block for > 24-ET from a <3,5,11>-prime-space by using > the following unison-vectors: > > > [2, 3, 5, 11]-monzo ratio > > > [ -4, 4, -1, 0] 81:80 > > [ 7, 0, -3, 0] 128:125 > > [-17, 2, 0, 4] 131769:131072 > > > > but instead of getting a 24-tone periodicity-block, > i got a 48-tone torsional-block. > > > 24-ET represents ratios-of-11 so well that there > has to be a periodicity-block hiding in here somewhere. > can you help? > > > > -monz

i'm not sure what you're asking gene. you'd like to remove the torsion? simple -- note that the sum of the three rows in the matrix above is [-14 6 -4 4] which is the square of [-7 3 -2 2] 3267:3200 using this for the third row of the matrix, you get [ -4, 4, -1, 0] 81:80 [ 7, 0, -3, 0] 128:125 [-7, 3, -2, 2] 3267:3200 and the torsion is gone.

Message: 7955 Date: Tue, 04 Nov 2003 15:59:17 Subject: Re: hey Paul From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> I'm interested in these scales... >

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

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

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

yup. anyway, this is a negri scale, yes?

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

> > > >// > >

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

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

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

it is. but the answer is yes, you can so reduce by the periods. gene was just trying to give a "hermite-reduced basis" or some such abstractly interesting form for the generators.

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

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

> > By the way, do these temperaments have names?

many of them do, thanks to gene . . .

Message: 7956 Date: Tue, 04 Nov 2003 16:51:45 Subject: Re: More TM base postings From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> ... of course, noting that the chromatic semitone disappears > because 2^(7/24) is 350 cents, and is thus a neutral-3rd > which represents both the major-3rd and minor-3rd on this > particular bingo-card, which in turn means that it's not > really doing much in the way of representing 5-limit JI > in the first place.

well, it's still doing so at least in theory, just like all the ETs on the 'dicot' line on the zoom-1 and zoom-10 charts: Definitions of tuning terms: equal temperament... * [with cont.] (Wayb.) (this mapping of 24-equal would appear at the intersection of the aristoxenean and dicot lines; i didn't label it because 24-equal is consistent in the 5-limit with the same approximations as 12-equal, so you'll see 24 on the same point as 12 instead.) and like those with a bingo card where the chromatic semitone vanishes, for example Yahoo groups: /tuning/files/perlich/10.gif * [with cont.] Yahoo groups: /tuning-math/files/Paul/7p.gif * [with cont.]

Message: 7957 Date: Tue, 04 Nov 2003 08:52:57 Subject: Re: Some 11-limit TM reduced et bases From: Carl Lumma

>he leaves out the most complex, which is intuitive. the simplest will >have the most effect on harmonic progressions in the tuning.

But isn't this also true for chromatic vectors? -Carl

Message: 7958 Date: Tue, 04 Nov 2003 09:02:43 Subject: Re: hey Paul From: Carl Lumma

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

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

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

> >yup. anyway, this is a negri scale, yes?

Yep.

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

>> > >> >// >> >

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

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

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

> >it is. but the answer is yes, you can so reduce by the periods. gene >was just trying to give a "hermite-reduced basis" or some such >abstractly interesting form for the generators.

Tx.

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

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

>> >> By the way, do these temperaments have names?

> >many of them do, thanks to gene . . .

Is there a way for people to look them up? -Carl

Message: 7959 Date: Tue, 04 Nov 2003 17:07:58 Subject: Re: Some 11-limit TM reduced et bases From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >he leaves out the most complex, which is intuitive. the simplest

will

> >have the most effect on harmonic progressions in the tuning.

> > But isn't this also true for chromatic vectors? > > -Carl

not really, chromatic vectors only determine how far the temperament is carried out to form a scale, and can look the same for unrelated temperaments and scales, but the temperament itself is characterized by the commatic vectors. if you take one of the simplest commas which vanishes in the equal temperament and re-interpret it as a chromatic vector, you'll end up with a system that differs more strongly from the 'native harmony' of the equal temperament than when you do this with a more complex comma. for example, 31 in the 5-limit is [81/80, 393216/390625], and making the 393216/390625 chromatic maintains the meantone character that dominates 31-equal's 5-limit behavior, while making 81/80 chromatic yields the more tenuous würschmidt system . . .

