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Message: 9200 - Contents - Hide Contents Date: Sat, 17 Jan 2004 14:26:28 Subject: Re: summary -- are these right? From: Carl Lumma>>>> >n a unit-length odd-limit lattice both 9 and 11 have length 1. >>>>>> Doesn't 9 have length 2? >>>> 9 occurs in two places on an odd-limit lattice of odd-limit >= 9. >>Is this "lattice" one of those goofy things people insist on calling >lattices even though they are not?I don't know.>What in the world do you mean?It's a rectangular Thing with an axis for each odd number.>9 is always 3^2, so it necessarily is twice as far from the origin as >3, and in the same direction. Right. >If 9 and 11 are the same length, 3 is half that length.There are two different routes to 9 in this Thing, with two different lengths. Maybe you can tell me what nice properties such a setup violates. Only if you impose log-weighting is the above true. -Carl
Message: 9201 - Contents - Hide Contents Date: Sat, 17 Jan 2004 14:44:41 Subject: Re: Dual space example From: Carl Lumma>I had thought of this business of subspaces and the JIP as a way of >formulating what we were already doing, but didn't realize it led to >anything new. I wish I'd seen the possible link to Paul's heuristic, >but another interesting aspect I didn't consider is this bounded >relative error business. Here's a basic example of what I mean. > >As I've pointed out from time to time, the norm which gives the >equilateral triangular lattice for 5-limit octave classes is > >||3^a 5^b|| = sqrt(a^2 + ab + b^2) > >where the class is represented by 3^a 5^b.I can believe that.>The norm on the dual space is then > >||(x3, x5)|| = sqrt(x3^2 - x3 x5 + x5^2) > >where x2 is the tuning of 3 (in log2, cents or whatever your favorite >log is terms) and x5 is the tuning of 5.I'll take your word on that.>The nearest point to [log(3), log(5)] on a subspace corresponding >to a temperament is the 5-limit rms tuning. If we look on the line >x5=4x3 - 4, for instance, we get the Woolhouse tuning. Yes!!Is there anything fundamentally keeping the 2s out of this? -Carl
Message: 9202 - Contents - Hide Contents
Date: Sat, 17 Jan 2004 22:48:57
Subject: Duals to ems optimization
From: Gene Ward Smith
For any set of consonances C we want to do an rms optimization for, we
can find a corresponding Euclidean norm on the val space (or
octave-excluding subspace if we are interested in the odd limit) by
taking the sum of terms
(c2 x2 + c3 x3 + ... + cp xp)^2
for each monzo |c2 c3 ... cp> in C. If we want something corresponded
to weighted optimization we would add weights, and if we wanted the
odd limit, the consonances in C can be restricted to quotients of odd
integers, in which case c2 will always be zero. We then may form the
symmetric matrix corresponding to the quadratic form we get from the
above sum and invert it; using this inverted matrix to define a
quadratic form on monzos and taking the square root gives us a
Euclidean norm on monzo space. We can of course normalize either norm
by multiplying by any positive constant.
If we do this in the 5-limit using {3, 5, 5/3} for C, we get
sqrt(x2^2 - x2 x3 + x3^2)
for the norm on the val (sub)space, and correspondingly
sqrt(e2^2 + e2 e3 + e3^2)
for the norm on octave classes--the triangular lattice norm.
Similarly, if C = {3, 5, 7, 5/3, 7/3, 7/5} we get
sqrt(3(x3^2 + x5^2 + x7^2) - 2(x3 x5 + x3 x7 + x5 x7))
as the norm on vals, and the familiar
sqrt(e3^2 + e5^2 + e7^2 + e3 e5 + e3 e7 + e5 e7)
as the norm on octave classes.
In the 11-limit and beyond, of course, things become more complicated
because we will want to introduce ratios of odd numbers which are not
necessarily primes. If we take ratios of odd numbers up to 11 for our
set of consonances, we get
sqrt(20x3^2+5x5^2-2x7x11-6x3x5+5x7^2+5x11^2-6x3x7-2x5x7-2x5x11-6x3x11)
as our norm on vals, and correspondingly,
sqrt(18e3^2+36e3e5+36e3e7+36e3e11+62e5^2+58e5e7+58e5e11+62e7^2+58e11e7+62e11^2)
as our norm on octave classes. This norm is not altogether
satisfactory; for instance it gives a length of sqrt(44) to 5/3 and
6/5, and a length of sqrt(62) to 5/4. This suggests to me that there
is something a little dubious in theory about using unweighted rms
optimization, at least in the 11 limit and beyond. An alternative rms
optimization scheme would be to use dual of the norm I've been using
on octave classes as the norm for a weighted rms optimization.
