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Message: 5651 - Contents - Hide Contents Date: Wed, 27 Nov 2002 00:38:04 Subject: Re: Paul's new names From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> including equal temperaments? can you give a sorting of ETs in some > limit? does GC reduce to something simpler for ETs?Indeed it does--for any et n, the geometric complexity is proportional to n.
Message: 5652 - Contents - Hide Contents Date: Wed, 27 Nov 2002 00:43:29 Subject: Re: Even more ridiculous 5-comma list From: Gene Ward Smith --- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> Regarding the link below. What are the vectors calculated from? Are they > wedge-invariants? Why are there two for each comma? Sorry if this seems > like a dumb question, but I'd like to be able to calculate these myself. > THANKSThe two vectors are actually to be thought of as two columns of a matrix; the matrix is the period/generator mapping, the first column giving how many periods, and the second how many generators, for the primes 2, 3 and 5. Wedging them together gives a wedge-invariant, but in the 5-limit case this can be identified with the comma. There are various ways to calculate such a matrix--for example, take two ets which define the temperament and reduce it to Hermite normal form.
Message: 5653 - Contents - Hide Contents Date: Wed, 27 Nov 2002 05:02:52 Subject: Re: spiral lattices From: Carl Lumma Indeed; Erv Wilson once sent me home with a pine cone the size of my head. -Carl
Message: 5654 - Contents - Hide Contents Date: Wed, 27 Nov 2002 05:06:18 Subject: Re: Paul's new names From: Carl Lumma>What are the main formualtions, > >we have graham's (unweighted minimax), unweighted rms, >weighted rms, geometric (both unweighted and weighted??) . . .For complexity? What are they? -Carl
Message: 5655 - Contents - Hide Contents Date: Thu, 28 Nov 2002 11:10:04 Subject: Re: Paul's new names From: Carl Lumma>> >inimax, rms, etc. of what? The numbers in the map? >>the number of generators comprising each consonant interval.Ok, thanks; as I thought. Conceptually, if we're thinking in terms of Partchian limits, I prefer simply the number of generators needed to span all the identities (consonant intervals). This can be 'weighted' by simply dividing by the number of identities. Reason being, I view the choice of a complete Partchian limit as a statement that all the identities in that limit will be treated as consonances in the music, and thus are all equally important in that sense. If we're not talking about Partchian limits, we can just omit the identities that make the range bad. Why we would want to smooth such bad approximations out with something like rms I cannot guess. I can see weighted error, but not weighted complexity, where the weighting proporational to the identity. Why should we expect more generators to be required to render 7 than 5? -Carl
Message: 5657 - Contents - Hide Contents Date: Thu, 28 Nov 2002 15:37:32 Subject: Re: Paul's new names From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>> Minimax, rms, etc. of what? The numbers in the map? >>>> the number of generators comprising each consonant interval. >> Ok, thanks; as I thought.It doesn't cover geometric complexity, for which you should see my postings on this list.
Message: 5658 - Contents - Hide Contents Date: Thu, 28 Nov 2002 17:43:20 Subject: Re: Paul's new names From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>> Minimax, rms, etc. of what? The numbers in the map? >>>> the number of generators comprising each consonant interval. >> Ok, thanks; as I thought. > > Conceptually, if we're thinking in terms of Partchian limits, > I prefer simply the number of generators needed to span all > the identities (consonant intervals).that's what i meant by "minimax" -- it's graham's way.> This can be 'weighted' > by simply dividing by the number of identities.that has no effect on the rankings.> Reason being, > I view the choice of a complete Partchian limit as a statement > that all the identities in that limit will be treated as > consonances in the music, and thus are all equally important > in that sense.**but if modulation is usually accomplished by making a small number of "chromatic" changes to the basic "diatonic" scale, shouldn't extra points be awarded if the modulations more often move one by the *simpler* consonances, particularly the 3-limit ones?> If we're not talking about Partchian limits, > we can just omit the identities that make the range bad. Why > we would want to smooth such bad approximations out with > something like rms I cannot guess.smooth such bad approximations out? i'm not sure what you mean. rms is similar to graham's method, but takes into account the second- longest, third-longest, etc. chains of generators to a small extent too. that seems like a good thing to me.> I can see weighted error, but not weighted complexity, where > the weighting proporational to the identity. Why should we > expect more generators to be required to render 7 than 5?see ** above.
