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).

6000
6050
6100
6150
**6200**
6250
6300
6350
6400
6450
6500
6550
6600
6650
6700
6750
6800
6850
6900
6950

**6200 -**
6225 -

Message: 6200 - Contents - Hide Contents Date: Sun, 26 Jan 2003 15:48:38 Subject: Re: Graham's top 20, with standard vals From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Gene Ward Smith wrote: >>> Vals are consistent by definition. What I meant was, wedging the val with the wedgie will give a zero vector. >> Say what? I thought a val was the complement of a vector.It's the dual of a vector, if by a vector you mean an interval in Monzo notation. Due to the magic of Poincare duality, you can wedge with either a vector or a val.>> The missing ones are the two on the list with only one standard val; you may not be using only standard vals. I wish you would explain this. By "standard" I mean the mapping is determined by rounding >> log2(p) for prime p to the nearest integer. >> I did explain this -- I'm not only taking nearest prime approximations > any more.So what are you taking?> Tricontaheximal is h72&h36. I don't see why you missed that.Because I was using a program which gave me a val from a wedgie plus a number, and for 36 it gave me [36, 57, 83, 101, 124, 133], which also works. The right way to do it would be to wedge with the standard val and see if you get a zero vector.

Message: 6201 - Contents - Hide Contents Date: Sun, 26 Jan 2003 03:47:43 Subject: Re: Graham's Top 20 13-limit temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> i have access to a 2.4 GHz machine for running Matlab overnight or >> for however long it takes. i'd be happy to try whatever algorithms >> you wish to spell out. >> One page claims Matlab is implemented in C. I seem to think Maple > is implemented in Maple, but I can't find that in the manual now. > I'd be surprised if either of them were faster than python, but > I could very well be wrong.Maple is in C also, but it isn't designed for speed. For instance, the float data type has a precision defined by "Digits", and the int data type allows for ints as big as the machine can handle.

Message: 6202 - Contents - Hide Contents Date: Sun, 26 Jan 2003 22:36:27 Subject: Re: Graham's top 20, with standard vals From: Carl Lumma>> >ay what? I thought a val was the complement of a vector. >>It's the dual of a vector, if by a vector you mean an interval >in Monzo notation. Due to the magic of Poincare duality, you >can wedge with either a vector or a val.I may be dangerously close to understanding vals. I've read the definition in monz's dictionary. Anybody care to give an example? Maybe Paul could shed some light on a layman's definition. -Carl

Message: 6203 - Contents - Hide Contents Date: Sun, 26 Jan 2003 05:59:28 Subject: Vals vs commas From: Gene Ward Smith To find a linear temperament wedgie we can use two vals, pi(n)-2 commas. As n increases, the number of combinations of a list of N commas taken pi(n)-2 at a time increases, whereas the number of standard vals and the number taken in pairs is steady, so one question is, do we need to worry about commas? We run into a problem with the strictly val approach if fewer than two standard vals cover the temperament in question; this can happen because it is of relatively low complexity, or because it has a small period. One way out of this is to include non-standard vals--that is, look at various second best choices for mappings to primes; this however, increases the computational burden also. Graham reports that he didn't find he needed the comma list approach. I can't quite see how this happened, since the Tricontaheximal temperament, with a period of 33 1/3 cents, has only one standard val, namely 72. Other temperaments are skating close to the wire by having only two standard vals--Hemififths with 41 and 58, and Diaschismic with 46 and 58 among the top four, which is all I have checked.

