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Message: 8100 Date: Tue, 11 Nov 2003 03:07:58 Subject: Re: Enharmonic diesis? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:

> > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> > > > > > i'm also trying to establish here a new standard usage > > > of brackets. since using commas to separate groups of > > > exponents in a monzo eliminates the need for contrasting > > > brackets to indicate the presence or absence of prime-factor 2, > > > i propose that we use angle-brackets for the prime-factors > > > themselves, and square-brackets for the actual monzo of > > > the exponents.

> > > > Well I suppose you could, but I don't think this is necessary, > > and we could save the angle-brackets for something more > > important. There is obviously no need for selective use of > > commas between the primes themselves. They provide no > > additional information there.

> > > huh? they would simply separate the groups of primes to > show how the exponents are grouped. i know that that's > "a given", so i guess maybe you're right.

I assumed that's what you intended, and there's no harm in a bit of redundancy. But I see this as _doubly_ redundant. If you know the system with the commas in the monzos, then you know that e.g. [1 1, 1] is a 2,3,5-monzo and not a 3,5,7-monzo, so adding the words "2,3,5-monzo" is redundant (but still a good idea). And if you're told it's a 2,3,5-monzo then you can ignore the commas and just line up exponents with primes. So "2,3 5-monzo" is doubly redundant and forces us to use something like the angle-brackets to hold it together: <2,3 5>-monzo. But while we intend the monzo [1, 2 3] to be quite different from the monzo [1 2, 3], a 2,3,5-monzo is of course no different from a <2,3 5>-monzo. Someone seeing the term "<2,3 5>-monzo" might wonder if there were such things as <2 3,5>-monzos and <2 3 5>-monzos, as different categories.

> > I would simply call these 2,3,5-monzos.

> > > simply using a comma between *every* prime, and no spaces. > i suppose i like that. (i don't sound too convinced, tho.)

I think mathematicians name things like that all the time. Don't they Gene? Don't forget to add the stuff about the selective comma(punctuation sense) convention to your "monzo" dictionary entry when you get a chance.

> > Maybe a good use for angle-brackets would be for wedgies > > since, as I understand it, these are in a _very_ different > > domain from that of monzos, and the angle-brackets are > > suggestive of wedges themselves.

> > great minds thinking alike! ;-) > > i had actually already thought of that too ... but since my > understanding of wedgies is lagging far behind that of many > of you others, i'll refrain from commenting further.

I'm afraid I don't understand them either. I couldn't even tell you the difference between a wedgie and a val. :-) I read Gene's "val" definition in your dictionary but I'm none the wiser. I don't see an entry for "wedgie". I guess some time I'd like to see some carefully explained examples of how vals and wedgies are used. I've never managed to follow it on tuning-math. I fully expect that when I eventually put in the effort to understand these things, I'll say "Is that all they are? Well why didn't you _say_ so?". :-) -- Dave Keenan

Message: 8101 Date: Tue, 11 Nov 2003 03:08:07 Subject: A Riemann zeta peek at 75 equal From: Gene Ward Smith I've put a jpg of Z(x) for values of x near 75 in the Photos directory. Here Z is the Riemann zeta function along the critical line, twisted in the usual way to get a real function of a real variable, and then rescaled by the factor ln(2)/2 pi in the argument, so as to make it correspond to divisions of the octave. The interest of 75 is that there is a zero of the Reimann zeta function near 75, and two reasonably large peaks, rather than the usual one, near to it. The highest peak is the flat octaves peak, which for a range of values around the peak corresponds to the 19-limit val [75, 119, 174, 211, 260, 278, 307, 319] which has 7-limit comma basis [225/224, 1728/1715, 15625/15309] The other peak is the sharp octaves peak; it has 19-limit val [75, 119, 174, 210, 259, 277, 306, 318] and comma basis [686/675, 875/864, 5120/5103] The standard val, in case anyone cares, would be [75, 119, 174, 211, 259, 278, 307, 319] The three are already distinct in the 11-limit.

Message: 8102 Date: Tue, 11 Nov 2003 03:21:23 Subject: Re: Enharmonic diesis? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> > Maybe a good use for angle-brackets would be for wedgies > > since, as I understand it, these are in a _very_ different > > domain from that of monzos, and the angle-brackets are > > suggestive of wedges themselves.

