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Message: 7451 Date: Tue, 23 Sep 2003 17:35:12 Subject: Re: Please remind me From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote: If anybody has canonical lists of unison

> vectors for different limits, that would help.

What makes a list canonical?

Message: 7453 Date: Tue, 23 Sep 2003 21:17:38 Subject: Re: Please remind me From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> I see. In the 7-limit for meantone, is 126/125 used instead of >81/80? > Or are they both used somehow...

they are both used, and together they form a basis for the kernel of 7-limit meantone. you need two independent vanishing unison vectors to get a linear temperament out of a 3-dimensional space like 7-limit JI, since 3-2=1 (this language ignores the 'octave' dimension).

Message: 7454 Date: Tue, 23 Sep 2003 09:42:46 Subject: Re: Please remind me From: Graham Breed Paul G Hjelmstad wrote:

>Thanks. I notice running, for example, meantone.optimizeRMS also >works. Have tried this for various Linear Temperaments. I notice >RMS is only slightly different from Minimax in Meantone. Both >are ~ 503.4 cents. I was expecting 503.8 cents, (7/26-comma >temperament) for RMS. Isn't RMS "root-mean-squared?" > >

Yes, the 5-limit RMS is 503.8 cents. The examples I gave were 7-limit, where the two optima are much closer. Graham

Message: 7455 Date: Tue, 23 Sep 2003 22:31:12 Subject: Re: Please remind me From: Graham Breed Gene Ward Smith wrote:

>What makes a list canonical? >

I thought you were working on algorithms to produce simple lists of unison vectors that produce all siginificant temperaments in a given limit. But anything would to for testing, as long as there aren't too many unison vectors, and enough interesting temperaments come out. Graham

Message: 7456 Date: Wed, 24 Sep 2003 06:45:41 Subject: Re: Please remind me From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> Gene Ward Smith wrote: >

> >What makes a list canonical? > >

> I thought you were working on algorithms to produce simple lists of > unison vectors that produce all siginificant temperaments in a

given

> limit. But anything would to for testing, as long as there aren't

too

> many unison vectors, and enough interesting temperaments come out.

You are probably thinking of my idea to simply list everything in a p- limit with an "epimeric exponent" less than a given figure, where if p/q is in reduced form, e = ln(p-q)/ln(q) is the epimeric exponent. If p/q is superparticular, this is zero, but even if the bound E satisfies 0 < E < 1, we have only a finite number of p-limited solutions. Hence it generalizes the idea of defining the canonical list as being simply the superparticular commas. To make it canonical, we would need a canonical way of choosing e.

Message: 7458 Date: Mon, 29 Sep 2003 20:17:44 Subject: Sagittal accidentals for Decimal/MIRACLE notation From: David C Keenan I wrote: "You can check whether they are valid accidentals for decimal notation by running their prime exponents thru the MIRACLE mapping to find out how many generators each corresponds to." When I did so, I found that I'd got it wrong for the multi-shaft sagittals (by wrongly assuming 72-ET). I'm now using the following 23-limit mapping to secors, [0, 6, -7, -2, 15, -34, -30, 54, 26] based on Gene's 19-limit mapping from Yahoo groups: /tuning-math/message/6450 * [with cont.] <chew, chew, gulp> That was me eating my words. I was silly enough back then, to say that this mapping was uninteresting. Sorry. So here's another attempt at sagittal versions of Graham's accidentals at Decimal Notation * [with cont.] (Wayb.) Graham's ASCII-Sagittal Comma interp. secors short long (prime exp, no 2) ----------------------------------------------------- ^^^ or m^ $# ~||| [5,0,0,0,1,0,0,-1] -30 ^^ or m _# )||( [-1,2] apotome-25S -20 /^ ^ /|\ [1,0,0,1] 11-M-diesis 21 ^ f |) [-2,0,-1] 7-comma -10 / / /| [4,-1] 5-comma 31 \ \ \! [-4,1] -31 v t !) [2,0,1] 10 \v v \!/ [-1,0,0,-1] -21 vv or w =b )!!( [1,-2] 20 vvv or wv sb ~!!! [-5,0,0,0,-1,0,0,1] 30 These single (multishaft) symbols for ^^ and ^^^ and their inverses are fairly unsatisfying (particularly the apotome+13:23_kleisma for ^^^), so I tend to agree that it may be better to just use multiple copies of the sagittal 7-comma symbol. Graham's /^ could also be represented by the single sagittal /|), but I chose /|\ for compatibility with 72-ET. I attempted to push on and find possible single accidentals for degrees of 175-ET in decimal/MIRACLE: ~| /| )|( //| |) ? /|\ ? ||) )||( //|| ||\ ||~) /||\ ~||| /||| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 deg -72 31 -41 62 -10 -82 21 -51 52 -20 83 11 -61 42 -30 73 sec 17 degrees of 175-ET is one secor. You'll see I failed to find single symbols for 6 and 8 degrees (-82 and -51 secors). But I've never thought about using sagittal like this before and it's early days. If you or Graham can find simpler kommas corresponding to these numbers of secors, I can tell you the sagittal symbols for them. However I rejected several because they either, (a) had a very large exponent for some prime (usually for 3 or 5), or (b) required an accented sagittal symbol, or (c) introduced inconsistencies in flag-degree arithmetic in 175-ET. Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 7459 Date: Mon, 29 Sep 2003 22:15:48 Subject: Re: Please remind me From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

