Tuning-Math Archive Section 1: 175

Message: 175 - Contents - Hide Contents Date: Fri, 08 Jun 2001 22:19:58 Subject: Re: The Unidala From: Paul Erlich --- In tuning-math@y..., JGill99@i... wrote:

> Requests for clarifications are welcome.

I request clarification. But I don't recall seeing your name before, so perhaps I should first say, welcome! It's going to be difficult to communicate at first since you come to this (as we all did initially) with your own language. So my suggestion is, stick around for a while, check out the posts on the main tuning list, and eventually you may be able to better express your ideas in a way which more people will be able to understand. I look forward to that day!

Message: 176 - Contents - Hide Contents Date: 8 Jun 2001 15:48:42 -0700 Subject: Fokker periodicity blocks corresponding to blackjack and canasta From: paul@s... When I plug the set of unison vectors 35:36 224:225 2400:2401 into my Fokker periodicity block program, I get: cents numerator denominator 0 1 1 35.697 49 48 119.44 15 14 147.43 49 45 231.17 8 7 266.87 7 6 350.62 60 49 386.31 5 4 463.07 98 75 498.04 4 3 582.51 7 5 617.49 10 7 701.96 3 2 736.93 75 49 813.69 8 5 849.38 49 30 933.13 12 7 968.83 7 4 1052.6 90 49 1080.6 28 15 1164.3 96 49 Clearly there are a lot of "broken" consonances here. Tempering out the 224:225 and 2400:2401 would convert this into the blackjack scale as we've presented and latticed it. So this supports my Hypothesis. Here are the step sizes: cents numerator denominator 35.697 49 48 83.746 360 343 27.985 686 675 83.746 360 343 35.697 49 48 83.746 360 343 35.697 49 48 76.756 392 375 34.976 50 49 84.467 21 20 34.976 50 49 84.467 21 20 34.976 50 49 76.756 392 375 35.697 49 48 83.746 360 343 35.697 49 48 83.746 360 343 27.985 686 675 83.746 360 343 35.697 49 48

Message: 178 - Contents - Hide Contents Date: Fri, 08 Jun 2001 23:54:28 Subject: Re: CS From: Dave Keenan --- In tuning-math@y..., carl@l... wrote:

> Really, Dave? No comment the CS alternative I fashioned at > your request? I'm surprised.

Sorry Carl, but when I didn't know if this list was going to continue ... So you're proposing a method that will give the degree of "CS-ness" of a scale. The completely general one would probably work but is way too complicated for the original purpose. For MOS only, you say L/s but don't say how to use it. A MOS can fail to be CS when L/s = 2 or when L/s = 3 etc, but not every MOS will do so. On Wilson's definition, no irrational generator can result in a non-CS MOS. But clearly some will be "nearly non-CS". Maybe this can be refined, but I've lost interest I'm afraid. Or maybe you mean to use purely the magnitude of L/s as a measure of how improper a scale is (nothing to do with CS)? The original purpose was in Grahams temperament-finder program, to rank the temperaments. So we don't bother looking at complete junk we use a Figure-of-Demerit that attempts to (crudely) include both melodic and harmonic factors. Currently it is a normalised max-abs error combined with the size of the smallest MOS containing a complete otonality. Of course some of these MOS could be wildly improper. We wondered whether we should use the smallest strictly proper MOS, but this seemed too strong. Paul suggested CS. I had a wrong idea about CS. Turns out it's pretty meaningless for irrational-fraction-of-an-octave generators. I'll forget it. Carl maybe you're suggesting we use the size of the smallest otonality-containing MOS that has L/s < 3 (or something), to make sure we're not counting a MOS that's completely useless melodically? Good idea. -- Dave Keenan

Message: 179 - Contents - Hide Contents Date: Sat, 09 Jun 2001 00:23:43 Subject: Re: CS From: carl@l...

> For MOS only, you say L/s but don't say how to use it.

An MOS is not CS iff L/s = 2. This is a corollary of a formula by John Chalmers.

>A MOS can fail to be CS ... when L/s = 3 Example? > The original purpose was in Grahams temperament-finder program, > to rank the temperaments. So we don't bother looking at complete > junk we use a Figure-of-Demerit that attempts to (crudely) > include both melodic and harmonic factors.

