This is an
**
Opt In Archive
.
**
We would like to hear from you if you want your posts included. For the contact address see
About this archive. All posts are copyright (c).

5000
5050
5100
5150
5200
5250
5300
5350
**5400**
5450
5500
5550
5600
5650
5700
5750
5800
5850
5900
5950

**5400 -**
5425 -

Message: 5400 - Contents - Hide Contents Date: Tue, 22 Oct 2002 05:03:53 Subject: Re: NMOS From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote: >>> Has anyone paid attention to scales which have a number of steps a >> multiple of a MOS? >> the torsional scales do!I meant a chain of generators where the number of generators is a multiple of a number giving a MOS--or in other words, is a multiple of something arising from a semiconvergent.

Message: 5401 - Contents - Hide Contents Date: Tue, 22 Oct 2002 11:54:45 Subject: Re: Epimorphic From: Gene Ward Smith --- In tuning-math@y..., manuel.op.de.coul@e... wrote:> So doesn't that need to be added to the definition > in Joe's dictionary too, since all components being > integer doesn't follow automatically from > h(qn)=n for n = 1 .. (number of notes - 1)?It follows automatically from the definition of a val, which is also in Joe's dictionary: Yahoo groups: /monz/files/dict/val.htm * [with cont.] I've also called it a finite Z-linear combination of p-adic valuations defined additively, but I doubt that helps anybody.

Message: 5402 - Contents - Hide Contents Date: Tue, 22 Oct 2002 05:07:13 Subject: Re: Epimorphic From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> 2401/2400 is not between 5/3 and 2/1, gene!OK, so be picky about it. :) 2401/1280 is, though. I intended to put up a slightly modified version of 1--9/8--5/4--4/3--3/2--5/3--15/8--2, with the 9/8 adjusted down by 2400/2401 and the 15/8 adjusted up by the same amount. Since h7(2401/2400)=2, this throws a spanner in the works.

Message: 5403 - Contents - Hide Contents Date: Tue, 22 Oct 2002 11:55:49 Subject: Re: Epimorphic From: Gene Ward Smith --- In tuning-math@y..., manuel.op.de.coul@e... wrote:> So I may conclude that the simplest example of a JI, > CS and non-epimorphic scale is this one: 1/1--4/1If you want to call that a scale.

Message: 5404 - Contents - Hide Contents Date: Tue, 22 Oct 2002 05:11:49 Subject: Fwd: [tuning] Re: Everyone Concerned From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> i think it's still a lattice!If you take a lattice (p-limit, let's say) and then take a sublattice (defined by commas, lets say) of the same rank n, if you mod out R^n by the sublattice you get a discrete group on the compact quotient. As I remarked, you should not assume every mathematican would call this a lattice! Of couse the rule is that you can change definitions as long as you make it clear what you are doing.

Message: 5406 - Contents - Hide Contents Date: Tue, 22 Oct 2002 05:17:04 Subject: Fwd: [tuning] Re: Everyone Concerned From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> i recall a certain list member who at first insisted that all words > should be understood according to their "math" definitions, ignoring > that fact that many of the same words already had "music" definitions > which should take precedence at least on the tuning list.Eh, I think you remember wrong. There are two completely different *math* definitions in very common use in mathematics. Obviously, I wouldn't suggest using both of them, or the wrong one.

Message: 5407 - Contents - Hide Contents Date: Tue, 22 Oct 2002 12:47:17 Subject: Re: epimorphic From: Pierre Lamothe Carl wrote: Pierre didn't say that. He said Epimorphic -> CS, which is exactly what Gene said. Clearly CS /-> Epimorphic. See my post in this thread. Sorry Carl, it's not what I said. I justly wrote the first post in the "CS implies EPIMORPHISM" thread for I saw much ambiguities in many posts about CS and epimorphic. Gene has defined epimorphic for a scale as "... there is a val h such that if qn is the nth scale degree, then h(qn) = n " and a val (in a context of rational tone group) as "... an homomorphism from the tone group to the integers. How are interconnected, by definition, scale and tone group ? Subgroup, subset, periodicity block ?? I used simply the well-known mathematical term EPIMORPHISM as surjective morphism and I shown: If a scale S has the CS property then there exist an epimorphism D applying each interval x, in the space of all intervals spanned by S, onto an integer corresponding to its scale degree, not only in S but in any derived scale S' obtained by tonic rotation and/or duality. Epimorphism don't imply CS I hope the following counterexample will suffice. It is well-known that the Zarlino scale is both CS and epimorphic. The unison vectors 81/80 (about 22 cents) and 25/24 (about 70 cents) generate the kernel of that epimorphism. It is very easy to transform that CS scale in a non-CS scale but respecting that epimorphism. The Zarlino degree 6 == 15/8 has approximately 1088 cents. It misses 112 cents to reach the octave. If you multiply 15/8 by a combination of unison vectors like 81/80 and 25/24, you dont change the class of the epimorphism, since class 6 + n (class 0) = class 6. If you add at 1088 cents, for instance, two comma of 22 cents and a chroma of 70 cents, the result is a degree 6 which is about 2 cents over the octave. If you don't see immediately that a such scale forcely reordered 0 2 204 386 498 702 884 1200 is non-CS, consider that 204 is subtended by 2 steps between 0 and 204 and only one step between 498 (4/3) and 702 (3/2). Pierre [This message contained attachments]

