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Message: 0 - Contents - Hide Contents Date: Mon, 21 May 2001 04:54:45 Subject: Hypothesis From: paul@s... No one responded to my Hypothesis on the tuning list. Search for "hypothesis".
Message: 1 - Contents - Hide Contents Date: Mon, 21 May 2001 04:54:45 Subject: Hypothesis From: paul@s... No one responded to my Hypothesis on the tuning list. Search for "hypothesis".
Message: 3 - Contents - Hide Contents Date: Mon, 21 May 2001 11:32:45 Subject: Re: Hypothesis From: Graham Breed Paul wrote:> No one responded to my Hypothesis on the tuning list. Search for "hypothesis".If you accidentally hit Ctrl-W in IE, the window you're typing in disappears. What a crock! So, I'll have to start this again. Hypotheses got mentioned a lot, so this is the relevant post: <Yahoo groups: /tuning/message/22135 * [with cont.] > "Hypothesis: If you temper out all but one of the unison vectors in a periodicity block, you get a distributionally even scale." Cross-reference with the tuning dictionary <Definitions of tuning terms: JI, (c) 1998 by J... * [with cont.] (Wayb.)> """ distributional evenness The scale has no more than two sizes of interval in each interval class. """ Does this mean the hyperparallelopiped has to become the distributionally even scale? Or only that the relevant linear temperament with that number of notes can be distributionally even? The example of the 24 note periodicity block from the schisma and diesis might be a counterexample. |-8 -1| | 0 -3| It won't be an MOS anyway, and I think Carey and Clampitt showed that an MOS is always distributionally even. Graham
Message: 4 - Contents - Hide Contents Date: Mon, 21 May 2001 17:43:09 Subject: Re: Hypothesis From: paul@s... --- In tuning-math@y..., "Graham Breed" <graham@m...> wrote:> Does this mean the hyperparallelopiped has to become the > distributionally even scale?Not necessarily -- I was thinking more along the lines of, the form of the periodicity block with the most consonances.> Or only that the relevant linear > temperament with that number of notes can be distributionally even?It might not be a linear temperament!> > The example of the 24 note periodicity block from the schisma and > diesis might be a counterexample. > > |-8 -1| > | 0 -3| > > It won't be an MOS anyway,Uhh . . . which unison vector are you tempering out?> and I think Carey and Clampitt showed that > an MOS is always distributionally even.But not all distributionally even scales are MOS!
Message: 5 - Contents - Hide Contents Date: Mon, 21 May 2001 18:59 +0 Subject: Re: Hypothesis From: graham@m... In-Reply-To: <9ebk3d+5dqs@e...> Paul wrote:> --- In tuning-math@y..., "Graham Breed" <graham@m...> wrote: >>> Does this mean the hyperparallelopiped has to become the >> distributionally even scale? >> Not necessarily -- I was thinking more along the lines of, the form > of the periodicity block with the most consonances.How would that relate to the unison vectors?>> Or only that the relevant linear >> temperament with that number of notes can be distributionally even? >> It might not be a linear temperament!One fewer unison vectors than you need for an ET will always give a linear temperament of some kind. That follows from my matrix definitions.>> The example of the 24 note periodicity block from the schisma and >> diesis might be a counterexample. >> >> |-8 -1| >> | 0 -3| >> >> It won't be an MOS anyway, >> Uhh . . . which unison vector are you tempering out?I don't know, I haven't tried. How are you defining "interval class"?>> and I think Carey and Clampitt showed that >> an MOS is always distributionally even. >> But not all distributionally even scales are MOS!But if we could prove that all linear temperaments give something like an MOS, that would prove the hypothesis. Graham
Message: 6 - Contents - Hide Contents Date: Mon, 21 May 2001 18:33:54 Subject: Re: Hypothesis From: paul@s... --- In tuning-math@y..., graham@m... wrote:>> >> Not necessarily -- I was thinking more along the lines of, the form >> of the periodicity block with the most consonances. >> How would that relate to the unison vectors?For example, the melodic minor scale and the major scale are both periodicity blocks of the unison vectors 25:24 and 81:80, with the 81:80 tempered out. But the melodic minor scale is not distributionally even. The major scale has more consonances . . .>>>> Or only that the relevant linear >>> temperament with that number of notes can be distributionally even? >>>> It might not be a linear temperament! >> One fewer unison vectors than you need for an ET will always give a linear > temperament of some kind. That follows from my matrix definitions.Something must be wrong with your definitions then. For example, my decatonic system comes from the unison vectors 64:63, 50:49, and 49:48, with 64:63 and 50:49 tempered out. But it's not represented by any linear temperament. However, the decatonic with the most consonances is distributionally even.>>> The example of the 24 note periodicity block from the schisma and >>> diesis might be a counterexample. >>> >>> |-8 -1| >>> | 0 -3| >>> >>> It won't be an MOS anyway, >>>> Uhh . . . which unison vector are you tempering out? >> I don't know, I haven't tried. How are you defining "interval class"?Where did I use that term?>>>> and I think Carey and Clampitt showed that >>> an MOS is always distributionally even. >>>> But not all distributionally even scales are MOS! >> But if we could prove that all linear temperaments give something like an > MOS, that would prove the hypothesis.Something _like_ an MOS, yes.
