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Message: 8125 Date: Wed, 12 Nov 2003 21:58:19 Subject: Re: Definition of microtemperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:

> --- 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.

we were focusing on the 72-equal incarnation of miracle.

> 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.

why? there's no analogy there. 1/4-comma meantone was not used for music where 8:9 is used as a consonance.

> The beating harmonics in a tempered 8:9 are much more difficult to > hear than for 2:3,

shouldn't that consideration lower the weight of 8:9 in the calculation, compensating this next point?

> 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.

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

> > Yes.

noooooooooooooooooo! :)

Message: 8126 Date: Wed, 12 Nov 2003 00:44:40 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- 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?

we're talking about how to optimally map primes to a given et, right?

Message: 8127 Date: Wed, 12 Nov 2003 22:03:08 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- 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.

excuse me, but i think the answer to carl's question is "the complete 5-limit otonal chord has *3* notes". right?

Message: 8128 Date: Wed, 12 Nov 2003 01:17:48 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

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

> > we're talking about how to optimally map primes to a given et,

right? I don't count it as an et unless it has a mapping; anyway "optimal val" is shorter and sweeter.

Message: 8129 Date: Wed, 12 Nov 2003 22:03:26 Subject: Re: 7-limit optimal et vals From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:

> > what is the optimality criterion?

> > Minimax error in the 7-limit.

any differences if you use rms?

Message: 8130 Date: Wed, 12 Nov 2003 01:23:42 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >

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

> > > > we're talking about how to optimally map primes to a given et,

> right? > > I don't count it as an et unless it has a mapping; anyway "optimal > val" is shorter and sweeter.

ok.

Message: 8131 Date: Wed, 12 Nov 2003 22:04:44 Subject: Re: Definition of microtemperament From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"

<gdsecor@y...>

> wrote: >

> > I have a particular 13-limit temperament in mind...

> > Which is?

... more useful than Miracle, in my opinion. If you want to try it out in Scala, then be advised that, like Miracle, it comes in several sizes (of 17, 29, and 41 tones). The 41- tone size is a superset of all of the others, and the 17-tone version comes in several keys (none of which are a subset of the 29-tone version). "Set notation 41e" or "sa41" misses one of the tones, so you should "set notation sahtt" to get all of them (and if you want to see conventional sharps and flats, then "set sagittal mixed". Here are the file contents for a few of these: ! secor17htt1.scl ! George Secor's 17-tone high-tolerance temperament subset #1 on C (5/4 & 7/4 exact) 17 ! 30.08878 140.19633 207.15739 265.24719 347.35372 5/4 496.42131 554.51111 612.60091 703.57869 733.66747 843.77502 882.73506 7/4 1050.93241 1089.89245 2/1 ! secor17htt2.scl ! George Secor's 17-tone high-tolerance temperament subset #2 on Ev (5/4 & 7/4 exact) 17 ! 30.08878 116.17960 207.15739 237.24617 347.35372 5/4 496.42131 554.51111 612.60091 703.57869 733.66747 843.77502 910.73608 7/4 1050.93241 1089.89245 2/1 ! secor17htt3.scl ! George Secor's 17-tone high-tolerance temperament subset #3 on G (5/4 & 7/4 exact) 17 ! 58.08980 116.17960 207.15739 237.24617 347.35372 5/4 472.40458 554.51111 593.47114 703.57869 733.66747 843.77502 910.73608 7/4 1050.93241 1089.89245 2/1 ! secor29htt.scl ! George Secor's 29-tone 13-limit high-tolerance temperament (5/4 & 7/4 exact) 29 ! 58.08980 97.04984 140.19633 179.15637 207.15739 265.24719 296.73557 347.35372 5/4 414.31478 472.40458 496.42131 554.51111 593.47114 633.37025 679.56197 703.57869 761.66849 800.62853 843.77502 882.73506 910.73608 7/4 992.84261 1050.93241 1089.89245 1117.89347 1175.98327 2/1 ! secor41htt.scl ! George Secor's 13-limit high-tolerance temperament superset (5/4 & 7/4 exact) 41 ! 30.08878 58.08980 97.04984 116.17960 140.19633 179.15637 207.15739 237.24617 265.24719 296.73557 323.33699 347.35372 5/4 414.31478 444.40355 472.40458 496.42131 526.51008 554.51111 593.47114 612.60091 636.61764 679.56197 703.57869 733.66747 761.66849 800.62853 819.75830 843.77502 882.73506 910.73608 940.82486 7/4 992.84261 1026.91568 1050.93241 1089.89245 1117.89347 1147.98225 1175.98327 2/1 Have fun analyzing these, and let me know if you think they sound like JI! --George