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

> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" > > <paul.hjelmstad@u...> wrote: > >

> > > I see. In the 7-limit for meantone, is 126/125 used instead of > > >81/80? > > > Or are they both used somehow...

> > > > they are both used, and together they form a basis for the kernel

> of

> > 7-limit meantone. you need two independent vanishing unison

vectors

> > to get a linear temperament out of a 3-dimensional space like 7-

> limit

> > JI, since 3-2=1 (this language ignores the 'octave' dimension).

> > Interesting. Could you give me the formula (or point me to a

message)

> that would extract a generator for a linear temperament like this > (that uses 2 unison vectors vanishing...)

i don't even know the formula in the case of only 1 unison vector vanishing! i've seen some of your posts, but what happens in cases where the period comes out to a fraction of an octave (such as 2048:2025, where the period comes out to 1/2 octave)? you can probably handle this, i just wasn't paying enough attention :(

Message: 7461 Date: Tue, 30 Sep 2003 02:38:50 Subject: Re: Sagittal accidentals for Decimal/MIRACLE notation From: Dave Keenan --- In tuning@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:

> BTW (Dave, are you listening?), back when these messages were > exchanged, I actually tried figuring out what sagittal accidentals > might be used for 10 nominals for Miracle. What I ended up with was > so unconventional, both with respect to our present way of thinking > harmonically (by prime factors in ratios) and to the harmonic > meanings of the sagittal symbols themselves, that I could only > conclude that it was not worth the trouble even for a composer to use > decimal notation.

I don't understand this at all. Where's the problem? Here's the simplest way to do it. The basic chromatic comma is simply the 7-comma 63:64 (In ASCII shorthand "t" down, "f" up), and the enharmonic equivalent of 3 stacked 7-commas uses the 5-comma 80:81 in the opposite direction ("\" down, "/" up). Here are the degrees of 72-ET between the nominals 0 and 1. 0 0/ 0f 0f/ 1t\ 1t 1\ 1 1tt 0ff Carl:

> > But also in that thread, I assert that by forcing 7 nominals, you > > ruin the mapping from commas to accidentals. Care to rebut that?

George:

> I answer with the following: In a Pythagorean sequence of tones > (with 7 nominals), the position in the sequence determines the power > of 3, and the prime factors above 3 may all be expressed in the > accidentals.

Yes, but almost all accidental commas also contain powers of 2 and 3.

> In a sequence of tones separated by some other > generator (such as a secor), the prime factors contained or implied > in the relationships of those tones vary from one tone to another, so > any comma-to-accidental relationship is much less useful (and much > more complicated.)