Message: 5659 - Contents - Hide Contents Date: Thu, 28 Nov 2002 00:30:45 Subject: Re: Paul's new names From: Carl Lumma>>> >e have graham's (unweighted minimax), unweighted rms, >>> weighted rms, geometric (both unweighted and weighted??) . . . >> >> For complexity? >>yup, doze R dem. >>> What are they? > >dem. doze.Minimax, rms, etc. of what? The numbers in the map? -Carl
Message: 5660 - Contents - Hide Contents Date: Thu, 28 Nov 2002 18:16:09 Subject: Re: Paul's new names From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> smooth such bad approximations out? i'm not sure what you mean. rms > is similar to graham's method, but takes into account the second- > longest, third-longest, etc. chains of generators to a small extent > too. that seems like a good thing to me.It really depends on whether or not you are interested in incomplete n-limit chords; normally, I think we would be.
Message: 5661 - Contents - Hide Contents Date: Thu, 28 Nov 2002 00:43:59 Subject: Re: Even more ridiculous 5-comma list From: Gene Ward Smith --- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> 250/243 (2)*(5)^3/(3)^5 porcupine[[1, 2, 3] [0,-3,-5]]> comp 5.948285733 rms 7.975800816 bad 1678.609846 > generators [1200.,162.9960265]1*1200 - 0*163 = 1200 (2) 2*1200 - 3*163 = 1911 (3) 3*1200 - 5*163 = 2785 (5)
Message: 5662 - Contents - Hide Contents Date: Thu, 28 Nov 2002 18:40:46 Subject: Re: Paul's new names From: Carl Lumma>> >onceptually, if we're thinking in terms of Partchian limits, >> I prefer simply the number of generators needed to span all >> the identities (consonant intervals). >>that's what i meant by "minimax" -- it's graham's way.Thought so. Isn't minimax a bad term for this?>> This can be 'weighted' by simply dividing by the number of >> identities. >> that has no effect on the rankings.But it allows you to compare different limits. Actually I haven't checked if something stronger than division would be needed, but you get the idea.>**but if modulation is usually accomplished by making a small >number of "chromatic" changes to the basic "diatonic" scale, >shouldn't extra points be awarded if the modulations more often >move one by the *simpler* consonances, particularly the 3-limit >ones?Not in my view. I'm thinking we should not even assume tonal composition at this level. -Carl
Message: 5663 - Contents - Hide Contents Date: Thu, 28 Nov 2002 04:57:15 Subject: Re: Even more ridiculous 5-comma list From: wallyesterpaulrus --- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> > Thanks. Actually I spoke too soon about the negative numbers. Upon further > examination they make perfect sense. I still don't understand the "period" > vector though. What is it measuring? (The "generator" vector makes sense.) > For example: > > > > 250/243 (2)*(5)^3/(3)^5 porcupine > [[1, 2, 3], [0, -3, - 5]] > > > > > comp 5.948285733 rms 7.975800816 bad 1678.609846 > generators [1200., 162.9960265] > > > > > > What does [1,2,3] measure in terms of the primes 2,3 and 5?you need both columns together. the first column (this one) refers to the first generator, or period. the second column ([0 -3 -5]) refers to the second generator, usually referred to simply as *the* generator. so in this case, prime 2 is approximated by 1*1200 + 0*162.9960265 prime 3 is approximated by 2*1200 + -3*162.9960265 prime 5 is approximated by 3*1200 + -5*162.9960265
Message: 5664 - Contents - Hide Contents Date: Thu, 28 Nov 2002 18:55:25 Subject: Re: Paul's new names From: Carl Lumma>> >mooth such bad approximations out? i'm not sure what you mean. >> rms is similar to graham's method, but takes into account the >> second-longest, third-longest, etc. chains of generators to a >> small extent too. that seems like a good thing to me. >>It really depends on whether or not you are interested in >incomplete n-limit chords; normally, I think we would be.If I'm interested in complete 7-limit tetrads on every beat, have a temperament with really simple 3s and 7s but complex 5s, rms will punish this less than I'd like. At least, the history of Western music seems to assert that for music employing n-limit harmony, < n-limit harmony sounds too different to fall back on for any length of time. -Carl
Message: 5665 - Contents - Hide Contents Date: Thu, 28 Nov 2002 05:00:34 Subject: Re: Paul's new names From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>>> we have graham's (unweighted minimax), unweighted rms, >>>> weighted rms, geometric (both unweighted and weighted??) . . . >>> >>> For complexity? >>>> yup, doze R dem. >>>>> What are they? >> >> dem. doze. >> Minimax, rms, etc. of what? The numbers in the map? > > -Carlthe number of generators comprising each consonant interval.