Message: 7960 Date: Tue, 04 Nov 2003 17:18:58 Subject: Re: Some 11-limit TM reduced et bases From: Paul Erlich moreover, when there are three or more commatic vectors, the reduction definition is more arbitrary -- the simplest (or shortest in the lattice) comma is uniquely and unambiguously defined, but the rest depend on the precise reduction definition -- for example minkowski reduction may lead to a very simple second comma and a more complex third comma, while another basis may sacrifice the simplicity of the second comma so that the third comma comes out less complex . . . --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

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

> > >he leaves out the most complex, which is intuitive. the simplest

> will

> > >have the most effect on harmonic progressions in the tuning.

> > > > But isn't this also true for chromatic vectors? > > > > -Carl

> > not really, chromatic vectors only determine how far the

temperament

> is carried out to form a scale, and can look the same for unrelated > temperaments and scales, but the temperament itself is

characterized

> by the commatic vectors. if you take one of the simplest commas

which

> vanishes in the equal temperament and re-interpret it as a

chromatic

> vector, you'll end up with a system that differs more strongly from > the 'native harmony' of the equal temperament than when you do this > with a more complex comma. for example, 31 in the 5-limit is

[81/80,

> 393216/390625], and making the 393216/390625 chromatic maintains

the

> meantone character that dominates 31-equal's 5-limit behavior,

while

> making 81/80 chromatic yields the more tenuous würschmidt

system . . .

Message: 7961 Date: Tue, 04 Nov 2003 09:19:20 Subject: Re: Some 11-limit TM reduced et bases From: Carl Lumma

>>>he leaves out the most complex, which is intuitive. the simplest >>>will have the most effect on harmonic progressions in the tuning.

>> >> But isn't this also true for chromatic vectors? >> >> -Carl

> >not really, chromatic vectors only determine how far the temperament >is carried out to form a scale, and can look the same for unrelated >temperaments and scales,

But still seems important, in light of the current "hey paul" thread, and Gene's T[n] thread.

>but the temperament itself is characterized >by the commatic vectors. if you take one of the simplest commas which >vanishes in the equal temperament and re-interpret it as a chromatic >vector, you'll end up with a system that differs more strongly from >the 'native harmony' of the equal temperament than when you do this >with a more complex comma. for example, 31 in the 5-limit is [81/80, >393216/390625], and making the 393216/390625 chromatic maintains the >meantone character that dominates 31-equal's 5-limit behavior, while >making 81/80 chromatic yields the more tenuous würschmidt system . . .

Ok, I'll buy that. -Carl

Message: 7962 Date: Tue, 04 Nov 2003 19:09:07 Subject: Re: ennealimmal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:

> You probably mean 18-tone. The generator doesn't need to be > exactly 50 cents, but if I understand your question correctly, yes.

In fact, a generator of 25/612 makes more sense.

Message: 7963 Date: Tue, 04 Nov 2003 19:10:45 Subject: Re: Some 11-limit TM reduced et bases From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> he leaves out the most complex, which is intuitive. the simplest

will

> have the most effect on harmonic progressions in the tuning.

Of course in the case of 41 that came down to choosing 243/242 over 245/242.