Ah. I used a different method. My approach was to find out how many generators the desired accidentals would need to correspond to, (-10 and 31). Then I calculated the MIRACLE mappings of the available commas in order of decreasing popularity and chose the first with unaccented symbols that corresponded to these numbers of generators. |) and /|. But I see now that this just notates the MIRACLE temperament itself (maybe from 3/31 oct to 4/41 oct generators), i.e. it lets one extend the chain beyond the 10 nominals, but doesn't say how to notate things that are _off_ the chain. For example strict rational pitches, or 22-ET using twin chains of rather extreme-valued secors (1/11 octave). This is analogous to looking at meantone with 7 nominals and seeing that the basic chromatic comma corresponds to 7 generators (the apotome) and that an enharmonic equivalent to the double apotome must use a comma corresponding to -12 generators. This only says how to notate meantone. To notate rational pitches (e.g. JI), we must choose a rational value for our notational generator, and it should be as simple a ratio as possible (2:3). What value of secor would one choose for notating rational pitches in decimal? It would have to be either 14:15 or 15:16. I suppose we should use 15:16 since it has the lower prime limit. Then to notate any ET in decimal we should determine what the best approximation is to 15:16 in that ET and if this is not between 1/10 and 1/11 octave, we should look at multiples of the ET. We should also consider multiples if the tuning is not consistent in the sense that two of the best 15:16 approximations are not the same as the best 225:256 approximation. Notice that this is violated in 72-ET. Maybe 14:15 would work better, but the problem is that the secor is intended to approximate _both_ of these ratios. This looks very messy to me, but maybe Carl and Graham can work on it and come up with a list of commas they would like to use for various purposes, and we can supply the sagittal symbols for them. So I now see that, even if diatonic scales had never been invented or had never been so popular, but assuming only that we agreed we wanted to reuse the same nominals for pitches an octave apart, we would still find that the fifth was the best overall choice of notational generator simply because it has the simplest rational value possible within the octave. Then, given human short-term memory, and the good Dr Miller to tell us about "The magical number 7 plus or minus 2", we would still have chosen 7 nominals, although we might have been tempted to use 12. Carl:

> >> But also in that thread, I assert that by forcing 7 nominals, you > >> ruin the mapping from commas to accidentals. Care to rebut that?

George:

> >I answer with the following: In a Pythagorean sequence of tones > >(with 7 nominals), the position in the sequence determines the power > >of 3, and the prime factors above 3 may all be expressed in the > >accidentals. In a sequence of tones separated by some other > >generator (such as a secor), the prime factors contained or implied > >in the relationships of those tones vary from one tone to another, so > >any comma-to-accidental relationship is much less useful (and much > >more complicated.)

Carl:

> I disagree. In the meantone diatonic scale, the position in the > sequence of nominals tells you the powers of 3 and 5. There's > one accidental, representing 25:24.

But these are not uniquely defined since syntonic commas may be included or removed at will, and in any case, sagittal is not based on meantone.

Message: 7462 Date: Tue, 30 Sep 2003 17:11:59 Subject: Re: Please remind me From: Manuel Op de Coul

>Graham? I know how to do one unison vector vanishing, like (1.5/ >(81/80)^7/26 for meantone, using RMS. How is it done for two unison >vectors? Thanks!

There isn't one formula because you can temper out unison vectors in different ways. With Scala you can calculate a generator however, and with Graham's programs too. One way is to create a lattice big enough so it covers the unison vectors (using the EULERFOKKER command for example). Then temper out the unison vectors with PROJECT/TEMPER. Don't forget to add 2/1 as a unison vector otherwise it will also be tempered. Then do UNIQUE and then SHOW DATA will give you a generator. Another way is with the CALCULATE/LEASTSQUARE command. You give it a set of intervals to approximate and their mapping. Then the generator is given directly, both for RMS and minimax. Manuel

Message: 7463 Date: Tue, 30 Sep 2003 01:10:31 Subject: Re: Sagittal accidentals for Decimal/MIRACLE notation From: Carl Lumma

>To notate rational pitches (e.g. JI), we must choose a rational value >for our notational generator,

Why?