Message: 7 - Contents - Hide Contents Date: Mon, 21 May 2001 15:03:06 Subject: Re: Hypothesis From: Joseph Pehrson Thanks, Paul, for inviting me to participate in your new group! Good luck with it! Joseph>From: paul@s... >Reply-To: tuning-math@xxxxxxxxxxx.xxx >To: tuning-math@xxxxxxxxxxx.xxx >Subject: [tuning-math] Hypothesis >Date: Mon, 21 May 2001 04:54:45 -0000 > >No one responded to my Hypothesis on the tuning list. Search for >"hypothesis". > > >To unsubscribe from this group, send an email to: >tuning-math-unsubscribe@xxxxxxxxxxx.xxx > > > >Your use of Yahoo! Groups is subject to Yahoo! Terms of Service * [with cont.] (Wayb.) > > _________________________________________________________________Get your FREE download of MSN Explorer at Explorer.MSN.com * [with cont.] (Wayb.)
Message: 8 - Contents - Hide Contents Date: Mon, 21 May 2001 19:11:01 Subject: Re: Hypothesis From: jpehrson@r... --- In tuning-math@y..., paul@s... wrote: Yahoo groups: /tuning-math/message/1 * [with cont.]> No one responded to my Hypothesis on the tuning list. Search for "hypothesis".I remember this interesting hypothesis mentioned on the Tuning List... So, I can take the "hint..." Somehow it can be "proven" mathematically... Where's Keenan or Walker?? ______ _____ ______ Joseph Pehrson
Message: 9 - Contents - Hide Contents Date: Tue, 22 May 2001 11:41 +0 Subject: Re: Hypothesis From: graham@m... In-Reply-To: <9ebn2i+9ksu@e...> Paul wrote:> --- In tuning-math@y..., graham@m... wrote: >>>>>> Not necessarily -- I was thinking more along the lines of, the > form>>> of the periodicity block with the most consonances. >>>> How would that relate to the unison vectors? >> For example, the melodic minor scale and the major scale are both > periodicity blocks of the unison vectors 25:24 and 81:80, with the > 81:80 tempered out. But the melodic minor scale is not > distributionally even. The major scale has more consonances . . .Does it matter which major scale we take? Or are we contracting the lattice to one dimension?>> One fewer unison vectors than you need for an ET will always give a > linear>> temperament of some kind. That follows from my matrix definitions. >> Something must be wrong with your definitions then. For example, my > decatonic system comes from the unison vectors 64:63, 50:49, and > 49:48, with 64:63 and 50:49 tempered out. But it's not represented by > any linear temperament. However, the decatonic with the most > consonances is distributionally even.It is, the equivalence interval's a half-octave and the generating interval's a fourth.>>>> |-8 -1| >>>> | 0 -3| >>>> >>>> It won't be an MOS anyway, >>>>>> Uhh . . . which unison vector are you tempering out? >>>> I don't know, I haven't tried. How are you defining "interval > class"? >> Where did I use that term?It's part of Monz's definition of "distributionally equal": "The scale has no more than two sizes of interval in each interval class.">> But if we could prove that all linear temperaments give something > like an>> MOS, that would prove the hypothesis. >> Something _like_ an MOS, yes.Usually that comes out fine. The unison vectors define a linear temperament, which forms an MOS with the right number of notes. Hopefully this will maximise the consonances (another hypothesis?). The diaschismic temperaments give an MOS with the half-octave as a generator. In general, for octave-equivalent unison vectors, the equivalence interval will always be some fraction of an octave. So the only outstanding problems are those temperaments where the determinant comes out as a multiple of the number of notes in the relevant ET. In that case, you get overcounting. The linear temperament can still be calculated, but not in its lowest terms. And the periodicity block contains twice as many notes as it needs to. It's not clear to me what's going on here, especially after one vector gets tempered out, but I'm sure the MOS concept can be expanded to cover it. Graham
Message: 10 - Contents - Hide Contents Date: Tue, 22 May 2001 18:58:32 Subject: Re: Hypothesis From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9ebn2i+9ksu@e...> > Paul wrote: >>> --- In tuning-math@y..., graham@m... wrote: >>>>>>>> Not necessarily -- I was thinking more along the lines of, the >> form>>>> of the periodicity block with the most consonances. >>>>>> How would that relate to the unison vectors? >>>> For example, the melodic minor scale and the major scale are both >> periodicity blocks of the unison vectors 25:24 and 81:80, with the >> 81:80 tempered out. But the melodic minor scale is not >> distributionally even. The major scale has more consonances . . . >> Does it matter which major scale we take? Or are we contracting the > lattice to one dimension?When the 81:80 is tempered out, the lattice curls into a cylinder. Then all major scales are identical.