Message: 7964 Date: Tue, 04 Nov 2003 00:48:00 Subject: Some 11-limit TM reduced et bases From: Gene Ward Smith This gives the basis for the corresponding standard val, and then the characteristic temperament (the linear temperament obtained from the first three of the four basis commas.) The name "Subminortone" I tacked onto the characteristic temperament for 41 is new; now would be the time to comment, complain, claim, or suggest changes. 12: [36/35, 45/44, 50/49, 56/55] [4, 4, 4, 12, -3, -5, 5, -2, 14, 20] [[4, 6, 9, 11, 13], [0, 1, 1, 1, 3]] "Diminished" 22: [50/49, 55/54, 64/63, 99/98] [2, -4, -4, 10, -11, -12, 9, 2, 37, 42] [[2, 3, 5, 6, 6], [0, 1, -2, -2, 5]] "Pajara" 31: [81/80, 99/98, 121/120, 126/125] [4, 16, 9, 10, 16, 3, 2, -24, -32, -3] [[1, 3, 8, 6, 7], [0, -4, -16, -9, -10]] "Squares" 41: [100/99, 225/224, 243/242, 245/242] [4, 9, 26, 10, 5, 30, 2, 35, -8, -62] [[1, 1, 1, -1, 2], [0, 4, 9, 26, 10]] "Subminortone" 46: [121/120, 126/125, 176/175, 245/243] [9, 5, -3, 7, -13, -30, -20, -21, -1, 30] [[1, 1, 2, 3, 3], [0, 9, 5, -3, 7]] "Alpha" 58: [126/125, 176/175, 243/242, 896/891] [10, 9, 7, 25, -9, -17, 5, -9, 27, 46] [[1, -1, 0, 1, -3], [0, 10, 9, 7, 25]] "Nonkleismic" 72: [225/224, 243/242, 385/384, 4000/3993] [6, -7, -2, 15, -25, -20, 3, 15, 59, 49] [[1, 1, 3, 3, 2], [0, 6, -7, -2, 15]] "Miracle" 118: [385/384, 441/440, 3136/3125, 4375/4374] [15, -2, -5, 22, -38, -50, -17, -6, 58, 79] [[1, 4, 2, 2, 7], [0, -15, 2, 5, -22]] "Hemithird" 152: [540/539, 1375/1372, 4375/4374, 5120/5103] [24, 32, 40, 24, -5, -4, -45, 3, -55, -71] [[8, 13, 19, 23, 28], [0, -3, -4, -5, -3]] "Octoid"

Message: 7965 Date: Tue, 04 Nov 2003 19:14:02 Subject: Re: hey Paul From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Is there a way for people to look them up?

Should I put up a web page? Dave, do you have an objection?

Message: 7966 Date: Tue, 04 Nov 2003 19:58:54 Subject: Re: reduced basis for 24-ET?? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> but instead of getting a 24-tone periodicity-block, > i got a 48-tone torsional-block.

You've caught the dreaded torsion disease.

> 24-ET represents ratios-of-11 so well that there > has to be a periodicity-block hiding in here somewhere. > can you help?

The TM basis for 3-5-11 24-et is a good starting place: <81/80, 121/120, 128/125> By definition, this has no torsion, and your software should now give you a 24 block. The three commas can be considered the basis for an 11-limit linear temperament, which means 7 is now included. The wedgie is [0, 0, 24, 0, 0, 38, 0, 56, 0, -83] the prime mapping is [[24, 38, 56, 67, 83], [0, 0, 0, 1, 0]] and the rms generators are [50.00000000, 20.12657339]

Message: 7967 Date: Tue, 04 Nov 2003 02:18:02 Subject: Re: Some 11-limit TM reduced et bases From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> It seems you order these largest-to-smallest. Why do we want to

leave

> out the smallest comma in the basis -- my guess was we'd want to

omit

> the largest.

No, I ordered them by Tenney height, but why in the world would we ditch the largest comma?

Message: 7968 Date: Tue, 04 Nov 2003 20:25:01 Subject: Re: reduced basis for 24-ET?? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> The TM basis for 3-5-11 24-et is a good starting place: > > <81/80, 121/120, 128/125>

If you add 49/48, you get the TM basis for the 11-limit standard val h24: <49/48, 81/80, 121/120, 128/125>. The characteristic temperament for this has basis <49/48, 81/80, 121/120>, the wedgie is [4, 6, 2, 10, 16, -8, 2, -40, -32, 21] and the mapping [[2,4,8,6,9], [0,-2,-8,-1,-5]] It is Hemifourth with the period reduced to a half-octave, so I suggest Bihemifourth as a name. Now is a good time to object. If we toss 121/120 from the mix, we get the TM basis for the 7-limit standard 24-et val, namely <49/48, 81/80, 128/125>. The first two commas of this, <49/48, 81/80> are the TM basis for the characteristic temperament, in this case Hemifourth. Tossing out 49/48 now gives us the TM basis for 5-limit 12-equal, with characteristic temperament Meantone.