I wasn't here for the beginning of the thread, so I don't know what your after. I doubt CS (or anything like it) is what you're looking for. CS and propriety have no direct meaning for scales with more than about 14 notes. They may simplify things as far as transposing and chord patterns are concerned, but it makes no sense to talk of melodically useful or not scales with 21 notes. Any reasonably octave-periodic scale with 21 tones will be full of both melodically useful and melodically useless scales, and propriety hardly matters for chromatic ornamentation.

> Currently it is a normalised max-abs error combined with the > size of the smallest MOS containing a complete otonality.

That sounds better, though I still don't know what you're trying to find.

> Carl maybe you're suggesting we use the size of the smallest > otonality-containing MOS that has L/s < 3 (or something), to > make sure we're not counting a MOS that's completely useless > melodically? Not me. > Good idea.

Maybe; I don't know. -Carl

Message: 181 - Contents - Hide Contents Date: Sat, 09 Jun 2001 04:08:07 Subject: Re: big "mean" math question From: jpehrson@r... --- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote: Yahoo groups: /tuning-math/message/180 * [with cont.]

> Is it my imagination, or has nobody already caught the error in this? > Paul E, even you??? > > Before you take the square root, you divide by the number of values > whose square has been summed. Thus, the RMS of 3 and 4 is: > > RMS = sqrt((3^2 + 4^2) / 2) > = sqrt((9 + 16) / 2) > = sqrt(12.5) > =~ 3.54 > > NOT sqrt(25) = 5!! > > JdL

Actually, John... this is interesting because, if I'd known this, I probably wouldn't have been quite as "mystified" as I was after Graham's original post. The method you outline immediately above seems somewhat "averagy" to me... so it would have seemed more sensible. Here was Graham's original quote from post 24541:

>Averages are trickier, you do need to consider all intervals then. >The most popular is the root mean squared (RMS). So you take the >errors in all intervals, square them all, add them together and >return the square root.

________ ______ _____ Joseph Pehrson

Message: 183 - Contents - Hide Contents Date: Sat, 8 Jun 2030 23:04:05 Subject: Re: The Unidala From: monz Hi Jay, Thanks for mentioning me so prominently in your work! I'm writing to you privately because I really want to study your posts a little more before commenting publicly, but felt that I really *had* to respond at least to you. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 184 - Contents - Hide Contents Date: Sat, 8 Jun 2030 23:27:34 Subject: Re: The Unidala From: monz

> I'm writing to you privately Oops. -monz Yahoo! GeoCities * [with cont.] (Wayb.)

"All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 189 - Contents - Hide Contents Date: Sat, 09 Jun 2001 12:54:38 Subject: Re: big "mean" math question From: jpehrson@r... --- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote: Yahoo groups: /tuning-math/message/186 * [with cont.]

> [Joseph Pehrson wrote:]

>> Actually, John... this is interesting because, if I'd known this, I >> probably wouldn't have been quite as "mystified" as I was after >> Graham's original post. The method you outline immediately above >> seems somewhat "averagy" to me... so it would have seemed more >> sensible. > >> Here was Graham's original quote from post 24541: >

>>> Averages are trickier, you do need to consider all intervals then. >>> The most popular is the root mean squared (RMS). So you take the >>> errors in all intervals, square them all, add them together and >>> return the square root. >

> Right. That'd be the "Root Sum Square", which, as you've surmised, > wouldn't be very "averagy". In fact, I'm not sure what it would be > useful for. I'm sure Graham, and probably all the other people who > responded to your post yesterday, _do_ know the correct definition,

but all had a "brain fart" (which I know a lot about, 'cause I get them all the time!).

> > The RMS value will always be less than the largest absolute value

which goes into its calculation (or equal if all input values are the same or -same). I can see that you were grasping for that in your original post. So, you have a better math sense than you realized!

> > JdL

Actually, John... that's pretty funny and, frankly, encouraging. It's never too late to learn at least *something!* I think part of the problem was the math "training" I had as a student. Math was always presented as a "practical" study with ugly and dull- looking "engineering type" books. No wonder I would read art and music books instead. The subject was *ruined* for me... or at least for *my* particular sensibilities... _______ _____ ________ Joseph Pehrson

Message: 190 - Contents - Hide Contents Date: Sat, 09 Jun 2001 18:45:05 Subject: Re: linear approximation From: Paul Erlich --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

>> It seems to me the opposite would be true. Allowing the whole line to >> shift would allow, in general, a better approximation to the step >> sizes, wouldn't it? >

> But then 1/1 is not included in the approximation.