Message: 5408 - Contents - Hide Contents Date: Tue, 22 Oct 2002 05:50:20 Subject: Re: Digest Number 497 From: Gene Ward Smith --- In tuning-math@y..., John Chalmers <JHCHALMERS@U...> wrote:> Gene asked: >>> Has anyone paid attention to scales which have a number of steps a >> multiple of a MOS? They inherit structure from the MOS, and using a 2MOS >> or a 3MOS seems like a good way to fill in those annoying gaps.Thanks for your reply, which was interesting. I seem to have sowed confusion with my question, however, so let me give an example. Orwell has a 13-tone MOS which in its 19/84 version is [3, 8, 8, 3, 8, 8, 3, 8, 8, 3, 8, 8, 8] It also has a 22-tone MOS, [3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 3, 5, 3, 5, 3, 5] This is all well an good, but what if you want a scale of consecutive generators right in the middle of this fairly large gap? Consider the 9-tone MOS: [11, 8, 11, 8, 11, 8, 11, 8, 8] If we replace every "11" by a "3, 8" and every "8" by a "3, 5" we get the 18-tone 2MOS: [3, 8, 3, 5, 3, 8, 3, 5, 3, 8, 3, 5, 3, 8, 3, 5, 3, 5] This, it seems to me, is slightly but discernably more regular than its neighbors, of 17 and 19 generators. 17: [3, 8, 3, 5, 3, 8, 3, 5, 3, 8, 3, 5, 3, 8, 8, 3, 5] 19: [3, 3, 5, 3, 5, 3, 8, 3, 5, 3, 8, 3, 5, 3, 8, 3, 5, 3, 5] I would say it also is more regular than the 3MOS of 15 tones or the 4MOS of 16 tones. 15: [3, 8, 3, 5, 3, 8, 8, 3, 8, 8, 3, 8, 8, 3, 5] 16: [3, 8, 3, 5, 3, 8, 3, 5, 3, 8, 8, 3, 8, 8, 3, 5]

Message: 5409 - Contents - Hide Contents Date: Tue, 22 Oct 2002 19:38:31 Subject: Re: Digest Number 497 From: wallyesterpaulrus --- In tuning-math@y..., John Chalmers <JHCHALMERS@U...> wrote:> Gene asked: >>> Has anyone paid attention to scales which have a number of steps a >> multiple of a MOS? They inherit structure from the MOS, and using a 2MOS >> or a 3MOS seems like a good way to fill in those annoying gaps. >> I think most of Messiaien's "Modes of Limited Transposition" in 12- tet > are multiple MOS's of 3, 4 and 6-tet. I don't have a list handy on this > computer to check, unfortunately. IIRC, William Lyman Young (in his > "Report > to the Swedish Royal Academy of Music" etc.) proposed a decatonic scale > in > 24-tet which was two 5-tone MOS's of 12 (2322323223) and a 14-tone scale > of 2 sections of the 7-tone diatonic sequence as 22122212212221 in > 24-tet. > He considered these as generated from cycles of half-fourths or > half-fiths. > > I suspect that some of Wyschnegradski's scales might be multiple MOS's > too, > but I don't have a list either. > > --Johnjohn, most of these scales are themselves MOSs. what gene was after, i think, was multiple-MOSs that are not themselves MOSs, such as a 24- note meantone or pythagorean chain.

Message: 5410 - Contents - Hide Contents Date: Tue, 22 Oct 2002 06:39:10 Subject: Re: NMOS From: Carl Lumma>>> >as anyone paid attention to scales which have a number of >>> steps a multiple of a MOS? >>>> the torsional scales do! >>I meant a chain of generators where the number of generators is >a multiple of a number giving a MOS--or in other words, is a >multiple of something arising from a semiconvergent.How would the multiple property justify itself againt scales that were two MOSs superposed at some other interval (besides the comma)? In the case of Messiaien, the octatonic scale is an NMOS. And as pointed out here before, it becomes Blackwood's decatonic in 15-tET. For the interlaced diatonic scales in 24-tET, Paul has pointed out that this has excellent 7-limit harmony in 26. I forget at what interval this is, but I don't think it's the comma. But Paul's excellent decatonics in 22 are two pentatonic MOSs apart by a non-comma (the half-octave). In short, regular double-period linear temperaments, or torsional ones, or whatever (I haven't been following) is so far as I can see the strongest constraint justified here. -Carl

Message: 5411 - Contents - Hide Contents Date: Tue, 22 Oct 2002 19:39:15 Subject: Re: NMOS From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote: >>>>> Has anyone paid attention to scales which have a number of steps a >>> multiple of a MOS? >>>> the torsional scales do! >> I meant a chain of generators where the number of generators is a >multiple of a number giving a MOS--or in other words, is a multiple >of something arising from a semiconvergent.yup, the torsional scales do this.