Message: 7969 Date: Tue, 04 Nov 2003 20:30:27 Subject: Re: Eponyms From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

wrote:

> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith"

<gwsmith@s...> wrote:

> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan"

<d.keenan@b...> wrote:

> > >

> > > > Have you actually read any of the several descriptions I've

given of

> > > > the proposed komma naming algorithm and its inverse? Are they

all

> > > > really that unclear?

Hey, what happened to this thread, anyway? If things start getting taken a little too personally, do we just pack up our stuff and go home? The comma-naming system that Dave was presenting is something that he and I have been using for quite some time. It evolved over the course of more than a year during our Sagittal development project, and we found these names very useful in discussing commas that we were trying to symbolize, so the names are something that both Dave and I can take credit for. The specific boundaries between categories were worked out by Dave, and I agree with them. The monzo comma-naming system is so cumbersome (i.e., unfriendly), that I can't imagine how anyone could follow a name above the 11- limit unless it's written down. Imagine trying to mention to someone in spoken conversation that you're trying to decide whether to use the symbol for "<-6, 0, 0, 0, 0, 0, 0, 1> or <6, 0, 0, 0, 0, 0, 0, 1>" in a composition -- are you expecting me to be mentally prepared to count all those zeros so I know what prime number you mean when you finally get to the "1" that matters? The problem is, the "name" (if you can call it that) emphasizes *powers* rather than *primes*, so it tends to get rather cryptic. Now if Dave says, "Should I use a 23-comma or 23-small-diesis (23S) symbol?", I immediately know what he's talking about: these are commas that alter tones in a 1/1 pythagorean chain to arrive at tones in a 23/16 and/or a 32/23 pythagorean chain. The difference in using a 23C vs. a 23S (how's that for brevity?) amounts to nothing more than how you wish to spell the pitch, i.e., which nominal you wish to modify. Likewise, a 7:11-comma vs. a 7:11 kleisma (or 7:11C vs. 7:11k) notates tones in 11/7 and 14/11 pythagorean chains with alternate spellings. Discarding powers of 2 and 3 from the *names*, therefore, is in keeping with the fact that Sagittal symbols may (in a heptatonic notation) modify *many* (and *only*) tones in a pythagorean chain containing 1/1. If you want to notate a certain ratio in JI, then its comma-symbol is determined *only* by the primes above 3 in the ratio, and the nominal being modified is determined by the powers of 2 and 3. And there's also a clue to help you remember which symbol represents which comma, because the comma name also tells you the size range.

> > > I don't buy kommas.

> > > > Do you mean you don't like spelling it with a "k" when it's being

used

> > as a generic term. That's fine. That's not part of the naming > > algorithm. That's just me.

> > (1) I don't like having two words which sound the same and with > related meanings > > (2) I think the goofy spelling, if you do this, should be for your > new meaning, and not imposed on an established one. "Comma in a > particular range of cents"="komma", in other words.

And please, let's not get hung up on "comma" vs. "komma". In the first place, we don't want "komma" to specify a particular size, because the "k" won't distinguish it from "kleisma" in the abbreviations that we're using. In the second place, "comma" should be okay for both the generic and specific-size-range terms, because the context should make it obvious how the term is being used; as evidence of this, read my preceding paragraph again. But don't get me wrong -- I'm not objecting to monz's HEWM factored- ratio notation, which is very useful in a lot of instances (as should be evident by just reading a few other current messages). But I would call this a (theoretical) *notation* for ratios rather than (systematic) *names* for commas. A name should be something that is simple enough to be easily recognized. --George

Message: 7970 Date: Tue, 04 Nov 2003 06:46:54 Subject: Re: More TM base postings From: monz --- 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 "Paul Erlich" <perlich@a...>

> wrote:

> >

> > > the first one assumes the standard val (or something) > > > for inconsistent ets; i'd at least show the results > > > for other vals.