Message: 6204 - Contents - Hide Contents Date: Sun, 26 Jan 2003 22:37:35 Subject: Re: Graham's Top 20 13-limit temperaments From: Graham Breed Gene Ward Smith wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote: >>>> i have access to a 2.4 GHz machine for running Matlab overnight or >>> for however long it takes. i'd be happy to try whatever algorithms >>> you wish to spell out. >>>> One page claims Matlab is implemented in C. I seem to think Maple >> is implemented in Maple, but I can't find that in the manual now. >> I'd be surprised if either of them were faster than python, but >> I could very well be wrong. > >> Maple is in C also, but it isn't designed for speed. For instance, the float data type has a precision defined by "Digits", and the int data type allows for ints as big as the machine can handle.This has come up on comp.lang.python. People who've used both say that Numeric Python is slightly faster than Matlab, although Matlab's matrix operations are faster. Also, if there's a native library for Matlab but not Python then Matlab's much faster. There are also people using Python to drive Matlab. Probably Maple is similar. As Python doesn't come with arbitrary precision floating point, Maple will be faster when you need it. Python does have arbitrary sized integers, and they now interact seemlessly with the normal integers. So it looks comparable to Maple for what we need. However, the Numeric extensions use a Fortran library that only works with floating point -- there aren't any routines to efficiently find the adjoint of an integer matrix. There's also nothing for wedge products. Graham

Message: 6205 - Contents - Hide Contents Date: Sun, 26 Jan 2003 07:24:43 Subject: Re: Temperament finder update From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote: >>> takes about 50 seconds. Those results >> >> 1/2, 16.4 cent generator * [with cont.] (Wayb.) >> >> are the same as for the unison vector search! >> Not that surprising, but the comma search seems more likely to >catch oddball systems. I suppose I could try to identify these.by duality, isn't the val search as likely to catch oddball systems, though "differently odd", as the comma search?

Message: 6206 - Contents - Hide Contents Date: Sun, 26 Jan 2003 07:28:06 Subject: Re: Graham's Top 20 13-limit temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> i have access to a 2.4 GHz machine for running Matlab overnight or >>> for however long it takes. i'd be happy to try whatever algorithms >>> you wish to spell out. >>>> One page claims Matlab is implemented in C. I seem to think Maple >> is implemented in Maple, but I can't find that in the manual now. >> I'd be surprised if either of them were faster than python, but >> I could very well be wrong. >> Maple is in C also, but it isn't designed for speed. For instance, >the float data type has a precision defined by "Digits", and the int >data type allows for ints as big as the machine can handle.well then will somebody *please* *please* . . . pretty please :) :) :) calculate the numerators and denominators here which came out in scientific notation, making it impossible for yahoo to sort by denominator: Yahoo groups: /tuning/database? * [with cont.] method=reportRows&tbl=10&sortBy=4

Message: 6207 - Contents - Hide Contents Date: Sun, 26 Jan 2003 07:40:52 Subject: hi everybodyyyyyyyyyyyyyyyyyyyyyyyyy From: wallyesterpaulrus on this list, one group of people is talking about equal temperaments which belong to important families of tunings (each family being where a particular set of unison vectors vanishes), another group is concerned with notating equal temperaments according to where the potential unison vectors lie relative to their chains of fifths, and the two groups are not talking to one another. am i perceiving the situation correctly?

Message: 6208 - Contents - Hide Contents Date: Sun, 26 Jan 2003 08:06:36 Subject: Re: Temperament finder update From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> by duality, isn't the val search as likely to catch oddball systems, > though "differently odd", as the comma search?The comma system is likely to catch low complexity systems missed by the val search; one idea is to do two searches, one of which is a comma search using only relatively large commas. The comma search might miss systems with one or more dud commas, meaning equivalences which don't actully contribute much of practical use.