>>>> One fewer unison vectors than you need for an ET will always give a >> linear>>> temperament of some kind. That follows from my matrix definitions. >>>> Something must be wrong with your definitions then. For example, my >> decatonic system comes from the unison vectors 64:63, 50:49, and >> 49:48, with 64:63 and 50:49 tempered out. But it's not represented by >> any linear temperament. However, the decatonic with the most >> consonances is distributionally even. >> It is, the equivalence interval's a half-octave and the generating > interval's a fourth.OK! If that falls out of your matrix formalism, then let's go with it!> >>>>> |-8 -1|>>>>> | 0 -3| >>>>> >>>>> It won't be an MOS anyway, >>>>>>>> Uhh . . . which unison vector are you tempering out? >>>>>> I don't know, I haven't tried. How are you defining "interval >> class"? >>>> Where did I use that term? >> It's part of Monz's definition of "distributionally equal": "The scale > has no more than two sizes of interval in each interval > class."In this case, an interval class is the set of all intervals subtended by n consecutive scale degrees in a given scale, for some whole number n.>>>> But if we could prove that all linear temperaments give something >> like an>>> MOS, that would prove the hypothesis. >>>> Something _like_ an MOS, yes. >> Usually that comes out fine. The unison vectors define a linear > temperament, which forms an MOS with the right number of notes.Let's prove this.> Hopefully this will maximise the consonances (another hypothesis?).It would be good to prove that too, but I'm afraid it won't always work. It works if all the consonances come out to simple powers of the generator, though.> > The diaschismic temperaments give an MOS with the half-octave as a > generator.You mean, the half-octave as an equivalence interval?> In general, for octave-equivalent unison vectors, the > equivalence interval will always be some fraction of an octave.Great. But you can't take "equivalence interval" too far here -- the strongest consonances become dissonances when altered by the half- octave.> > So the only outstanding problems are those temperaments where the > determinant comes out as a multiple of the number of notes in the > relevant ET. In that case, you get overcounting. The linear temperament > can still be calculated, but not in its lowest terms. And the > periodicity block contains twice as many notes as it needs to. > > It's not clear to me what's going on here, especially after one vector > gets tempered out, but I'm sure the MOS concept can be expanded to cover > it.We'll figure it out!
Message: 11 - Contents - Hide Contents Date: Wed, 23 May 2001 01:40:11 Subject: the best tuning list From: monz Hey Paul, I *LOVE* THIS LIST!!!!!! Thanks for being wise enough to create it! -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0"
Message: 12 - Contents - Hide Contents Date: Wed, 23 May 2001 03:08:42 Subject: Re: the best tuning list From: jpehrson@r... --- In tuning-math@y..., "monz" <joemonz@y...> wrote: Yahoo groups: /tuning-math/message/11 * [with cont.]> > Hey Paul, > > > I *LOVE* THIS LIST!!!!!! > > > Thanks for being wise enough to create it! > > > -monz > Yahoo! GeoCities * [with cont.] (Wayb.) > "All roads lead to n^0"This is a really cool Tuning List, but so far, there is very little math on it... not that I would understand it, anyway... but it's always nice to look at, in MY opinion... _________ ______ _______ Joseph Pehrson
Message: 13 - Contents - Hide Contents Date: Wed, 23 May 2001 12:22 +0 Subject: Re: Hypothesis From: graham@m... In-Reply-To: <9eecso+amc3@e...> Paul wrote:>> It is, the equivalence interval's a half-octave and the generating >> interval's a fourth. >> OK! If that falls out of your matrix formalism, then let's go with it!You should always be able to define a linear temperament using two intervals. Finding the right two can be tricky.>> >>>>>> |-8 -1|>>>>>> | 0 -3| >>>>>> >>>>>> It won't be an MOS anyway, >>>>>>>>>> Uhh . . . which unison vector are you tempering out? >>>>>>>> I don't know, I haven't tried. How are you defining "interval >>> class"? >>>>>> Where did I use that term? >>>> It's part of Monz's definition of "distributionally equal": "The > scale>> has no more than two sizes of interval in each interval >> class." >> In this case, an interval class is the set of all intervals subtended > by n consecutive scale degrees in a given scale, for some whole > number n.Temper out the schisma from the periodicity block above. You end up with a 24-note schismic scale. No way can that have two step sizes! That looks like a refutation with the definitions I have. How about a weaker hypothesis using propriety instead? Schismic-24 is still proper, but not strictly proper. I'm sure some even hairier examples would break this. Remember unison vectors don't even have to be small intervals.>> Usually that comes out fine. The unison vectors define a linear >> temperament, which forms an MOS with the right number of notes. >> Let's prove this.I'm sure you can always get the linear temperament. You can describe it with fractions of the octave and chromatic unison vector if needs be. Getting to the MOS is more difficult, if you have a formula for that it would be useful anyway. Seeing as this is the mathematical list, I'll give the matrix equation: (R1) (R2) (R2) (R2) (M1) (00) (. )H' = (. )H' (. ) (. ) (. ) (. ) (Mn) (00) Where R1 and R2 are the chromatic unison vectors (one of which will usually be the octave) as row vectors. M1 to Mn are the commatic unison vectors. 