> > > > The 5-limit needs to be what it is in order to include > > 81/80, and for some of these, the 7-limit needs to be > > what it is to avoid torsion.

> > yes, that process should be made explicit though -- i'd > rather explain that 24-equal sometimes doesn't have a > reasonable basis due to torsion, and in those cases is > best understood as an equal halving of each 12-equal step, > instead of using hidden rules to provide a nice-looking > answer.

yes, i agree totally with paul. in fact, the precise thing that prompted me to write my original post requesting these TM-reduced bases was that i tried to create 41-ET with our software, and while it did give 41edo as the temperament, it gave an 82-tone periodicity-block for the JI scale. as soon as i saw that, i suspected that it was due to torsion, and sure enough, that turned out to be the case. in the specific case of 24-ET in a [3,5]-prime-space, it's nice to be able to see how choosing a val of h(5)=8 results in a "double 12-ET", whereas h(5)=7 results in a true 24-tone periodicity-block ... ... of course, noting that the chromatic semitone disappears because 2^(7/24) is 350 cents, and is thus a neutral-3rd which represents both the major-3rd and minor-3rd on this particular bingo-card, which in turn means that it's not really doing much in the way of representing 5-limit JI in the first place. -monz

Message: 7971 Date: Tue, 04 Nov 2003 13:16:53 Subject: Re: hey Paul From: Carl Lumma

>> Is there a way for people to look them up?

> >Should I put up a web page? Dave, do you have an objection?

Graham's catalog is neither complete or up to date, last I checked. The existence of a single resource is a lot to ask, I know... -Carl

Message: 7972 Date: Tue, 04 Nov 2003 07:44:29 Subject: reduced basis for 24-ET? From: monz hi Gene, i tried deriving a periodicity-block for 24-ET from a <3,5,11>-prime-space by using the following unison-vectors: [2, 3, 5, 11]-monzo ratio [ -4, 4, -1, 0] 81:80 [ 7, 0, -3, 0] 128:125 [-17, 2, 0, 4] 131769:131072 but instead of getting a 24-tone periodicity-block, i got a 48-tone torsional-block. 24-ET represents ratios-of-11 so well that there has to be a periodicity-block hiding in here somewhere. can you help? -monz

Message: 7973 Date: Tue, 04 Nov 2003 22:51:06 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:

> Now if Dave says, "Should I use a 23-comma or 23-small-diesis (23S) > symbol?", I immediately know what he's talking about: these are > commas that alter tones in a 1/1 pythagorean chain to arrive at

tones

> in a 23/16 and/or a 32/23 pythagorean chain.

This is fine for your very specialized purposes, but what about the rest of us? Do you have a name for every superparticular ratio up to the 23-limit?

Message: 7974 Date: Tue, 04 Nov 2003 07:53:43 Subject: Re: reduced basis for 24-ET?? From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> hi Gene, > > > i tried deriving a periodicity-block for > 24-ET from a <3,5,11>-prime-space by using > the following unison-vectors: > > > [2, 3, 5, 11]-monzo ratio > > > [ -4, 4, -1, 0] 81:80 > > [ 7, 0, -3, 0] 128:125 > > [-17, 2, 0, 4] 131769:131072 > > > > but instead of getting a 24-tone periodicity-block, > i got a 48-tone torsional-block. > > > 24-ET represents ratios-of-11 so well that there > has to be a periodicity-block hiding in here somewhere. > can you help?

i tried using [14 -3 1 2] = 16384:16335 for the third unison-vector, along with 81:80 and 125:128, and it worked beautifully. i got one 12-tone PB in the 11^0 [3,5]-plane, and another very much like it in the 11^1 plane. -monz

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