>and it should be as simple a ratio as >possible (2:3).

Why?

>What value of secor would one choose for notating >rational pitches in decimal?

Huh? Steps to make a notation: 1. Pick your nominals and put them on a lattice. This is equivalent to selecting a basic scale as far as I'm concerned. 2. Pick accidentals that allow your basic scale to tile the lattice. If your nominals correspond to a linear temperament, the value of the generator shouldn't matter within the bounds set by the mapping to primes that your tiling is based upon.

>This looks very messy to me, but maybe Carl and Graham can work on it >and come up with a list of commas they would like to use for various >purposes, and we can supply the sagittal symbols for them.

Just work your way down any of the comma lists that Gene and Paul have published.

>So I now see that, even if diatonic scales had never been invented or >had never been so popular, but assuming only that we agreed we wanted >to reuse the same nominals for pitches an octave apart, we would still >find that the fifth was the best overall choice of notational >generator simply because it has the simplest rational value possible >within the octave.

Why? As with temperaments, the simplicity of the overall map is the important thing. Using the generators to hit primes seems a good strategy for reducing complexity, but we still see a host of excellent temperaments with half-octave generators; in MIRACLE itself the secor isn't a rational approximation.

>George:

>> >I answer with the following: In a Pythagorean sequence of tones >> >(with 7 nominals), the position in the sequence determines the power >> >of 3, and the prime factors above 3 may all be expressed in the >> >accidentals. In a sequence of tones separated by some other >> >generator (such as a secor), the prime factors contained or implied >> >in the relationships of those tones vary from one tone to another, so >> >any comma-to-accidental relationship is much less useful (and much >> >more complicated.)

> >Carl:

>> I disagree. In the meantone diatonic scale, the position in the >> sequence of nominals tells you the powers of 3 and 5. There's >> one accidental, representing 25:24.

> >But these are not uniquely defined since syntonic commas may be >included or removed at will, and in any case, sagittal is not based >on meantone.

I didn't say it was. I was providing a counterexample to George's statement that the simplicity of the accidental system depends on the generator. It depends on the entire map. -Carl

Message: 7465 Date: Tue, 30 Sep 2003 21:50:03 Subject: Re: Please remind me From: Graham Breed Paul G Hjelmstad wrote:

>Graham? I know how to do one unison vector vanishing, like (1.5/ >(81/80)^7/26 for meantone, using RMS. How is it done for two unison >vectors? Thanks! >vanishing? > >

I don't know what yo want to do, but here's how you construct a linear temperament from two unison vectors:

>>> temper.temperOut((

... temper.factorizeRatio(81,80), ... temper.factorizeRatio(64,63))) 0/1, 1902.9 cent generator basis: (1.0, 1.5857621954716261) mapping by period and generator: [(1, 0), (0, 1), (-4, 4), (6, -2)] mapping by steps: [(1, 0), (-1, 1), (-8, 4), (8, -2)] highest interval width: 6 complexity measure: 6 (7 for smallest MOS) highest error: 0.021121 (25.345 cents) Graham

Message: 7467 Date: Tue, 30 Sep 2003 14:52:37 Subject: hey gene From: Carl Lumma What's this:

># h2 scale blocks > >cm1 := [9/7, 6/5, 8/7]; >c1 := [[-1, 0, 0], [-1, 0, 1], [-1, 1, 1], [0, 0, 0]]; >s1 := [1, 15/14, 6/5, 5/4, 9/7, 3/2, 12/7, 7/4];

And did anything ever become of this: FreeLists / tuning-math / [tuning-math] 38 lin... * [with cont.] (Wayb.) Oh, and I don't see anything explaining TM reduction on your website, or anywhere else for that matter. -Carl

Message: 7468 Date: Tue, 30 Sep 2003 21:54:51 Subject: Re: Please remind me From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

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

> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" > > <paul.hjelmstad@u...> wrote:

> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich"

<perlich@a...>

> > > wrote:

> > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" > > > > <paul.hjelmstad@u...> wrote: > > > >

> > > > > I see. In the 7-limit for meantone, is 126/125 used instead

> of

> > > > >81/80? > > > > > Or are they both used somehow...