Message: 5412 - Contents - Hide Contents Date: Tue, 22 Oct 2002 07:32:22 Subject: Re: NMOS From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> How would the multiple property justify itself againt scales > that were two MOSs superposed at some other interval (besides > the comma)?What's that again? Did you look at my example--its an example where the period is the octave; no octatonics or decatonics need apply.

Message: 5413 - Contents - Hide Contents Date: Tue, 22 Oct 2002 19:40:24 Subject: Fwd: [tuning] Re: Everyone Concerned From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >>> i think it's still a lattice! >> If you take a lattice (p-limit, let's say) and then take a >sublattice (defined by commas, lets say) of the same rank n, if you >mod out R^n by the sublattice you get a discrete group on the >compact quotient. As I remarked, you should not assume every >mathematican would call this a lattice! Of couse the rule is that >you can change definitions as long as you make it clear what you are >doing.how about the term "point lattice", which is what mathworld suggests?

Message: 5414 - Contents - Hide Contents Date: Tue, 22 Oct 2002 10:40:15 Subject: Re: Epimorphic From: manuel.op.de.coul@xxxxxxxxxxx.xxx Gene wrote:>Is that what I wrote? It should be >1/1--2700/2401--5/4-4/3--3/2--5/3--2401/1280--2/1My routine claims this scale is epimorphic. Being puzzled, I found out why. The val was [7, 11, 16, 19.5] and not the [7, 11, 16, 20] which was printed out. So, do the components need to be integer, if so, why? You said that h7(2401/2400)=2, but 2401/2400 isn't in the scale so this is irrelevant? Manuel

Message: 5415 - Contents - Hide Contents Date: Tue, 22 Oct 2002 19:42:58 Subject: Fwd: [tuning] Re: Everyone Concerned From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >>> i recall a certain list member who at first insisted that all words >> should be understood according to their "math" definitions, ignoring >> that fact that many of the same words already had "music" definitions >> which should take precedence at least on the tuning list. >> Eh, I think you remember wrong. There are two completely different >*math* definitions in very common use in mathematics. Obviously, I >wouldn't suggest using both of them, or the wrong one.i wasn't talking about "lattice" in the paragraph above. did you think i was?

Message: 5416 - Contents - Hide Contents Date: Tue, 22 Oct 2002 10:43:41 Subject: Re: Epimorphic From: manuel.op.de.coul@xxxxxxxxxxx.xxx Carl wrote:>The stronger argument against CS /-> Epimorphic is >that CS doesn't require JI, as Gene pointed out.Right, I was thinking that all scales being both CS and RI are epimorphic. Now we still need a watertight definition of epimorphic. I must have misunderstood Pierre's definition of it. Manuel

Message: 5417 - Contents - Hide Contents Date: Tue, 22 Oct 2002 19:43:56 Subject: Fwd: [tuning] Re: Everyone Concerned From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >>> i think it's still a lattice! >> If you take a lattice (p-limit, let's say) and then take a >sublattice (defined by commas, lets say) of the same rank n, if you >mod out R^n by the sublattice you get a discrete group on the >compact quotient. As I remarked, you should not assume every >mathematican would call this a lattice!what's the musical analogue? i'm lost.>Of couse the rule is that >you can change definitions as long as you >make it clear what you are >doing.how about the term "point lattice", which mathworld suggests?

Message: 5418 - Contents - Hide Contents Date: Tue, 22 Oct 2002 09:17:14 Subject: Re: Epimorphic From: Gene Ward Smith --- In tuning-math@y..., manuel.op.de.coul@e... wrote:> Gene wrote:>> Is that what I wrote? It should be >> 1/1--2700/2401--5/4-4/3--3/2--5/3--2401/1280--2/1 >> My routine claims this scale is epimorphic. Being > puzzled, I found out why. The val was [7, 11, 16, 19.5] > and not the [7, 11, 16, 20] which was printed out. > So, do the components need to be integer, if so, why?It needs to define an epimorphic mapping to the integers; that means the components must be integers, and the gcd must be 1. The second part your routine presumably takes care of automatically, but it means that you can't multiply the above mapping by 2 and get [14, 22, 32, 40].> You said that h7(2401/2400)=2, but 2401/2400 isn't in the > scale so this is irrelevant?It's not an accident that your mapping tempers out 2401/2400, since this is what I mundged by. However, a more convincing example is one where there isn't a mapping, for instance ! nonepi.scl ! Non epimorphic scale 7 ! 2701/2400 5/4 4/3 98415/65536 5/3 2401/1280 2/1