In the 5-limit, this norm on octave classes is
sqrt(p3e3^2 + p3e3e5 + p5e5^2)
where p3 = log2(3), p5 = log2(5). The dual norm on vals is
sqrt(p5x3^2 - p3x3x5 + p3x5^2)
These norms will weigh lower prime errors a little higher than higher
prime errors, which of course is also what TOP does. Now I need a
catchy name for them.
Message: 9203 - Contents - Hide Contents Date: Sat, 17 Jan 2004 18:23:49 Subject: Re: summary -- are these right? From: Carl Lumma>> >an you demonstrate how to get length log(9) out >> of 9/5? >>9/5 is a ratio of 9.I meant on the lattice.>OK, which part were we talking about?You were looking for Paul Hahn's algorithm, which is like the 2nd or 3rd message in there. It isn't that long in any case. -Carl
Message: 9204 - Contents - Hide Contents Date: Sat, 17 Jan 2004 21:08:33 Subject: Re: summary -- are these right? From: Carl Lumma>>> > graph (as in graph-theory) but with lengths for each rung? >>>> That could be a "directed graph" I think. >>Directed means each rung has a specific beginning point and ending >point.Whoops, got my mix crossed up. I meant "network".>> But all the flavors >> of graph I'm aware of lack orientation, fixed dimensionality, >> and so forth. Maybe "space" would work here? >>A space has an infinite number of points between any two points. Crap.>>>> So summing up, can we say that we're happy with our >>>> octave-specific concordance heuristic and associated >>>> lattice/metric, and that we have an octave-equivalent >>>> concordance heuristic but *no* associated lattice/metric? >>>>>> I'd prefer not to say 'concordance heuristic', but yes. >>>> What would you say? > >concordance function? Ok. -Carl
Message: 9205 - Contents - Hide Contents Date: Sat, 17 Jan 2004 22:50:21 Subject: Re: summary -- are these right? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> There are two different routes to 9 in this Thing, with two > different lengths. Maybe you can tell me what nice properties > such a setup violates.Uniqueness and linearity.
Message: 9206 - Contents - Hide Contents Date: Sat, 17 Jan 2004 22:51:34 Subject: Re: Dual space example From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Yes!! > > Is there anything fundamentally keeping the 2s out of this?No; you get something similar in each even limit, for instance.
Message: 9207 - Contents - Hide Contents Date: Sat, 17 Jan 2004 14:58:05 Subject: Re: summary -- are these right? From: Carl Lumma>> >here are two different routes to 9 in this Thing, with two >> different lengths. Maybe you can tell me what nice properties >> such a setup violates. >>Uniqueness and linearity.And does this interfere with having... () a lattice () a metric () a norm ? With log weighting we restore linearity, I'm guessing. Then do we have... () a lattice () a metric () a norm ? -Carl
Message: 9208 - Contents - Hide Contents Date: Sat, 17 Jan 2004 18:32:36 Subject: Re: summary -- are these right? From: Carl Lumma>> >he two obvious variations are rectangular odd-limit >>How can odd-limit be rectangular? Makes no sense to me.One can certainly have a rectangular lattice with a 9-axis.>> and triangular octave-specific. >>Then the metric is not log(n*d) anymore.We actually haven't specified how to find the lengths of rungs like 9:5... -Carl
Message: 9209 - Contents - Hide Contents Date: Sat, 17 Jan 2004 23:13:14 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> By "unweighted" I probably mean a norm without coefficents for >>>>> an interval's coordinates. >>> //>>>>> The norm on Tenney space... >>>>> >>>>> || |u2 u3 u5 ... up> || = >>>>> log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up| >>>>> >>>>> The 'coefficients on the intervals coordinates' here are >>>>> log2(2), log2(3) etc. >>>>>>>> So 'unweighted', 9 has a length of 2 but 11 has a length of >>>> 1 . . . :( >>>>>> Unless you use odd-limit. >>>> Please elaborate on how that's 'unweighted' in your view. >> On a unit-length odd-limit lattice both 9 and 11 have length 1. > I'm not claiming anything necessarily good about this, note. > I am asking for comments about it however.OK . . . this, I suppose, is Paul Hahn's lattice or something?>>>>> The taxicab distance on this lattice is log(odd-limit). >>>>>>>> No it isn't -- try 9:5 for example. >>>>>> This is what you were claiming in 1999. >>> >>> "" >>> But the basic insight is that a triangular lattice, with >>> Tenney-like lengths, a city-block metric, and odd axes or >>> wormholes, agrees with the odd limit perfectly, and so is >>> the best octave-invariant lattice representation (with >>> associated metric) for anyone as Partchian as me. >>> "" >>>> Right -- you need those odd axes, which screws up uniqueness, >> and thus most of how we've been approaching temperament. >> But does the metric agree with log(odd-limit) or not? > For 9:5, log(oddlimit) is log(9). If you run it through > the "norm" you get... 2log(3) + log(5).No, because 9 has its own axis.