> > > great minds thinking alike! ;-)

I see no point in it myself; one is hardly likely to confuse a wedgie with an interval, though telling an 11-limit linear wedgie from an 11- limit planar wedgie is another matter. There are also vals to consider; it's too hard writing these as column vectors, which is really the clearest way to do it.

Message: 8103 Date: Wed, 12 Nov 2003 00:03:19 Subject: Re: Vals? From: Carl Lumma

>I've explained what a val is numerous times. I can't insist you pay >attention to everything I say; these days you and George tend to lose >me, after all, which is fair enough.

But you haven't explained how it works.

>If the standard 5-limit val >for 12-equal is [12 19 28] or something, how does it come from >round(n log2(p))? Oh n is 12, eh? So vals are uniquely identified >by this n? > >So how does one find a standard val for an odd limit (the start of >this thread, which perhaps George is still following)? Where do you >get your n?

-C.

Message: 8105 Date: Wed, 12 Nov 2003 09:04:50 Subject: Re: Definition of microtemperament From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: >

> > Gene would like the limit set at 1 c, although I haven't read why.

> > "Micro" to me means small enough that the error hardly matters.

Exactly what I said in my definition. It's nice that we agree on something. :-)

> > It's all pretty arbitrary, but I think we need to draw such a line > > somewhere.

> > There's always my magnitude scale, with lines differing by a factor > of two.

I can't find this by searching the archive. I tried all kinds of things. Please explain or give a URL. The term microtemperament has a long history of referring to temperaments with errors less than half that of 1/4-comma meantone. So the magnitude scale could go down by factors of two using the syntonic comma as the basic unit. But it would be better to "carve nature at its joints" if possible. That is, look at the minimax errors of large numbers of the best temperaments and see where the gaps are, near to these binary fractions of the comma. There seems to be one such gap between about 2.8 c and 3.1 c for linear temperaments up to 15-limit. Here's a definition from Feb 2000. Yahoo groups: /tuning/message/8589 * [with cont.] And here's the first use (Feb 1999) of the earlier term "wafso-just" that "microtemperament" replaced. Yahoo groups: /tuning/message/1012 * [with cont.]

Message: 8107 Date: Wed, 12 Nov 2003 19:01:57 Subject: Re: Vals? From: Carl Lumma

>Actually, if you need a shorter term than "prime-mapping", it seems >like "mapping" would do. What other kinds of mappings do we use in >tuning-math?

A "mapping", as it has been used, is sufficient to define a linear temperament. A val is not. But choo got me as to the exact relationship/difference between the two. -Carl

Message: 8108 Date: Wed, 12 Nov 2003 16:54:16 Subject: Re: Vals? From: Graham Breed Dave Keenan wrote:

> But it appears that little or no explanation would have been necessary > if you had simply called them prime mappings.

I think I call them "equal mappings" to mean mappings of equal temperaments that don't imply you have an actual equal temperament. That is, I think this concept is the same as Gene's "Val". Graham

Message: 8109 Date: Wed, 12 Nov 2003 17:17:34 Subject: Re: Definition of microtemperament From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> > wrote:

> > I would have said "would always be less than about 3 cents"

or "...

> > less than 3.5 cents" in order to include Miracle. Or don't you > > consider that a microtemperament, and if not, then what should we > > call it?

> > I've always considered miracle to be a microtemperament at the 7-

limit

> (2.4 c) but not at the 9 or 11 limits (3.3 c).

I don't follow this. The error of 4:5 in Miracle (with minimax generator) is ~3.323c. Anyway, what should I call a temperament in which all of the consonances are within 3.33 cents and all of the 7-limit consonances are within half of that amount, which I believe sounds like "JI brought to life" (i.e., with the "stagnation" or "deadness" eliminated with a small amount of tempering)? I have a particular 13- limit temperament in mind, and I could send you a recording of a live 1975 performance on the Scalatron in that temperament, so you could judge whether it sounds like JI or not. Or I could make an mp3 file from the recording available, if anyone else wants to hear it.

> I originally said "less than half the 5-limit error of 1/4-comma > meantone", i.e. less than 2.7 c.