> > > > > > > > they are both used, and together they form a basis for the

> kernel

> > > of

> > > > 7-limit meantone. you need two independent vanishing unison

> > vectors

> > > > to get a linear temperament out of a 3-dimensional space like

7-

> > > limit

> > > > JI, since 3-2=1 (this language ignores the 'octave'

dimension).

> > > > > > Interesting. Could you give me the formula (or point me to a

> > message)

> > > that would extract a generator for a linear temperament like

this

> > > (that uses 2 unison vectors vanishing...)

> > > > i don't even know the formula in the case of only 1 unison vector > > vanishing! i've seen some of your posts, but what happens in

cases

> > where the period comes out to a fraction of an octave (such as > > 2048:2025, where the period comes out to 1/2 octave)? you can > > probably handle this, i just wasn't paying enough attention :(

> > Graham? I know how to do one unison vector vanishing, like (1.5/ > (81/80)^7/26 for meantone, using RMS.

in order to do that, you need to know the mapping from generators to primes first!

> How is it done for two unison > vectors?

once you know the mapping from generators to primes, it's no different, you can use rms or whatever in exactly the same way. i've done a huge number of such calculations for the case of 50:49 and 64:63 vanishing, on the main tuning list . . .

Message: 7469 Date: Tue, 30 Sep 2003 15:02:19 Subject: Re: Canonical homomorphisms revisted From: Carl Lumma Gene, Any news on this front? Might it be useful for the notation effort? -Carl

>From: "Gene Ward Smith" <gwsmith@xxxxx.xxx> >Date: Sun, 13 Apr 2003 >Subject: [tuning-math] Canonical homomorphisms > >The problem with my first definition is that it works beautifully when >it works, but it doesn't always work (1/2 time in the 5-limit, 1/4 of >the time in the 7-limit, and so forth.) Taking it as a source of >inspiration, here is a definition which works in general. >Unfortunately it no longer is as slick as goose grease. > >If T is a temperament, call q a "subgroup comma" for T if q>1 is not a >power of anything else, and if only three primes are involved in its >factorization (these commas are easily found from the wedgie.) Call a >prime "reducible" if it appears by itself in either numerator or >denominator for the factorization of a subgroup comma. If T is >p-limit, let P be the set of primes <= p, and let R be the set of >reducible primes. Then N = P\R is the set of non-reducible primes. If >card(N) = g, where g is the number of generators (2 for a linear >temperament, 3 for a planar temperament, and so forth) then set G = N. >If card(N)<g, then fill it out by adding the smallest of the remaining >primes, and set the result to G; if card(N)>g, then reduce it by >removing the largest of the primes in N, and call that G. We end with >a set G of primes such that card(G)=g, and we may use G as a set of >generators for the temperament T. > >Strange as it may seem, this definition actually corresponds to my >previous one in those cases where the previous one gives a >homomorphism. It also seems to give us reasonable results, or at least >so it seems to me, YMMV. Here is what we get for meantone and miracle; >I give the mapping of the generators (2 and 3/2 in the case of >meantone, 2 and 16/15 in the case of miracle), and the mapping applied >to primes: > >5-limit meantone 81/80 >Generators [2, 5^(1/4)] >Prime mapping [2, 2*5^(1/4), 5] > >7-limit meantone [1, 4, 10, 4, 13, 12] >Generators [2, 2^(3/10)*7^(1/10)] >Prime mapping [2, 2*2^(3/10)*7^(1/10), 2*2^(1/5)*7^(2/5), 7] > >11-limit meantone [1, 4, 10, 18, 4, 13, 25, 12, 28, 16] >Generators [2, 2^(3/10)*7^(1/10)] >Prime mapping [2, 2*2^(3/10)*7^(1/10), 2*2^(1/5)*7^(2/5), 7, >7/4*2^(2/5)*7^(4/5)] > >11-limit meanpop [1, 4, 10, -13, 4, 13, -24, 12, -44, -71] >Generators [7^(13/71)*11^(10/71), 7^(11/71)*11^(3/71)] >Prime mapping [7^(13/71)*11^(10/71), 7^(24/71)*11^(13/71), >7^(44/71)*11^(12/71), 7, 11] > >5-limit miracle 34171875/33554432 >Generators [3^(7/25)*5^(6/25), 1/5*3^(3/25)*5^(24/25)] >Prime mapping [3^(7/25)*5^(6/25), 3, 5] > >7-limit miracle [6, -7, -2, -25, -20, 15] >Generators [3^(7/25)*5^(6/25), 1/5*3^(3/25)*5^(24/25)] >Prime mapping [3^(7/25)*5^(6/25), 3, 5, 3^(3/5)*5^(4/5)] > >11-limit miracle [6, -7, -2, 15, -25, -20, 3, 15, 59, 49] >Generators [5^(15/59)*11^(7/59), 1/5*5^(57/59)*11^(3/59)] >Prime mapping [5^(15/59)*11^(7/59), 5^(3/59)*11^(25/59), 5, >5^(49/59)*11^(15/59), 11]