> Not the same, > it seems. However if you followed the > lumma.org/stuff/latice1999.txt link,The page cannot be found.> apparently Paul Hahn > did present a metric that agrees with log(odd-limit).Can you re-present it?>>>>>> I was thinking stuff like ||9|| = ||3|| = 1 >>>>> and thus ||3+3|| < ||3|| + ||3|| but that's ok. It seems >>>>> bad though, since the 3s are pointed in the same direction. >>>>>>>> What lattice/metric was this about? >>>>>> Unweighted odd-limit taxicab. >>>> In which 9 has its own axis . . . so the following: >>>>> By "3+3" I meant adding two 3 >>> vectors. The equation is 1 < 2. >>>> does not apply. >> Sure it does. As you say, the 9 appears in two places. If > the metric comes out the same either way,It doesn't -- see above.> I don't see how this > fact would "screws up uniqueness, and thus most of how we've > been approaching temperament."Even if the 'metric came out the same either way', every note would appear in an infinite number of places in the lattice, since for every integer n the ratio p/q can also be expressed as p/q*3^(-2n)*9^n .
Message: 9210 - Contents - Hide Contents Date: Sat, 17 Jan 2004 21:48:58 Subject: Re: A new graph for Paul? From: Carl Lumma>> >1 -3| 125/128 > >135/128Gene and all, What if, instead of issuing a correction post like this, we were to post a full corrected version and delete the original from the archives? Posterity may thank us... Just a thought. -Carl
Message: 9211 - Contents - Hide Contents Date: Sat, 17 Jan 2004 23:20:16 Subject: Re: TOP history From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Paul Erlich wrote: >>> It's exactly what I've been pleading to you guys to help me figure >> out last year and probably even earlier, except without octave- >> equivalence. The idea was to temper out commas uniformly over their >> length in the lattice, to see what error function this was optimal >> with respect to, and to then apply this same error function to >> optimize temperaments with more than one comma. The posts asking >> about this can be found in the archives here. >> Was it? Oh. Well, I found this thread: > > Yahoo groups: /tuning-math/message/2857 * [with cont.] > > The new thing is the concept of weighted minimax. > > > GrahamRight, because I didn't know that weighted minimax is exactly what results from 'applying' the octave-specific 'heuristic' (meaning tempering such that it's exactly correct) until I actually thought about it. Being sick (though mentally impaired by fever) and away from the list for a while seemed to help revive independent thought.
Message: 9212 - Contents - Hide Contents Date: Sat, 17 Jan 2004 15:29:52 Subject: Re: summary -- are these right? From: Carl Lumma>> >n a unit-length odd-limit lattice both 9 and 11 have length 1. >> I'm not claiming anything necessarily good about this, note. >> I am asking for comments about it however. >>OK . . . this, I suppose, is Paul Hahn's lattice or something?For his diameter measure I think it is.>>>>>> The taxicab distance on this lattice is log(odd-limit). >>>>>>>>>> No it isn't -- try 9:5 for example. >>>>>>>> This is what you were claiming in 1999. >>>> >>>> "" >>>> But the basic insight is that a triangular lattice, with >>>> Tenney-like lengths, a city-block metric, and odd axes or >>>> wormholes, agrees with the odd limit perfectly, and so is >>>> the best octave-invariant lattice representation (with >>>> associated metric) for anyone as Partchian as me. >>>> "" >>>>>> Right -- you need those odd axes, which screws up uniqueness, >>> and thus most of how we've been approaching temperament. >>>> But does the metric agree with log(odd-limit) or not? >> For 9:5, log(oddlimit) is log(9). If you run it through >> the "norm" you get... 2log(3) + log(5). >>No, because 9 has its own axis.It's still different than log(odd-limit), and in fact log(5) + log(9) = 2log(3) + log(5).>> Not the same, >> it seems. However if you followed the >> lumma.org/stuff/latice1999.txt link, >>The page cannot be found.Typo here; "lattice".>> apparently Paul Hahn >> did present a metric that agrees with log(odd-limit). >>Can you re-present it?See the above url.>> I don't see how this >> fact would "screws up uniqueness, and thus most of how we've >> been approaching temperament." >>Even if the 'metric came out the same either way', every note would >appear in an infinite number of places in the lattice, since for >every integer n the ratio p/q can also be expressed as >p/q*3^(-2n)*9^n >. Hmm. -Carl