Message: 8132 Date: Wed, 12 Nov 2003 02:05:19 Subject: 7-limit optimal et vals From: Gene Ward Smith 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: 8133 Date: Wed, 12 Nov 2003 14:16:14 Subject: Re: Vals? From: Carl Lumma

>> 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.

I've read everything you've ever posted to this list, much of it more than once, and much of it I've saved locally. -Carl

Message: 8134 Date: Wed, 12 Nov 2003 03:18:51 Subject: Re: Vals? 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: >

> > Couldn't I (in fact didn't I) just define an (unqualified) ET-mapping > > in exactly the same way?

> > Go ahead and do so, however a val is not necessarily an et-mapping of > any kind.

In this message Yahoo groups: /tuning-math/message/7528 * [with cont.] you said the two could be identified (except for a question about finiteness)? Here is a possible Monz dictionary definition of an ET-prime-mapping (improved from the one I already gave in the above message, that you seem to have missed): ---------------------------------------------------------------------- ET-prime-mapping A list of the (whole) numbers of steps of some equal temperament (ET) (not necessarily octave based) used to approximate each prime number (considered as a frequency ratio). An "n-limit" ET-prime-mapping (where n is a whole number) only lists numbers of steps for primes no greater than n. The "standard" prime mapping for an ET is the one that gives the best approximation for each prime, but note that this is not guaranteed to give the best approximation for all ratios, and other mappings may be more useful in some cases. To find the number of steps approximating some ratio in some ET, express the ratio as a prime-exponent-vector and multiply its elements by the corresponding elements in the chosen ET-prime-mapping and sum the products. This is called the dot product or inner product or scalar product of the two vectors. For example the standard 7-limit ET-prime-mapping for 12-EDO is [12 19 28 34]. These numbers can be calculated for any EDO as Round(N*ln(p)/ln(2)) where N is the number of divisons per octave and p is the prime number. To find how many steps of 12-EDO approximate a 7/5 frequency ratio, first express the ratio as a prime-exponent-vector. 7/5 = 2^0 * 3^0 * 5^-1 * 7^1 = [0 0 -1 1] now find its dot product with the prime-mapping [0 0 -1 1].[12 19 28 34] = 0*12 + 0*19 + -1*28 + 1*34 = 6 So 7/5 is approximated by 6 steps, a tritone. ----------------------------------------------------------------------

> And then a "p-limit ET-mapping" would be a

> > restricted one. > > > > So they're exactly the same!!!!

> > I havn't seen your definition.

I suspect you just didn't recognise it as such, because it was in plain English.

> > So why have we been calling them "vals" all this time?

> > Because there wasn't a good word for "finitely generated homomorphic > mapping from Q+ to Z" or "Z-linear combination of padic valuations" > already in existence. > > A mathworld

> > search on the term finds nothing. Did you invent the term?

> > You bet. We needed a term for it, and there wasn't one.

_You_ might have needed a term for "finitely generated homomorphic mapping from Q+ to Z" or "Z-linear combination of padic valuations", but I don't think anyone else on this list did.

> Is it

> > merely an obscure synonym for "homomorphism", or "group homomorphism"? > > The fact that it _is_ a group homomorphism is far from being its most > > important characteristic as far as microtonality is concerned. The > > fact that it maps ratios to steps of ETs is of far more interest.

> > A val *does not necessarily* map ratios to steps of an ET, but it *is* > always a homomorphism.

So what would you estimate is the percentage of the vals posted to tuning-math that could not be read as mapping ratios to steps of an ET. Please give some examples of these and explain what they _do_ mean in tuning terms.