Message: 5419 - Contents - Hide Contents Date: Tue, 22 Oct 2002 19:47:50 Subject: Re: NMOS From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>>> Has anyone paid attention to scales which have a number of >>>> steps a multiple of a MOS? >>>>>> the torsional scales do! >>>> I meant a chain of generators where the number of generators is >> a multiple of a number giving a MOS--or in other words, is a >> multiple of something arising from a semiconvergent. >> How would the multiple property justify itself againt scales > that were two MOSs superposed at some other interval (besides > the comma)?huh? did you mean the generator?> In the case of Messiaien, the octatonic scale is an NMOS.it's also an MOS, plain and simple.> And as pointed out here before, it becomes Blackwood's > decatonic in 15-tET. another MOS. > For the interlaced diatonic scales in > 24-tET, Paul has pointed out that this has excellent 7-limit > harmony in 26. I forget at what interval this is, but I > don't think it's the comma. the comma? > But Paul's excellent decatonics in 22 are two pentatonic MOSs > apart by a non-comma (the half-octave).these are the symmetrical decatonics, and they _are_ MOSs. the pentachordal decatonics aren't. the same goes for the 14-note scales in 26 -- the symmetrical ones are MOS, the "tetrachordal" ones aren't.> In short, regular > double-period linear temperaments, or torsional ones, or > whatever (I haven't been following)well, those are two different things . . .

Message: 5420 - Contents - Hide Contents Date: Tue, 22 Oct 2002 19:51:58 Subject: Re: Epimorphic From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., manuel.op.de.coul@e... wrote:>> So I may conclude that the simplest example of a JI, >> CS and non-epimorphic scale is this one: 1/1--4/1 >> If you want to call that a scale.i'm confused. are we assuming a 2/1 interval of equivalence? then wouldn't be a scale at all, would it?

Message: 5421 - Contents - Hide Contents Date: Tue, 22 Oct 2002 00:20:25 Subject: Fwd: [tuning] Re: Everyone Concerned From: wallyesterpaulrus --- In tuning-math@y..., <Josh@o...> wrote:> Maybe I misunderstood the issue. > > Where I come from, if it looks like a lattice > and it quacks like a lattice, we call it a lattice. > > What is the reason people are objecting to this?in the field of mathematics known as abstract algebra, the term "lattice" is used to signify something completely different. i recall a certain list member who at first insisted that all words should be understood according to their "math" definitions, ignoring that fact that many of the same words already had "music" definitions which should take precedence at least on the tuning list. of course this is somewhat different, few musicians use the term "lattice" except of course just intonation theorists and the like . . . so i guess there already is a "music" definition of lattice and the issue should be moot. that's why i found it odd that monz was bringing it up at this late date!

Message: 5422 - Contents - Hide Contents Date: Tue, 22 Oct 2002 01:40:22 Subject: Re: Epimorphic From: Carl Lumma>> >t turns out the question was moot since Pierre showed that it's >> equivalent to CS. >>Not so far as I can see.Pierre didn't say that. He said Epimorphic -> CS, which is exactly what Gene said. Clearly CS /-> Epimorphic. See my post in this thread. -Carl

Message: 5423 - Contents - Hide Contents Date: Wed, 23 Oct 2002 12:13:09 Subject: Re: NMOS From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> I don't see how the symmetrical decatonics can be MOS, since > they don't have Myhill's property.They look awfully Myhill to me.

Message: 5424 - Contents - Hide Contents Date: Wed, 23 Oct 2002 12:21:29 Subject: Fwd: [tuning] Re: Everyone Concerned From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:>> If you take a lattice (p-limit, let's say) and then take a >> sublattice (defined by commas, lets say) of the same rank n, if you >> mod out R^n by the sublattice you get a discrete group on the >> compact quotient. As I remarked, you should not assume every >> mathematican would call this a lattice! >> what's the musical analogue? i'm lost.A 5-limit octave-reduced Fokker block painted on a donut, hopefully using something edible.>> Of couse the rule is that >you can change definitions as long as you >> make it clear what you are >doing. >> how about the term "point lattice", which mathworld suggests?Why not? It doesn't get used that much, but we needn't let that worry us. Will we become confused by the question of whether an array of points is a point lattice if it is not a subgroup of R^n?

5000
5050
5100
5150
5200
5250
5300
5350
**5400**
5450
5500
5550
5600
5650
5700
5750
5800
5850
5900
5950

**5400 -**
5425 -