Message: 9213 - Contents - Hide Contents Date: Sat, 17 Jan 2004 18:40:42 Subject: Re: summary -- are these right? From: Carl Lumma>>>> >an you demonstrate how to get length log(9) out >>>> of 9/5? >>>>>> 9/5 is a ratio of 9. >>>> I meant on the lattice. >>Yes, that's how this 'lattice' is defined, isn't it?I was asking for any way it could be defined to make it equal odd-limit, but this seems like cheating because you require odd-limit infinity, and thus you're never taking any multi-stop routes.>>> OK, which part were we talking about? >>>> You were looking for Paul Hahn's algorithm, which is >> like the 2nd or 3rd message in there. It isn't that >> long in any case. >>OK -- that's the algorithm when each consonance in a given >odd-limit is given a rung of length 1. Right. >So going back to the above, if the given >odd-limit is less than 9, 9/5 will have to be constructed out of 3 >and 3/5, thus has a length of 2, per Paul Hahn's lattice. No logs >get involved there.Right. It's easy. But it doesn't correspond to the "ratio-of" the ratio. My point, if any, is that I think this will be impossible with odd-limit < inf. on a triangular lattice. -Carl
Message: 9214 - Contents - Hide Contents Date: Sat, 17 Jan 2004 23:35:38 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> On a unit-length odd-limit lattice both 9 and 11 have length 1. >>>>>> Doesn't 9 have length 2? >>>> 9 occurs in two places on an odd-limit lattice of odd-limit >= 9. >> Is this "lattice" one of those goofy things people insist on calling > lattices even though they are not? What in the world do you mean? > > 9 is always 3^2, so it necessarily is twice as far from the origin as > 3, and in the same direction. If 9 and 11 are the same length, 3 is > half that length.What if you're dealing with a world where all sine wave components are constrained to be in 768-equal? And '9' is not the same as '3^2'?
Message: 9215 - Contents - Hide Contents Date: Sat, 17 Jan 2004 00:06:43 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:>>> By "unweighted" I probably mean a norm without coefficents for >>> an interval's coordinates. > //>>> The norm on Tenney space... >>> >>> || |u2 u3 u5 ... up> || = >>> log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up| >>> >>> The 'coefficients on the intervals coordinates' here are >>> log2(2), log2(3) etc. >>>> So 'unweighted', 9 has a length of 2 but 11 has a length of >> 1 . . . :( >> Unless you use odd-limit.Please elaborate on how that's 'unweighted' in your view.>>>>> This ruins the correspondence with taxicab distance on >>>>> the odd-limit lattice given by Paul's/Tenney's norm, >>>>>>>> Huh? Which odd-limit lattice and which norm? >>>>>> It's the same norm on a triangular lattice with a dimension >>> for each odd number. >>>> That's not a desirable norm. >>>>> The taxicab distance on this lattice is log(odd-limit). >>>> No it isn't -- try 9:5 for example. >> This is what you were claiming in 1999. > > "" > But the basic insight is that a triangular lattice, with > Tenney-like lengths, a city-block metric, and odd axes or > wormholes, agrees with the odd limit perfectly, and so is > the best octave-invariant lattice representation (with > associated metric) for anyone as Partchian as me. > ""Right -- you need those odd axes, which screws up uniqueness, and thus most of how we've been approaching temperament. Kees apparently saw the issue the same way I did, and I'm still puzzling over some issues with his framework, but no one here seemed able to help (last year).>>> It's also the same distance as on the Tenney lattice, >>> except perhaps for the action of 2s in the latter (I >>> forget the reasoning there). >>>> Try building up the reasoning from scratch. >> Here's what I was trying to remember... citation? > """ > The reason omitting the 2-axis forces one to make the lattice > triangular is that typically many more powers of two will be > needed to bring a product of prime factors into close position > than to bring a ratio of prime factors into close position. So > the latter should be represented by a shorter distance than the > former. Simply ignoring distances along the 2-axis and sticking > with a rectangular (or Monzo) lattice is throwing away > information. > // > ... a weight of log(axis) should be applied to all axes, and > if a 2-axis is included, a rectangular lattice is OK. If a > 2-axis is not included, a triangular lattice is better. > // > ... in an octave-specific rectangular (or parallelogram) > lattice, 7:1 and 5:1 are each one rung and 7:5 is two rungs. In > an octave-specific sense, 7:1 and 5:1 really are simpler than > 7:5; the former are more consonant. 7:4 and 5:4 are each three > rungs in the rectangular lattice, but they still come out a little > simpler than 7:5 since the rungs along the 2-axis are so short. > If you can buy that 35:1 is as simple as 7:5, then the octave- > specific lattice really should be rectangular, not triangular. > 35:1 is really difficult to compare with 7:5 -- it's much less > rough but also much harder to tune . . . > """ >>>> I was thinking stuff like ||9|| = ||3|| = 1 >>> and thus ||3+3|| < ||3|| + ||3|| but that's ok. It seems >>> bad though, since the 3s are pointed in the same direction. >>>> What lattice/metric was this about? >> Unweighted odd-limit taxicab.In which 9 has its own axis . . . so the following:> By "3+3" I meant adding two 3 > vectors. The equation is 1 < 2.does not apply.