00 is a row of zeros. So the things that look like column matrices are actually square. H' is the tempered equivalent of the list of prime axes, including 2. Multiply on the left by the inverse of the matrix with the unison vectors in, and you have an equation defining H' in terms of itself. You can then get your chromatic unison vectors in terms of H', and you have a two-dimensional system. Usually the chromatic vectors are an octave and a twelfth. For meantone and schismic this works fine. For diaschismic, they both have to be divided by two, but only the octave actually needs to be divided. For Miracle, the octave and twelfth both have to be divided by 6, so you have to re-define it with a fifth as a unison vector. For other scales it'll be hard to find the chromatic UV that leads nicely to the MOS.>> Hopefully this will maximise the consonances (another hypothesis?). >> It would be good to prove that too, but I'm afraid it won't always > work. It works if all the consonances come out to simple powers of > the generator, though.It probably would work for a sufficiently large MOS. But we could frame the hypothesis so that the MOS is used as the default PB. I think that would make sense.>> The diaschismic temperaments give an MOS with the half-octave as a >> generator. >> You mean, the half-octave as an equivalence interval? Yes, sorry.>> In general, for octave-equivalent unison vectors, the >> equivalence interval will always be some fraction of an octave. >> Great. But you can't take "equivalence interval" too far here -- the > strongest consonances become dissonances when altered by the half- > octave.I think this is in line with what Wilson intended by using "equivalence interval" instead of "octave".>> So the only outstanding problems are those temperaments where the >> determinant comes out as a multiple of the number of notes in the >> relevant ET. In that case, you get overcounting. The linear > temperament>> can still be calculated, but not in its lowest terms. And the >> periodicity block contains twice as many notes as it needs to. >> >> It's not clear to me what's going on here, especially after one > vector>> gets tempered out, but I'm sure the MOS concept can be expanded to > cover >> it. >> We'll figure it out!You still get a chain of generators, but not closed to give only two step sizes. Graham
Message: 14 - Contents - Hide Contents Date: Thu, 24 May 2001 01:43:55 Subject: Re: Hypothesis From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> Temper out the schisma from the periodicity block above. You end up with > a 24-note schismic scale. No way can that have two step sizes! > > That looks like a refutation with the definitions I have.I think the problem is that, as you said before, the scale really has 12 pitch classes, not 24, due to the syntonic comma squared vanishing.> How about a > weaker hypothesis using propriety instead?Blackjack it not proper.> Schismic-24 is still proper, > but not strictly proper. I'm sure some even hairier examples would break > this. Remember unison vectors don't even have to be small intervals. Exactly. > >>>> Usually that comes out fine. The unison vectors define a linear >>> temperament, which forms an MOS with the right number of notes. >>>> Let's prove this. >> I'm sure you can always get the linear temperament. You can describe it > with fractions of the octave and chromatic unison vector if needs be. > Getting to the MOS is more difficult, if you have a formula for that it > would be useful anyway. > > Seeing as this is the mathematical list, I'll give the matrix equation: > > (R1) (R2) > (R2) (R2) > (M1) (00) > (. )H' = (. )H' > (. ) (. ) > (. ) (. ) > (Mn) (00) > > Where R1 and R2 are the chromatic unison vectors (one of which will > usually be the octave) as row vectors. M1 to Mn are the commatic unison > vectors. 00 is a row of zeros. So the things that look like column > matrices are actually square. H' is the tempered equivalent of the list > of prime axes, including 2. > > Multiply on the left by the inverse of the matrix with the unison vectors > in, and you have an equation defining H' in terms of itself. You can > then get your chromatic unison vectors in terms of H', and you have a > two-dimensional system.OK, good so far.> > Usually the chromatic vectors are an octave and a twelfth.Lost me there. ? For> Miracle, the octave and twelfth both have to be divided by 6, so you have > to re-define it with a fifth as a unison vector. ?? > It probably would work for a sufficiently large MOS. But we could frame > the hypothesis so that the MOS is used as the default PB. I think that > would make sense.All right, as long as we don't get circular.