I was thinking that it was included. Just because the line doesn't have to pass exactly through 1/1 doesn't mean that the line is not _affected_ by the presence of the point at 1/1.

Message: 191 - Contents - Hide Contents Date: Sat, 09 Jun 2001 18:46:44 Subject: Re: uniqueness From: Paul Erlich --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> > Carl wrote:

>> What prompted the change? Did you find a bug? >

> Yes or at least I think I did. First I discovered > the wrong assumption of octave equivalence, so the > value for Bohlen-Pierce for example wouldn't be right. > Try for me the 19-tET case, I think it should be > 9, not 7 (without inversional equivalence).

In 19-tET, 6:7 and 7:8 are approximated by the same interval. So 9 is too high.

Message: 194 - Contents - Hide Contents Date: Sun, 10 Jun 2001 19:05:34 Subject: Re: proved From: Robert Walker Hi there, Sorry, gave wrong values in Herz for the c6-10 cents to c6 in steps of 2.5 cents. Should be 523.251 Hz 520.237 Hz 520.989 Hz 521.742 Hz 522.496 Hz 523.251 Hz Numbers I gave were for steps of 2 cents. Robert

Message: 196 - Contents - Hide Contents Date: Mon, 11 Jun 2001 12:43 +0 Subject: Re: Bobs one true tuning for the rest of his life... From: graham@m... In-Reply-To: <9g2ab8+cpvi@e...> Paul wrote:

> --- In tuning@y..., graham@m... wrote: >

>> FWIW, this gives the Miracle unison vector 540:539. >

> Hmm . . . how do you get this from 224:225 and 2400:2401? I know > 1024:1029 was one, but . . . I haven't slept so I'm not going to > attempt to prime-factorize in my head . . . you can reply to > tuning-math . . . BTW tuning-math lives on . . .

You can't, it's 11-prime limit. I haven't checked back, but I think it's one of the unison vectors I originally gave. I'm not sure offhand how you get it from the 385:384, 243:232 and 225:224 but I checked it with the defining matrix and it works. (2 3 1 -2 -1)(10 1) = (0 0) (16 1) (23 3) (28 3) (35 2) It's interesting that there are so many superparticular Miracle unison vectors. Is that unexpected? Graham

Message: 197 - Contents - Hide Contents Date: Mon, 11 Jun 2001 12:31:24 Subject: Re: Bobs one true tuning for the rest of his life... From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:

> In-Reply-To: <9g2ab8+cpvi@e...> > Paul wrote: >

>> --- In tuning@y..., graham@m... wrote: >>

>>> FWIW, this gives the Miracle unison vector 540:539. >>

>> Hmm . . . how do you get this from 224:225 and 2400:2401? I know >> 1024:1029 was one, but . . . I haven't slept so I'm not going to >> attempt to prime-factorize in my head . . . you can reply to >> tuning-math . . . BTW tuning-math lives on . . . >

> You can't, it's 11-prime limit. Duh! > I haven't checked back, but I think it's > one of the unison vectors I originally gave.

As I recall, those didn't seem to give Fokker periodicity blocks that quite agreed with the MIRACLE MOSs. I wonder why that is?

> I'm not sure offhand how > you get it from the 385:384, 243:232 and 225:224

I'll try that triplet along with 35:36 and see what kinds of blackjacks my FPB program gives.

> It's interesting that there are so many superparticular Miracle unison > vectors. Is that unexpected?

I think superparticulars are the smallest unison vectors for a given taxicab distance in the triangular lattice, if the lattice is constructed Kees' way. I was conjecturing to Dave Keenan that tempering out superparticular unison vectors, with number-size proportional to N, generally cause the constituent consonant intervals to be tempered by an amount proportional to 1/N^2. If a unison vector is not superparticular, but is instead K-particular, where K is the difference between numerator and denominator, then the amount of tempering of the consonances will be more like K/(N^2) -- i.e., more tempering.