Message: 9216 - Contents - Hide Contents Date: Sat, 17 Jan 2004 23:32:27 Subject: Re: A new graph for Paul? From: Carl Lumma>Not a bad plan; I do wonder how it works for people who read >via email.I just got your correction to the message in question by e-mail. It would help to have a standard header for the subject line. Maybe... Correction01: [old subject] ...to be followed by Correction02 and so forth. -Carl
Message: 9217 - Contents - Hide Contents Date: Sat, 17 Jan 2004 23:36:49 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> On a unit-length odd-limit lattice both 9 and 11 have length 1. >>>>>>>> Doesn't 9 have length 2? >>>>>> 9 occurs in two places on an odd-limit lattice of odd-limit >= 9. >>>> Is this "lattice" one of those goofy things people insist on calling >> lattices even though they are not? >> I don't know. >>> What in the world do you mean? >> It's a rectangular Thing with an axis for each odd number.It's actually triangular, and is what Erv Wilson uses to map out his CPSs.
Message: 9218 - Contents - Hide Contents Date: Sat, 17 Jan 2004 00:11:54 Subject: Re: Question for Dave Keenan From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>>> What does "yes" mean here? >>>> the sound holds together as a single pitch. >> My guess is that it will be experienced as a single pitch, but one > that cannot be accurately determined. The pitch will be fuzzy or vague > in a similar way to that of a harmonic note of very short duration.Thanks for that. Sounds to come.>>>> If I take any inharmonic timbre with one loud partial and some >> quiet,>>>> unimportant ones (very many fall into this category), and use a >>>> tuning system where >>>> >>>> 2:1 off by < 10.4 cents >>>> 3:1 off by < 16.5 cents >>>> 4:1 off by < 20.8 cents >>>> 5:1 off by < 24.1 cents >>>> 6:1 off by < 26.9 cents >>>> >>>> and play a piece with full triadic harmony, doesn't it follow >> that>>>> the harmony should 'hold together' the way 5-limit triads should? >>>>>> I don't know. What has the single loud partial got to do with it? Is >>> this partial one of those mentioned above? >>>> No, it essentially determines the pitch of the timbre. >> So the waveform is essentially sinusoidal? Why not use sinusoidal > waves for this thought experiment?They're not especially musical -- you'll have an easier time hearing chords as sets of separate notes when the timbre is not a pure sine wave.>>> We know that with quiet sine waves nothing special happens with any >>> dyad except a unison, and that loud sine waves work like harmonic >>> timbres presumably due to harmonics >>>> and combinational tones . . . > > Good point. >>>> being generated in the >>> nonlinearities of the ear-brain system. >>>> quiet harmonic timbres don't generate combinational tones, so they >> won't "work like" loud sine waves. >> >>> Don't we? >>>> That also ignores virtual pitch. A set of quiet sine waves can evoke >> a single pitch which does not agree with any combinationaltone . . .>> at certain intervals, the pitch evoked will be least ambiguous, which >> is certainly 'something special happening' . . . >> How many sine waves in an approximate harmonic series do you need for > this to be experienced? And what arrangements work? I was only > speaking of dyads.The effect has been observed with dyads, but most of the experiments concern three (or more) sine waves, since they evoke the phenomenon more readily.>> The fact is that, when using inharmonic timbres of the sort I >> described, Western music seems to retain all it meaning: certain >> (dissonant) chords resolving to other (consonant) chords, etc., all >> sounds quite logical. My sense (and the opinion expressed in >> Parncutt's book, for example) is that *harmony* is in fact very >> closely related to the virtual pitch phenomenon. We already know, >> from our listening tests on the harmonic entropy list, that the >> sensory dissonance of a chord isn't a function of the sensory >> dissonances of its constituent dyads. Furthermore, you seem to be >> defining "something special" in a local sense as a function of >> interval size, but in real music you don't get to evaluate each >> sonority by detuning various intervals various amounts, which >> this "specialness" would seem to require for its detection. >> >> The question I'm asking is, with what other tonal systems, besides >> the Western one, is this going to be possible in. >> If by "Western tonal systems", you mean any based on approximating > small whole number ratios of frequency,No, I meant diatonic/meantone.> What's your point?Did the above really not say anything to you?