I think you're comparing apples and oranges here. The max error of 9- limit 1/4-comma meantone is twice that of 5-limit meantone, simply because 8:9 will have twice the error of 2:3. If you're going to use anything on the order of half the error of meantone as your cutoff, then you should also extend this to half the error of 8:9 in meantone for a 9 limit. Otherwise, when you evaluate 9-limit (or higher) temperaments against your standard, you are really using a 1/4-the- error-of meantone standard. The beating harmonics in a tempered 8:9 are much more difficult to hear than for 2:3, hence that interval is more difficult to play in tune with flexible-pitch instruments, hence the actual error for that interval in a live performance is likely to be greater.

> I let it creep up already so a couple of temperaments with 2.8 c > errors could scrape in, and I went up to 3 for this definition just > because it seemed silly to be as precise as 2.8 c, so I definitely > wouldn't want it to creep _past_ 3 cents. > > Gene would like the limit set at 1 c, although I haven't read why. > However I believe this definition caters for that, by allowing the

ear

> to arbitrate, and mentionaing the context dependence. In some

contexts

> a temperament with an error between 1 and 3 cents may not be a > microtemperament. > > All I'm saying with the 3 cent thing is that there is no context in > which an error _greater_ than 3 cents would be considered a > microtemperament, ear or no ear.

Then I guess that (without knowing exactly how much 3 cents has been exceeded) you'll have to listen to my recording and then decide. But whatever you decide, I think that you would definitely agree that the accuracy is a magnitude or two better than meantone (and even noticeably better than 72-ET).

> It's all pretty arbitrary, but I think we need to draw such a line > somewhere.

Yes. --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> ... > The term microtemperament has a long history of referring to > temperaments with errors less than half that of 1/4-comma meantone.

So

> the magnitude scale could go down by factors of two using the

syntonic

> comma as the basic unit. But it would be better to "carve nature at > its joints" if possible. That is, look at the minimax errors of

large

> numbers of the best temperaments and see where the gaps are, near to > these binary fractions of the comma. There seems to be one such gap > between about 2.8 c and 3.1 c for linear temperaments up to 15-

limit.

> > Here's a definition from Feb 2000. > Yahoo groups: /tuning/message/8589 * [with cont.]

If you want to draw a boundary at ~2.8 cents, then draw it there, not at 3.0 cents, because you're inviting others to want the boundary to creep upward. BTW, what is the next (larger-error) gap? I am inclined to think that a factor of two is too large for establishing magnitudes of error. --George

Message: 8110 Date: Wed, 12 Nov 2003 20:37:42 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> > <0 1 4 10| should be familiar from the meantone temperament.

> > This looks like one row of the 7-limit prime-mapping for the

meantone

> linear temperament using a fifth as the generator, in particular the > row giving the mapping to fifth generators. Isn't it somewhat > incomplete without the other row that gives the mapping to octave > generators (periods)?

The octaves would be another val.

> Why do we want to give the same name to something which in one case

is

> the complete mapping for an ET (a 1D temperament), and in the other > case only a part of the mapping for an LT (a 2D temperament)?

Is it just barely possible I might know something about mathematics? To me this is a bit like asking why we would care about a single comma, when we need more than one of them for a Fokker block, BTW.

> But assuming that there's a good reason, I'd simply call them > "prime-mappings" or "1D-prime mappings".

Call them what you like, but clearly your names are clumsier.

> But it appears that little or no explanation would have been

necessary

> if you had simply called them prime mappings.

They aren't prime mappings per se; that's just a basis. I *did* explain they were homomorphic mappings, and give examples with column vectors, etc etc.

> So is a val, as applied to tuning theory, simply a prime-mapping,

or a

> 1D-prime-mapping?

More or less.

Message: 8111 Date: Wed, 12 Nov 2003 20:39:13 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >I've explained what a val is numerous times. I can't insist you pay > >attention to everything I say; these days you and George tend to

lose

> >me, after all, which is fair enough.

> > But you haven't explained how it works.

I have done just that many times.

Message: 8112 Date: Wed, 12 Nov 2003 20:48:59 Subject: Re: Definition of microtemperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> > There's always my magnitude scale, with lines differing by a

factor

> > of two.

> > I can't find this by searching the archive. I tried all kinds of > things. Please explain or give a URL.

Maximum error in cents Magnitude 0: 0.25-0.5 Magnitude 1: 0.5-1.0 Magnitude 2: 1.0-2.0 Magnitude 3: 2.0-4.0 -log2(4 * error) is the formula. Miracle is a third magnitude temperament by this.