Message: 6209 - Contents - Hide Contents Date: Sun, 26 Jan 2003 09:41:16 Subject: Re: Temperament finder update From: Graham Breed Carl Lumma wrote:> numpy or Numarray?The one you get from "ppm install Numeric" in ActivePython. I think that's numpy. Graham

Message: 6210 - Contents - Hide Contents Date: Sun, 26 Jan 2003 10:41:22 Subject: Re: Graham's Top 20 13-limit temperaments From: Graham Breed wallyesterpaulrus wrote:> calculate the numerators and denominators here which came out in > scientific notation, making it impossible for yahoo to sort by > denominator: > > Yahoo groups: /tuning/database? * [with cont.] > method=reportRows&tbl=10&sortBy=4 " large limma", "0 3 -2", ... * [with cont.] (Wayb.) Graham

Message: 6211 - Contents - Hide Contents Date: Mon, 27 Jan 2003 08:51:20 Subject: Re: Graham's top 20, with standard vals From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>>> my reasoning in not being particularly interested in this? it's that, >> like graham, i don't think the standard val is necessarily the best >> val for any ET. try 64-equal in the 5-limit. >> Do you have an alternative definition?how about the choice that minimizes the rms error?

Message: 6212 - Contents - Hide Contents Date: Mon, 27 Jan 2003 12:07:46 Subject: Calculating geometric complexity II From: Gene Ward Smith Here are Maple routines which have the exact coefficients. They are not, of course, computationally effiecient, but it would be easy to calculate the logarithms only once if that is a problem, though I havn't found complexity calculations to be a bottleneck. These should be readily translatable to Matlab, Python, or anything else. gc7 := proc(l) # l is 7-limit wedgie sqrt(evalf(1/4*(-ln(5)^4+4*ln(5)^2*ln(7)^2)*l[1]^2 +1/4*(-ln(3)^4+4*ln(3)^2*ln(7)^2)*l[2]^2 +1/4*(-ln(3)^4+4*ln(3)^2*ln(5)^2)*l[3]^2 +1/2*(ln(3)^2*ln(5)^2-2*ln(3)^2*ln(7)^2)*l[1]*l[2] -1/2*ln(3)^2*ln(5)^2*l[1]*l[3]+ 1/2*(ln(3)^4-2*ln(3)^2*ln(5)^2)*l[2]*l[3])) end: gc11 := proc(l) # l is 11-limit linear wedgie sqrt(evalf((-1/4*ln(5)^2*ln(7)^4-1/4*ln(11)^2*ln(5)^4+ln(7)^2*ln(11)^2*ln(5)^2)*l[1]^2 +(ln(11)^2*ln(7)^2*ln(3)^2-1/4*ln(3)^4*ln(11)^2-1/4*ln(3)^2*ln(7)^4)*l[2]^2 +(ln(3)^2*ln(5)^2*ln(11)^2-1/4*ln(3)^2*ln(5)^4-1/4*ln(3)^4*ln(11)^2)*l[3]^2 +(ln(7)^2*ln(3)^2*ln(5)^2-1/4*ln(3)^2*ln(5)^4-1/4*ln(7)^2*ln(3)^4)*l[4]^2 +(1/2*ln(3)^2*ln(5)^2*ln(11)^2-ln(11)^2*ln(7)^2*ln(3)^2+1/4*ln(3)^2*ln(7)^4)*l[1]*l[2] +(-1/2*ln(3)^2*ln(5)^2*ln(11)^2+1/4*ln(7)^2*ln(3)^2*ln(5)^2)*l[1]*l[3] -1/4*ln(7)^2*ln(3)^2*ln(5)^2*l[1]*l[4] +(-ln(3)^2*ln(5)^2*ln(11)^2+1/2*ln(7)^2*ln(3)^2*ln(5)^2-1/4*ln(7)^2*ln(3)^4+ 1/2*ln(3)^4*ln(11)^2)*l[2]*l[3]+(-1/2*ln(7)^2*ln(3)^2*ln(5)^2+1/4*ln(7)^2*ln(3)^4)*l[2]*l[4] +(-ln(7)^2*ln(3)^2*ln(5)^2+1/2*ln(3)^2*ln(5)^4+1/4*ln(7)^2*ln(3)^4)*l[3]*l[4])) end: gpc11 := proc(l) # l is 11-limit planar wedgie sqrt(evalf((-1/4*ln(7)^4+ln(7)^2*ln(11)^2)*l[1]^2 +(-1/4*ln(5)^4+ln(5)^2*ln(11)^2)*l[2]^2 +(-1/4*ln(5)^4+ln(5)^2*ln(7)^2)*l[3]^2 +(-1/4*ln(3)^4+ln(3)^2*ln(11)^2)*l[4]^2 +(-1/4*ln(3)^4+ln(3)^2*ln(7)^2)*l[5]^2 +(-1/4*ln(3)^4+ln(3)^2*ln(5)^2)*l[6]^2 +(-1/2*ln(5)^2*ln(7)^2+ln(5)^2*ln(11)^2)*l[1]*l[2] +(-1/2*ln(5)^2*ln(7)^2)*l[1]*l[3] +(-1/2*ln(3)^2*ln(7)^2+ln(3)^2*ln(11)^2)*l[1]*l[4] +(-1/2*ln(3)^2*ln(7)^2)*l[1]*l[5] +(1/2*ln(3)^2*ln(5)^2-1/2*ln(3)^2*ln(7)^2)*l[1]*l[6] +(-1/2*ln(5)^4+ln(5)^2*ln(7)^2)*l[2]*l[3] +(-1/2*ln(3)^2*ln(5)^2+ln(3)^2*ln(11)^2)*l[2]*l[4] +(-ln(3)^2*ln(5)^2+ln(3)^2*ln(7)^2)*l[2]*l[5] +(-1/2*ln(3)^2*ln(5)^2)*l[2]*l[6] +(-1/2*ln(3)^2*ln(5)^2+ln(3)^2*ln(7)^2)*l[3]*l[5] +(-1/2*ln(3)^2*ln(5)^2)*l[3]*l[6] +(-1/2*ln(3)^4+ln(3)^2*ln(7)^2)*l[4]*l[5] +(-1/2*ln(3)^4+ln(3)^2*ln(5)^2)*l[4]*l[6] +(-1/2*ln(3)^4+ln(3)^2*ln(5)^2)*l[5]*l[6])) end:

Message: 6213 - Contents - Hide Contents Date: Mon, 27 Jan 2003 09:17:34 Subject: Re: Graham's Top 20 13-limit temperaments From: Carl Lumma> better than any other complexity measure! cool, so you must > *really* like the heuristic for complexity . . . Apparently so. > my best recollection, off the top of my head: > > log-flat badness < 3500, rms error < 50 cents, geometric > complexity < 104-151 (doesn't matter where you draw the > line in this range).Ok, but two small nits: () Is that geometric complexity as Gene defines it? () Being that badness is just a combination error and complexity, why is it needed / how can it change the bounds on the list? -Carl

Message: 6214 - Contents - Hide Contents Date: Mon, 27 Jan 2003 12:34:21 Subject: Re: Calculating geometric complexity II From: Graham Breed Gene Ward Smith wrote:> Here are Maple routines which have the exact coefficients. They are not, of course, computationally effiecient, but it would be easy to > calculate the logarithms only once if that is a problem, though I havn't found complexity calculations to be a bottleneck. These should be readily translatable to Matlab, Python, or anything else.How are you indexing your wedgies? I use tuples, so that when multiplying 1-vectors, z[i, j] = x[i,] + y[j,] where x[0,] is the octave coefficient, x[1,] the 3:1 and so on. Can you provide conversion tables between your [i] and my [i,j]? I think I've got the idea of vals as well. A dual isn't the same as a complement! Using ^ for the wedge product, and ~ for the complement, we have h12^~comma = ~comma^h12 = {} where "h12" is the val for 5-limit 12-equal, "comma" is the unison vector for 81:80 and {} is the empty wedgie. Vals and unison vectors are both 1-vectors. For duality, I'll have to add a flag to each object. So the complement operation also inverts the flag. A unison vector has the dual flag set to 0 and a val has the dual flag set to 1. To compute a wedge product, both dual flags has to agree. So when you ask to calculate h12^comma, the function can look at the two dual flags, see they aren't the same, and take the complement of the second element to make it so. That means, asking for h12^comma means you get h12^~comma and it doesn't matter if you meant ~h12^comma because the dual flag gets set again for the next step of the calculation. And in this case the result is the same anyway. You can also say that h12^h7 == comma because h12^h7 will have its dual flag set, and the comparison function knows it really has to return h12^h7 == ~comma and the routine for calculating a linear temperament knows it needs to start with a wedgie that has its dual flag cleared, and so if you feed it h12^h7 it converts it to ~(h12^h7). I still don't know how to store a multivector so that it's its own dual which seems to be what you're doing. Graham