Message: 9219 - Contents - Hide Contents Date: Sat, 17 Jan 2004 15:38:20 Subject: Re: summary -- are these right? From: Carl Lumma>> >t's a rectangular Thing with an axis for each odd number. >>It's actually triangular, and is what Erv Wilson uses to map out his >CPSs.The Thing I was referring to here was most certainly rectangular. -Carl
Message: 9220 - Contents - Hide Contents Date: Sat, 17 Jan 2004 00:29:11 Subject: Re: 46 augmented scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: Sorry, I neglected to octave-reduce these. That should cut down on the number.
Message: 9221 - Contents - Hide Contents Date: Sat, 17 Jan 2004 23:44:40 Subject: Re: Duals to ems optimization From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> In the 11-limit and beyond, of course, things become more complicated > because we will want to introduce ratios of odd numbers which are not > necessarily primes. If we take ratios of odd numbers up to 11 for our > set of consonances, we get > > sqrt(20x3^2+5x5^2-2x7x11-6x3x5+5x7^2+5x11^2-6x3x7-2x5x7-2x5x11- 6x3x11) > > as our norm on vals, and correspondingly, > > sqrt (18e3^2+36e3e5+36e3e7+36e3e11+62e5^2+58e5e7+58e5e11+62e7^2+58e11e7+62e 11^2) > > as our norm on octave classes. This norm is not altogether > satisfactory; for instance it gives a length of sqrt(44) to 5/3 and > 6/5, and a length of sqrt(62) to 5/4. This suggests to me that there > is something a little dubious in theory about using unweighted rms > optimization, at least in the 11 limit and beyond.Gene, note that I've always counted 3/1 and 9/3, etc., separately in these optimizations. If you use that "weighting", do things look less dubious? (The weight is proportional to the number of ways the interval class can be represented by a ratio of odd numbers within the limit.)
Message: 9222 - Contents - Hide Contents Date: Sat, 17 Jan 2004 02:08:32 Subject: 43 augmented scales From: Gene Ward Smith Even after octave reducing, I'm getting 43 different scales. For these scales, the period is octave-reduced to a value of 0, 1 or 2, and the generator starts from 0 and goes however far it goes, which I then call the complexity. complexity 3 = Augmented[12] {[1, 2], [1, 1], [1, 0], [2, 1], [2, 2], [1, 3], [0, 3], [0, 2], [0, 1], [0, 0], [2, 0], [2, 3]} complexity 5 {[0, 4], [1, 5], [0, 5], [1, 2], [2, 4], [2, 2], [1, 3], [0, 3], [0, 2], [0, 1], [2, 0], [2, 3]} {[0, 4], [2, 5], [1, 2], [1, 1], [2, 1], [2, 2], [1, 3], [0, 3], [0, 2], [1, 4], [0, 0], [2, 3]} {[0, 4], [1, 5], [1, 2], [1, 1], [2, 4], [2, 2], [1, 3], [0, 3], [0, 2], [0, 1], [2, 0], [2, 3]} {[2, 5], [1, 2], [1, 1], [1, 0], [2, 1], [2, 2], [1, 3], [0, 3], [0, 2], [1, 4], [0, 0], [2, 3]} {[2, 5], [1, 2], [1, 1], [1, 0], [2, 4], [2, 1], [2, 2], [1, 3], [0, 3], [0, 2], [2, 0], [2, 3]} {[0, 4], [0, 5], [1, 2], [2, 1], [2, 2], [1, 3], [0, 3], [0, 2], [0, 1], [1, 4], [0, 0], [2, 3]} {[0, 4], [1, 5], [0, 5], [1, 2], [2, 2], [1, 3], [0, 3], [0, 2], [0, 1], [1, 