> The term microtemperament has a long history of referring to > temperaments with errors less than half that of 1/4-comma meantone.

I didn't know that. I do recall wafso-just.

Message: 8113 Date: Wed, 12 Nov 2003 22:56:59 Subject: Re: Vals? From: Carl Lumma

>Then a val is just a mapping-row,

What confuses the hell out of me is that Gene keeps using the word "column" re. vals, but they don't give successive approximations to the same prime, they give a single approx. to various primes.

>I don't see how >calling them vals adds anything to this. In fact I think it >just obscures things.

By now it should be no surprise that I'm utterly confused by your obsession over this word. Considered a career in postmodern critical theory? At the very least, I'd hope understand what vals are good for before trying to rename them. Or maybe you understand why the 11-limit has no standard val, and can explain it to the rest of us. -Carl

Message: 8114 Date: Wed, 12 Nov 2003 21:04:51 Subject: Re: Linear Temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> So, can a mapping (of primes in terms of generators) be derived from > two temperaments (say 12 & 19) without using commas? or wedgies?

How

> would that be done? Thanks

A method I often use is to reduce the matrix whose rows are the two vals in quesition to integer Hermite normal form; this means reducing the front square part, the rest of the matrix following along. The reason I so often use it is that in Maple, if the two vals are written in terms of lists, v1, and v2, and I make a lists of lists [v1, v2], then ihermite([v1, v2]) does this for me immediately. Hermite Normal Form -- from MathWorld * [with cont.]

Message: 8115 Date: Wed, 12 Nov 2003 23:06:50 Subject: Re: 7-limit optimal et vals From: Carl Lumma

>And we have ED3 for the BP tunings.

Who's we? I, for one, reject any and all EDx terminology with the Iron Fist of Discountenance... -Carl

Message: 8116 Date: Wed, 12 Nov 2003 21:08:06 Subject: Re: Gaps between Zeta function zeros and ets From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> Are these in order of the largest gap to the smallest for the first > 10000 zeros, or some other ordering. How do you derive 954, for > example. Thanx

954 is the largest in the sense that the size of the gap, times log2(954), is the largest up through 1089. If I'd cut things off at 900, 311 would have been the largest, which is kind of cool. I'm going to check and see what happens if I integrate Z(x) between the two zeros of the gap.

Message: 8117 Date: Wed, 12 Nov 2003 21:10:51 Subject: Re: Definition of microtemperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote: I have a particular 13-

> limit temperament in mind...

Which is?

Message: 8118 Date: Wed, 12 Nov 2003 13:40:26 Subject: Re: Vals? From: Carl Lumma

>> >I've explained what a val is numerous times. I can't insist you pay >> >attention to everything I say; these days you and George tend to >> >lose me, after all, which is fair enough.

>> >> But you haven't explained how it works.

> >I have done just that many times.

Here's what you've given us so far...

>Consider the otonal chord of the n-odd-limit. This has (n+1)/2 octave >reduced elements, 1 < q[i] <= 2, where the q[i], i from 1 to (n+1)/2, >are arranged in increasing size. The n-odd-limit has pi(n) primes; we >may solve the (n+1)/2 linear equations for the val which sends q[1] >to 1, q[2] to 2, up to q[(n+1)/2]=2 to (n+1)/2. These linear >equations have a unique solution in the 3, 5, 7, 9, and 13 odd limits. >For 3 we get [2, 3], for 5 [3, 5, 7] and so forth--the standard vals >in the respective prime limits 3, 5, 7, 7, 11 for 2, 3, 4, 5, and 7. >To give a simple example, in the 5-limit, (5+1)/2 = 3, and we may >start from the 3-chord [5/4, 3/2, 2]. If we solve for a val [a, b, c] >such that 5/4, or [-2, 0, 1] is mapped to 1, 3/2 is mapped to 2, and >2 is mapped to 3 we get the equations a5 - 2 a2 = 1, a3 - a2 = 2, and >a2 = 3, the solution of which is a2 = 3, a3 = 5, and a5 = 7, so the >val in question is uniquely determined to be [3, 5, 7], the standard >3-val for the 5-limit.

standard val-

>The vector consisting of round(n log2(p)) for primes p in ascending >order up to the chosen prime limit, considered as defining a val.