Message: 6215 - Contents - Hide Contents Date: Mon, 27 Jan 2003 09:37:43 Subject: Re: Graham's Top 20 13-limit temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> better than any other complexity measure! cool, so you must >> *really* like the heuristic for complexity . . . > > Apparently so. >>> my best recollection, off the top of my head: >> >> log-flat badness < 3500, rms error < 50 cents, geometric >> complexity < 104-151 (doesn't matter where you draw the >> line in this range). >> Ok, but two small nits: > > () Is that geometric complexity as Gene defines it? yes. > () Being that badness is just a combination error and > complexity, why is it neededbecause otherwise you'd have a huge number of temperaments, and not the same number in each complexity range. for example, imagine how many possible temperaments there must be with rms error < 50 cents and complexity between, say, 74 and 104. some huge number.

Message: 6216 - Contents - Hide Contents Date: Mon, 27 Jan 2003 13:16:03 Subject: Re: Calculating geometric complexity II From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Gene Ward Smith wrote:>> Here are Maple routines which have the exact coefficients. They are not, of course, computationally effiecient, but it would be easy to >> calculate the logarithms only once if that is a problem, though I havn't found complexity calculations to be a bottleneck. These should be readily translatable to Matlab, Python, or anything else. >> How are you indexing your wedgies? I use tuples, so that when > multiplying 1-vectors, > > z[i, j] = x[i,] + y[j,] > > where x[0,] is the octave coefficient, x[1,] the 3:1 and so on. Can you > provide conversion tables between your [i] and my [i,j]?I'm not sure if we are speaking the same language, but I'm using lexicographical order; that is, z[0,1], z[0,2] .... z[0,n] would be followed by z[1,2]...z[1,n] and so forth. This gives a linear temperament wedgie as the product of two vals, and puts the 2-part, which is related to the generators column of the period-generator matrix, at the beginning.> I think I've got the idea of vals as well. A dual isn't the same as a > complement! Using ^ for the wedge product, and ~ for the complement, we > have > > h12^~comma = ~comma^h12 = {} > > where "h12" is the val for 5-limit 12-equal, "comma" is the unison > vector for 81:80 and {} is the empty wedgie.Are you still using empty wedgies for zero vectors? I hope this isn't giving problems. In any case, the above is definitional; ~comma is the 3-product such that h ^ ~comma = h(comma), so that we can identify compliments with duals.> Vals and unison vectors are both 1-vectors. For duality, I'll have to > add a flag to each object. So the complement operation also inverts the > flag. A unison vector has the dual flag set to 0 and a val has the dual > flag set to 1. Makes sense. > I still don't know how to store a multivector so that it's its own dual > which seems to be what you're doing.I'm simply being unsophisticated about it--I store the wedgies as lists, and reverse the ordering when I compute from commas, etc. in order to get the lists to be the same.

Message: 6217 - Contents - Hide Contents Date: Mon, 27 Jan 2003 10:55:35 Subject: Re: Graham's Top 20 13-limit temperaments From: Carl Lumma>because otherwise you'd have a huge number of temperaments, and not >the same number in each complexity range. for example, imagine how >many possible temperaments there must be with rms error < 50 cents >and complexity between, say, 74 and 104. some huge number.Right on. -C.