4], [0, 0], [2, 3]} complexity 6 {[0, 4], [2, 5], [1, 2], [1, 1], [2, 1], [1, 3], [0, 3], [0, 2], [1, 4], [0, 0], [1, 6], [2, 3]} {[1, 5], [2, 6], [0, 5], [1, 2], [1, 0], [2, 4], [2, 2], [1, 3], [0, 3], [0, 1], [1, 4], [2, 3]} {[0, 4], [1, 5], [2, 6], [2, 5], [1, 2], [1, 1], [2, 2], [1, 3], [0, 3], [1, 4], [0, 0], [2, 3]} {[0, 4], [0, 5], [2, 5], [2, 4], [2, 1], [2, 2], [1, 3], [0, 6], [0, 3], [0, 2], [2, 0], [2, 3]} {[0, 4], [1, 5], [2, 6], [1, 2], [1, 1], [2, 2], [1, 3], [0, 3], [0, 1], [1, 4], [0, 0], [2, 3]} {[0, 4], [0, 5], [2, 4], [2, 1], [2, 2], [1, 3], [0, 6], [0, 3], [0, 2], [0, 1], [2, 0], [2, 3]} complexity 7 {[0, 5], [2, 5], [1, 0], [2, 4], [2, 1], [2, 2], [1, 3], [0, 6], [1, 7], [0, 3], [0, 2], [1, 4]} {[0, 4], [2, 5], [1, 1], [2, 7], [2, 1], [1, 3], [0, 6], [0, 2], [1, 4], [0, 0], [1, 6], [2, 3]} {[0, 4], [1, 5], [0, 5], [1, 2], [1, 3], [1, 7], [0, 3], [0, 2], [0, 1], [1, 4], [0, 0], [1, 6]} complexity 8 {[1, 5], [2, 6], [0, 5], [1, 2], [2, 8], [2, 4], [2, 2], [1, 7], [0, 3], [0, 1], [2, 0], [0, 7]} {[0, 4], [2, 5], [1, 1], [2, 7], [0, 8], [2, 1], [1, 3], [0, 6], [0, 2], [0, 0], [1, 6], [2, 3]} {[0, 4], [1, 5], [2, 5], [1, 1], [2, 7], [0, 8], [1, 3], [0, 6], [0, 2], [0, 0], [1, 6], [2, 3]} {[0, 4], [1, 5], [1, 8], [2, 5], [1, 1], [2, 7], [1, 3], [0, 6], [0, 2], [1, 6], [2, 0], [2, 3]} {[0, 4], [1, 5], [1, 8], [2, 5], [1, 1], [2, 7], [2, 2], [1, 3], [0, 6], [0, 2], [2, 0], [2, 3]} {[1, 5], [2, 6], [0, 5], [1, 2], [2, 8], [2, 4], [2, 2], [1, 3], [1, 7], [0, 3], [0, 1], [2, 0]} complexity 9 {[0, 9], [2, 6], [0, 5], [1, 8], [1, 2], [1, 0], [0, 3], [0, 1], [1, 4], [1, 6], [2, 3], [0, 7]} {[0, 9], [2, 6], [0, 5], [1, 2], [1, 0], [2, 4], [0, 3], [0, 1], [1, 4], [1, 6], [2, 3], [0, 7]} {[1, 5], [2, 6], [0, 5], [2, 8], [2, 4], [2, 2], [1, 3], [0, 6], [1, 9], [1, 7], [0, 3], [2, 0]} {[0, 9], [0, 5], [1, 2], [2, 8], [2, 4], [2, 7], [1, 3], [1, 7], [0, 2], [0, 1], [1, 6], [2, 0]} {[2, 6], [2, 9], [1, 8], [2, 5], [1, 0], [2, 1], [2, 2], [0, 6], [1, 7], [0, 3], [1, 4], [0, 7]} {[0, 4], [0, 9], [1, 8], [2, 5], [1, 1], [2, 7], [1, 3], [0, 6], [0, 2], [1, 6], [2, 0], [2, 3]} {[1, 5], [2, 6], [2, 9], [1, 8], [2, 5], [1, 0], [2, 2], [0, 6], [1, 7], [0, 3], [1, 4], [0, 7]} {[2, 9], [1, 8], [2, 5], [1, 0], [2, 1], [2, 2], [1, 3], [0, 6], [1, 7], [0, 3], [0, 2], [1, 4]} {[0, 4], [0, 9], [0, 5], [1, 2], [2, 4], [2, 7], [1, 3], [0, 2], [0, 1], [1, 6], [2, 0], [2, 3]} complexity 10 {[1, 5], [2, 6], [0, 5], [2, 8], [2, 4], [2, 10], [2, 2], [1, 9], [1, 7], [0, 3], [2, 0], [0, 7]} {[0, 4], [1, 5], [0, 10], [2, 5], [1, 1], [2, 7], [0, 8], [1, 3], [0, 6], [0, 2], [0, 0], [2, 3]} {[1, 5], [2, 6], [0, 5], [2, 8], [2, 4], [2, 10], [2, 2], [1, 3], [1, 9], [1, 7], [0, 3], [2, 0]} {[0, 4], [1, 5], [0, 10], [2, 5], [1, 1], [2, 7], [0, 8], [1, 3], [0, 6], [1, 7], [0, 2], [0, 0]} complexity 11 {[2, 9], [1, 8], [0, 10], [2, 5], [1, 0], [1, 11], [2, 1], [1, 3], [0, 6], [1, 7], [0, 2], [1, 4]} {[2, 6], [2, 9], [1, 8], [2, 5], [1, 0], [2, 