Message: 15 - Contents - Hide Contents Date: Thu, 24 May 2001 03:06:31 Subject: Re: Hypothesis From: monz --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: Yahoo groups: /tuning-math/message/14 * [with cont.]> --- In tuning-math@y..., graham@m... wrote: >>> Temper out the schisma from the periodicity block above. >> You end up with a 24-note schismic scale. No way can that >> have two step sizes! >> >> That looks like a refutation with the definitions I have. >> I think the problem is that, as you said before, the scale > really has 12 pitch classes, not 24, due to the syntonic > comma squared vanishing. > > <etc.>Can you guys please illustrate all this with lattices and other tables and diagrams? My math abilities are far below a lot of you others on *this* list, and it's all I can do to keep up with the other tuning lists now, to say nothing of the difficulties I have understanding what's written here. I know it would slow down the rate of discourse when those of you who speak the lingo start rapid-fire exchange, as happened over the past month with the MIRACLEs, but I for one would greatly appreciate help by seeing lots of visuals. Thanks.>> Seeing as this is the mathematical list, I'll give the matrix >> equation: >> >> (R1) (R2) >> (R2) (R2) >> (M1) (00) >> (. )H' = (. )H' >> (. ) (. ) >> (. ) (. ) >> (Mn) (00) >> >> Where R1 and R2 are the chromatic unison vectors (one of >> which will usually be the octave) as row vectors. M1 to Mn >> are the commatic unison vectors. 00 is a row of zeros. >> So the things that look like column matrices are actually >> square. H' is the tempered equivalent of the list >> of prime axes, including 2.Graham, again, could you please use lattices etc. to show this? I use vector addition for the prime factors in my work, and I thought at first that your matrix notation used here was similar, but I still don't understand it after having read your webpages. Could you (or someone else?) help guide me thru it? And explain how it's different from what I use, if it indeed is? My version of this "matrix addition" is here: Explanation of JustMusic theory and notation, ... * [with cont.] (Wayb.) I'd like detailed contributions like these I'm requesting to be made with a view toward incorporation into my Tuning Dictionary. I'm willing to make audio examples for as much of what we discuss as I can, but I have to understand what it is. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0"
Message: 16 - Contents - Hide Contents Date: Thu, 24 May 2001 03:45:18 Subject: Fwd: optimizing octaves in MIRACLE scale.. From: monz --- In tuning@y..., "Paul Erlich" <paul@s...> wrote: Monz wrote,> I'm interested in seeing what results > when we include the 2:1 in the optimization, and I can't > do that calculation.OK -- we now have to do a two-parameter optimization, where the two parameters are the size of the generator (let's call it G), and the size of the approximate 2:1 (let's call it O). So this requires multivariate calculus rather than the freshman univariate kind. Also, we'll now be minimizing the sum-of-squares of the errors of all the intervals in an integer limit, rather than an odd limit. This integer limit can be 7, 8, 9, 10, 11, or 12. Pick one and I'll try to work it out (it _will_ be hairy). If you will allow me to use MATLAB, I can perform the optimization numerically (that is, the computer will make repeated guesses and converge on the correct solution), which will at least reduce _somewhat_ the amount of work I have to do. --- End forwarded message --- OK, cool. MATLAB is fine with me. (If I didn't "allow" you to use it, does that mean that you'd go thru the drudgery of the hand calculations just to show me how it's done? If so, Paul, you're a really beautiful person. You can relax and use MATLAB.) Did we ever take a serious look at 11-odd-limit approximations in the MIRACLE family? I don't recall much about 11, other than Graham's discussions of neutral "3rds". So if most of our work so far is 7- or 9-based, I guess that's OK. I'd like to include 11 if you have no preference. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0"