Message: 6218 - Contents - Hide Contents Date: Mon, 27 Jan 2003 14:52:55 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote: >>> ***** HEY IF ANYBODY ELSE OUT THERE IS READING THIS, HERE'S A >> QUESTION: What other ETs above 494 besides 612 and 624 would you >> want to notate -- ones in which the 5' comma (a.k.a, historical 5- >> schisma, 32768:32805) is either a single degree of the ET or vanishes? >> 665, 684, 730, 742 and 836.Thanks, Gene. I'll also add 653 to that list. But we won't be able to notate 684, because the 5' comma vanishes and no other symbol in the sagittal notation would represent a single degree, either. --George

Message: 6219 - Contents - Hide Contents Date: Mon, 27 Jan 2003 15:15:59 Subject: Re: hi everybodyyyyyyyyyyyyyyyyyyyyyyyyy From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> on this list, > > one group of people is talking about equal temperaments which belong > to important families of tunings (each family being where a > particular set of unison vectors vanishes), > > another group is concerned with notating equal temperaments according > to where the potential unison vectors lie relative to their chains of > fifths, > > and the two groups are not talking to one another. > > am i perceiving the situation correctly?Hi, Paul. Thanks for checking up on us. I did participate in the discussions about hemififth and kleismic temperaments (but Gene eventually realized that I was really addressing catakleismic). Nothing in these discussions resulted in any changes to any ET notations that Dave and I have already agreed on. The catakleismic discussion did cover some larger divisions that we have not yet addressed, so I can't say yet how the results using these two approaches might differ. Another factor in all of this is our recent introduction of the 5' comma (traditional 5-schisma) into the notation, which is going to affect how some of these things are done. So there has been a limited amount of communication about these things, but we've been pretty busy working on our separate approaches, and eventually we're going to have to bring it all together and compare notes. --George

Message: 6220 - Contents - Hide Contents Date: Mon, 27 Jan 2003 15:33:53 Subject: Re: hi everybodyyyyyyyyyyyyyyyyyyyyyyyyy From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote: >>> Gene has contributed to the Common notation thread regarding notating >> linear temperaments. But I agree he seems to have been following the >> idea that a temperament can be notated adequately using the notation >> for its most representative ET, and apparently assuming that George >> and I have already found the best notation for that ET (within the >> constraints we have imposed upon ourselves). neither of which may e true. >> Oh. Well, darn. >>> Although chains of fifths are the backbone of the sagittal notation, >> this does not prevent it from notating ETs in LT-specific ways, so the >> same ET can be notated differently depending on which LT you are >> considering it as. I recently gave some examples in a "Notating Linear >> Temperaments" thread (or some such), but no one responded or carried >> it forward. >> I can't follow these things when they involve the symbols, not atleast unless I have a key handy (which I don't.) I mentioned in message #5403 that I made a quick-reference table for the most common symbols in a file when I had to answer your question about "what the 11-diesis is": Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/quickref.txt The actual symbols may be seen in these files: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/AdaptJI.gif Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/Symbols6.gif I hope that this is good enough for now, until we have a decent explanation of the notation available. --George

Message: 6221 - Contents - Hide Contents Date: Mon, 27 Jan 2003 08:08:37 Subject: Re: A common notation for JI and ETs From: monz> From: <gdsecor@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, January 27, 2003 6:52 AM > Subject: [tuning-math] Re: A common notation for JI and ETs > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith > <genewardsmith@j...>" <genewardsmith@j...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" > <gdsecor@y...> wrote: >>>>> ***** HEY IF ANYBODY ELSE OUT THERE IS READING THIS, >>> HERE'S A QUESTION: What other ETs above 494 besides >>> 612 and 624 would you want to notate -- ones in which >>> the 5' comma (a.k.a, historical 5-schisma, 32768:32805) >>> is either a single degree of the ET or vanishes? >>>> 665, 684, 730, 742 and 836. >> Thanks, Gene. I'll also add 653 to that list. > > But we won't be able to notate 684, because the 5' comma > vanishes and no other symbol in the sagittal notation > would represent a single degree, either. > > --Georgehow about 768? ... because it's the tuning resolution for a number of popular electronic instruments. -monz