10], [0, 11], [2, 1], [2, 2], [0, 3], [1, 4], [0, 7]} {[0, 5], [1, 0], [2, 4], [2, 7], [2, 10], [0, 11], [0, 8], [2, 1], [1, 3], [1, 9], [0, 2], [1, 6]} complexity 13 {[1, 13], [1, 5], [2, 6], [1, 8], [1, 0], [2, 10], [0, 11], [2, 2], [1, 9], [0, 3], [1, 4], [0, 7]} complexity 15 {[2, 9], [1, 8], [0, 10], [2, 5], [1, 0], [1, 11], [1, 15], [2, 1], [0, 6], [1, 7], [0, 2], [1, 4]} {[2, 9], [1, 8], [0, 10], [2, 5], [1, 0], [1, 11], [0, 14], [1, 15], [2, 1], [0, 6], [1, 7], [1, 4]} {[2, 9], [1, 8], [0, 10], [2, 5], [1, 0], [2, 13], [1, 11], [0, 14], [1, 15], [0, 6], [1, 7], [1, 4]}
Message: 9223 - Contents - Hide Contents Date: Sat, 17 Jan 2004 23:48:48 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> On a unit-length odd-limit lattice both 9 and 11 have length 1. >>> I'm not claiming anything necessarily good about this, note. >>> I am asking for comments about it however. >>>> OK . . . this, I suppose, is Paul Hahn's lattice or something? >> For his diameter measure I think it is. >>>>>>>> The taxicab distance on this lattice is log(odd-limit). >>>>>>>>>>>> No it isn't -- try 9:5 for example. >>>>>>>>>> This is what you were claiming in 1999. >>>>> >>>>> "" >>>>> But the basic insight is that a triangular lattice, with >>>>> Tenney-like lengths, a city-block metric, and odd axes or >>>>> wormholes, agrees with the odd limit perfectly, and so is >>>>> the best octave-invariant lattice representation (with >>>>> associated metric) for anyone as Partchian as me. >>>>> "" >>>>>>>> Right -- you need those odd axes, which screws up uniqueness, >>>> and thus most of how we've been approaching temperament. >>>>>> But does the metric agree with log(odd-limit) or not? >>> For 9:5, log(oddlimit) is log(9). If you run it through >>> the "norm" you get... 2log(3) + log(5). >>>> No, because 9 has its own axis. >> It's still different than log(odd-limit), and in fact > log(5) + log(9) = 2log(3) + log(5).You're forgetting that 5:3 has its own rung in this lattice, with length log(5), since the 'odd-limit' of 5:3 is 5 (more correctly, 5:3 is a ratio of 5).>>>> Not the same, >>> it seems. However if you followed the >>> lumma.org/stuff/latice1999.txt link, >>>> The page cannot be found. >> Typo here; "lattice".The page cannot be found.>>> apparently Paul Hahn >>> did present a metric that agrees with log(odd-limit). >>>> Can you re-present it? >> See the above url.The page still cannot be found.
Message: 9224 - Contents - Hide Contents Date: Sat, 17 Jan 2004 23:49:32 Subject: Re: summary -- are these right? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> It's a rectangular Thing with an axis for each odd number. >>>> It's actually triangular, and is what Erv Wilson uses to map out his >> CPSs. >> The Thing I was referring to here was most certainly rectangular. > > -CarlWell then it's no Thing that I've ever thought about or talked about or heard of before!
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