Message: 7470 Date: Tue, 30 Sep 2003 22:08:53 Subject: Re: hey gene From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Oh, and I don't see anything explaining TM reduction on > your website, or anywhere else for that matter. > > -Carl

it's been explained repeatedly here. t stands for tenney, m for minkowski.

Message: 7471 Date: Tue, 30 Sep 2003 15:51:32 Subject: Re: hey gene From: Carl Lumma

>> Oh, and I don't see anything explaining TM reduction on >> your website, or anywhere else for that matter. >> >> -Carl

> >it's been explained repeatedly here. t stands for tenney, m for >minkowski.

Yes, Paul, I know that. And I've just re-read the threads all through the LLL -> TM stuff, but I still haven't found any of the "repeated" explanations. -Carl

Message: 7472 Date: Tue, 30 Sep 2003 16:52:44 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> What gets wrecked if one's basis isn't distrib. even?

> >you yourself said that one begins a set of nominals forming a >periodicity block (on the tuning-math list; maybe you should reply to >this over there).

What does distrib. evenness have to do with PBs?

>that won't work once you get beyond the set of primes that you're >allowing the linear temperament to approximate.

Obviously.

>of course, george and >dave's proposal is not based on any temperament at all.

I don't mean to suggest that notations should be based on temperaments. But I do assert that the problem of finding notations is equivalent to the problem of finding PBs. -Carl

Message: 7473 Date: Tue, 30 Sep 2003 23:57:27 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> What gets wrecked if one's basis isn't distrib. even?

> > > >you yourself said that one begins a set of nominals forming a > >periodicity block (on the tuning-math list; maybe you should reply

to

> >this over there).

> > What does distrib. evenness have to do with PBs?

a fokker periodicity block, when all but one of the unison vectors are tempered out, becomes a distributionally even scale. that's my Hypothesis, anyway. i mentioned *altered* versions of DE scales because one may wish to start with a "non-fokker" periodicity block.

Message: 7474 Date: Tue, 30 Sep 2003 17:02:44 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma At 04:57 PM 9/30/2003, you wrote:

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

>> >> What gets wrecked if one's basis isn't distrib. even?

>> > >> >you yourself said that one begins a set of nominals forming a >> >periodicity block (on the tuning-math list; maybe you should reply

>to

>> >this over there).

>> >> What does distrib. evenness have to do with PBs?

> >a fokker periodicity block, when all but one of the unison vectors >are tempered out, becomes a distributionally even scale. that's my >Hypothesis, anyway.

So distrib. even and MOS are equivalent?

>i mentioned *altered* versions of DE scales because one may wish to >start with a "non-fokker" periodicity block.

What's a non-fokker PB -- a shape other than a parallelepiped? -Carl

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