Message: 6222 - Contents - Hide Contents Date: Mon, 27 Jan 2003 16:29:31 Subject: Re: Calculating geometric complexity II From: Graham Breed Gene Ward Smith wrote:> I'm not sure if we are speaking the same language, but I'm using lexicographical order; that is, z[0,1], z[0,2] .... z[0,n] would > be followed by z[1,2]...z[1,n] and so forth. This gives a linear temperament wedgie as the product of two vals, and puts the 2-part, which is related to the generators column of the period-generator matrix, at the beginning.Oh, that's good. It should be the same as my invariant. But are 7-limit wedge products taken from vectors or vals? I get 7-limit meantone as 21.97, 11-limit meantone as 31.72 and h12^h19^h22 in the 11-limit as 29.52. The planar temperament with 441:440 and 225:224 is 34.44.> Are you still using empty wedgies for zero vectors? I hope this isn't giving problems. In any case, the above is definitional; ~comma is the > 3-product such that h ^ ~comma = h(comma), so that we can identify > compliments with duals.Empty wedgies are empty wedgies. I haven't had any trouble with them.> I'm simply being unsophisticated about it--I store the wedgies as lists, and reverse the ordering when I compute from commas, etc. in order to get the lists to be the same.That sounds like taking the complement. I thought you said you didn't have to because you were using duality. And how can you be sure that reversing the list will do the trick? Some of the coefficients should be negated if you aren't using a special ordering. Graham

Message: 6223 - Contents - Hide Contents Date: Mon, 27 Jan 2003 16:38:22 Subject: Re: Calculating geometric complexity II From: Graham Breed Gene Ward Smith wrote:> I'm not sure if we are speaking the same language, but I'm using lexicographical order; that is, z[0,1], z[0,2] .... z[0,n] would > be followed by z[1,2]...z[1,n] and so forth. This gives a linear temperament wedgie as the product of two vals, and puts the 2-part, which is related to the generators column of the period-generator matrix, at the beginning.Oh, and I expect you're indexing from 1 as well. In which case I get 7-limit meantone 23.76 11-limit meantone 23.85 h12^h19^h22 22.77 441:440 ^ 225:224 28.57 Graham

Message: 6224 - Contents - Hide Contents Date: Mon, 27 Jan 2003 19:38:03 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:>>>> >>>> ***** HEY IF ANYBODY ELSE OUT THERE IS READING THIS, >>>> HERE'S A QUESTION: What other ETs above 494 besides >>>> 612 and 624 would you want to notate -- ones in which >>>> the 5' comma (a.k.a, historical 5-schisma, 32768:32805) >>>> is either a single degree of the ET or vanishes? >>>>>> 665, 684, 730, 742 and 836. >>>> Thanks, Gene. I'll also add 653 to that list. >> >> But we won't be able to notate 684, because the 5' comma >> vanishes and no other symbol in the sagittal notation >> would represent a single degree, either. >> >> --George >> how about 768? ... because it's the tuning resolution for a > number of popular electronic instruments. > > -monzSorry, that one can't be done with the commas that we have. The 5' comma (32768:32805) is actually -1 degrees, so it would be too confusing to use it. And while the 19 comma (512:513) could be used as 1 degree, there's nothing for 2, 3, and 4 degrees. Anyway, 768 isn't even 1,3,9-consistent, hence not very desirable musically. But after taking a quick look at 653, 665, 730, 742 and 836, I think those five are all do-able. --George

6000
6050
6100
6150
**6200**
6250
6300
6350
6400
6450
6500
6550
6600
6650
6700
6750
6800
6850
6900
6950

**6200 -**
6225 -