Message: 5666 - Contents - Hide Contents Date: Thu, 28 Nov 2002 19:24:31 Subject: Re: Paul's new names From: Carl Lumma>>> >he number of generators comprising each consonant interval. >>>> Ok, thanks; as I thought. >>It doesn't cover geometric complexity, for which you should >see my postings on this list.Msg. 4533 is the one, I'm guessing. Very cool. But I don't have the technique for choosing the defining commas. Is there any way this can be defined in terms of the map? -Carl
Message: 5667 - Contents - Hide Contents Date: Thu, 28 Nov 2002 19:30:22 Subject: Re: Paul's new names From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> If I'm interested in complete 7-limit tetrads on every beat, > have a temperament with really simple 3s and 7s but complex 5s, > rms will punish this less than I'd like. At least, the history > of Western music seems to assert that for music employing > n-limit harmony, < n-limit harmony sounds too different to > fall back on for any length of time.Then you want Graham complexity, which is nice, because it is very easy to compute.
Message: 5668 - Contents - Hide Contents Date: Thu, 28 Nov 2002 19:32:35 Subject: Re: Paul's new names From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> Msg. 4533 is the one, I'm guessing. Very cool. But I don't > have the technique for choosing the defining commas. Is > there any way this can be defined in terms of the map?My program extracts commas from the wedgie and then reduces them. Another way would be to run down a comma list using the map, and see which are mapped to [0,0].
Message: 5669 - Contents - Hide Contents Date: Thu, 28 Nov 2002 21:38:55 Subject: Re: Paul's new names From: wallyesterpaulrus --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>>> Conceptually, if we're thinking in terms of Partchian limits, >>> I prefer simply the number of generators needed to span all >>> the identities (consonant intervals). >>>> that's what i meant by "minimax" -- it's graham's way. >> Thought so. Isn't minimax a bad term for this?yup, sorry -- it's just "max".>> **but if modulation is usually accomplished by making a small >> number of "chromatic" changes to the basic "diatonic" scale, >> shouldn't extra points be awarded if the modulations more often >> move one by the *simpler* consonances, particularly the 3-limit >> ones? >> Not in my view. I'm thinking we should not even assume tonal > composition at this level.i agree. but even on the level of modal composition, i feel this weighting has some appeal, for the same kinds of reasons.
Message: 5670 - Contents - Hide Contents Date: Thu, 28 Nov 2002 21:41:49 Subject: Re: Paul's new names From: wallyesterpaulrus> --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote: >>> If I'm interested in complete 7-limit tetrads on every beat, >> have a temperament with really simple 3s and 7s but complex 5s, >> rms will punish this less than I'd like. At least, the history >> of Western music seems to assert that for music employing >> n-limit harmony, < n-limit harmony sounds too different to >> fall back on for any length of time.the incomplete chords need not be lower-limit. for your 7-limit example, how about 1:5:7, 1:3:7, etc.? in western music, you have plenty of two-part music (especially bach) which uses incomplete triads (that is, dyads) *all* the time, but typically these are still 5-limit.
Message: 5671 - Contents - Hide Contents Date: Thu, 28 Nov 2002 21:47:14 Subject: Re: Paul's new names From: wallyesterpaulrus> --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote: >>> Msg. 4533 is the one, I'm guessing. Very cooli like the idea of preserving some features of tenney's harmonic distance function . . . but i'm not sure i can fully visualize geometric complexity yet . . .
Message: 5673 - Contents - Hide Contents Date: Fri, 29 Nov 2002 03:19:11 Subject: Re: Paul's new names From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote: >>>>> If I'm interested in complete 7-limit tetrads on every beat, >>> have a temperament with really simple 3s and 7s but complex 5s, >>> rms will punish this less than I'd like. At least, the history >>> of Western music seems to assert that for music employing >>> n-limit harmony, < n-limit harmony sounds too different to >>> fall back on for any length of time.> the incomplete chords need not be lower-limit. for your 7-limit > example, how about 1:5:7, 1:3:7, etc.?I think Saxfare, which does not use 5, sounds like music.
Message: 5674 - Contents - Hide Contents Date: Sat, 30 Nov 2002 19:49:43 Subject: Re: Even more ridiculous 5-comma list From: wallyesterpaulrus --- In tuning-math@y..., "paulhjelmstad" <paul.hjelmstad@u...> wrote:> How would one use rms to get 317.079753? > Paulfor 5-limit temperaments, you minimize the rms error in the 3/1, the 5/1, and the 5/3 -- in other words, all three 5-limit consonant interval classes. -another paul
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