...It appears that in the case of the "standard 3-val for the 5-limit", n=3. Is that why you called it a 3-val? Where did 3 come from? Further, is the standard val supposed to be the val with the smallest possible numbers that works? Or, I don't get your criterion for deciding you want "the val which sends q[1] to 1, q[2] to 2, up to q[(n+1)/2]=2 to (n+1)/2". Further, why are you sending 5/4 to 1 and 3/2 to 2 and 2/1 to 3, instead of the reverse? I thought "the q[i], i from 1 to (n+1)/2, are arranged in increasing size". -Carl

Message: 8119 Date: Wed, 12 Nov 2003 23:18:45 Subject: Re: Vals? From: Carl Lumma

>I'm just disappointed we got this far with "val"

...with only 1 -- two if we're lucky -- persons who know how to use them for what they're capable of. Gene, since you won't say what's desirable about being a standard val, and you haven't said what the lack of a standard 11-limit val means about the 11-limit, I can only guess that the definition of standard val is an error, since the 11-limit is one of the more useful ways to get hexads. -Carl

Message: 8120 Date: Wed, 12 Nov 2003 21:41:37 Subject: Re: 7-limit optimal et vals From: Paul Erlich what is the optimality criterion? --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> Here is a list of all of them which are not already standard vals,

for

> n from 1 to 1 100. No torsion issues arise. In some cases other vals > scored nearly as well. > > <1 2 3 3] > > <3 5 7 9] > > <8 13 19 23] > > <11 18 26 31] > > <13 20 30 36] > > <14 22 32 39] > > <17 27 40 48] > > <20 31 46 56] > > <23 36 53 64] > > <28 44 65 78] > > <30 47 69 84] > > <33 52 76 92] > > <34 54 79 96] > > <39 62 91 110] > > <48 76 112 135] > > <52 83 121 146] > > <54 85 125 151] > > <64 102 149 180] > > <65 103 151 183] > > <66 104 153 185] > > <67 106 155 188] > > <71 112 165 199] > > <85 135 198 239] > > <86 136 199 241] > > <96 152 223 269] > > <98 155 227 275] > > <100 159 232 281]

Message: 8121 Date: Wed, 12 Nov 2003 23:24:43 Subject: Re: 7-limit optimal et vals From: Carl Lumma

>That's fine, but we still _have_ EDO and ED3 whether we want to use >them or not. Or are you able to erase them from your memory? :-)

Actually, only a tiny fraction of the tiny fraction of theorists who use these lists use this terminology.

>If so, sorry to remind you of them again, and don't ever look at the >index to Monz's dictionary. At least keep away from the E's, OK. ;-)

There are ways of attacking a terminology. Publishing papers with similar but subtly different terminology, for example. I don't think this will be necessary, though, as the worthlessness of "EDO" should be readily apparent to most onlookers. -Carl

Message: 8122 Date: Wed, 12 Nov 2003 21:52:34 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Here's what you've given us so far...

I've given way, way way more than that. I can't force anyone to read it.

> ...It appears that in the case of the "standard 3-val for the 5-

limit",

> n=3. Is that why you called it a 3-val? > Where did 3 come from?

A division of the octave into three parts, or in other words, a mapping of 2 to 3.

> > Further, is the standard val supposed to be the val with the

smallest

> possible numbers that works?

It's simply what you get by rounding log2(p) to the nearest integer. Or, I don't get your criterion for

> deciding you want "the val which sends q[1] to 1, q[2] to 2, up to > q[(n+1)/2]=2 to (n+1)/2". > > Further, why are you sending 5/4 to 1 and 3/2 to 2 and 2/1 to 3, > instead of the reverse? I thought "the q[i], i from 1 to (n+1)/2, > are arranged in increasing size".

Eh? clearly 5/4 < 3/2 < 2.

Message: 8123 Date: Wed, 12 Nov 2003 21:53:47 Subject: Re: 7-limit optimal et vals From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> what is the optimality criterion?

Minimax error in the 7-limit.

Message: 8124 Date: Wed, 12 Nov 2003 00:43:12 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> so how about "mapping" instead of "val" with the implication > (preferably stated along with n) that we are talking about ET.

I don't see the point. What about optimal et?

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