> If you insist, you could replace the term with > "finitely generated homomorphic mapping from Q+ to Z", I suppose, but > I imagine "a p-limit homomorphic mapping to the integers" or something > like that would suit you better.

Those certainly don't suit me. By all means use the word "val" to stand for this abstract mathematical category. But this is the _tuning_ math list. We want names that indicate their meaning as applied to _tuning_. For example, in the application area of electrical theory we use vectors to represent the magnitude and phase of sinusoidal voltages and currents. But we don't just call them all vectors. We want names that tell us what they _mean_. We call them "voltage phasors" and "current phasors". I assume that the val, strictly speaking, is the operation of taking the dot product with a vector of step numbers, not the actual vector of step numbers itself. That's ok. The term "mapping" is used similarly ambiguously. It doesn't usually cause any misunderstanding.

> It most certainly would not have served me better. You do what you > like, but please don't expect me to follow your lead.

So are you saying that you don't really care if only two or three people on this list understand how the things you write about apply to tuning?

Message: 8135 Date: Wed, 12 Nov 2003 22:36:40 Subject: Re: Definition of microtemperament From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"

<gdsecor@y...> wrote:

> > --- 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.

> > we were focusing on the 72-equal incarnation of miracle.

This then *requires* moving the ~2.8c boundary to 3.0 cents, not just putting it there because it looks less arbitrary. If Dave said there was a gap between ~2.8 and ~3.1 cents, and if it takes a special (non- 11-limit-optimal) version of Miracle to make the 7-limit cut, then I would say that the boundary is in the wrong place, and he should have left it at ~2.8 cents and allowed Miracle (in all of its incarnations) to fall into a different category (for which I'm still awaiting a name).

> > 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.

> > why? there's no analogy there. 1/4-comma meantone was not used for > music where 8:9 is used as a consonance.

No, it wasn't historically, but that doesn't mean that someone *couldn't* use an extended meantone temperament for 9-limit harmony. I will admit that there are better options for that purpose, but then that's the whole point -- this gives us a baseline against which we can compare those options.

> > The beating harmonics in a tempered 8:9 are much more difficult

to

> > hear than for 2:3,

> > shouldn't that consideration lower the weight of 8:9 in the > calculation, compensating this next point? >

> > 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.

That was precisely my point. More leeway can be allowed for some of these less-consonant consonances.

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

line

> > > somewhere.

> > > > Yes.

> > noooooooooooooooooo! :)

Do you mean noooooooooooooooooo categories or noooooooooooooooooot arbitrary, or booooooooooooooooooth? :-) --George

Message: 8136 Date: Wed, 12 Nov 2003 03:51:39 Subject: Re: Definition of microtemperament From: Dave Keenan --- 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 originally said "less than half the 5-limit error of 1/4-comma meantone", i.e. less than 2.7 c. 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. It's all pretty arbitrary, but I think we need to draw such a line somewhere.

Message: 8137 Date: Wed, 12 Nov 2003 22:43:16 Subject: Re: Definition of microtemperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:

> > why? there's no analogy there. 1/4-comma meantone was not used

for

> > music where 8:9 is used as a consonance.

> > No, it wasn't historically, but that doesn't mean that someone > *couldn't* use an extended meantone temperament for 9-limit

harmony. right, but then they'd be more likely to use something like 1/5-comma or 1/6-comma meantone.

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

> line

> > > > somewhere.

> > > > > > Yes.

> > > > noooooooooooooooooo! :)

> > Do you mean noooooooooooooooooo categories or noooooooooooooooooot > arbitrary, or booooooooooooooooooth? :-) > > --George

there's no way you could *hear* the point at which the line is drawn (nor should it necessarily be drawn according to minimax), so i'd prefer to use 'microtemperament' in a looser way -- if anyone cares to check on the 'microtemperedness' of a particular temperament, the exact numbers should be readily available. maybe 2.8 to 3.1 can be considered a 'gray zone', where *context* will determine whether the effect is one of microtemperament or not.

Message: 8138 Date: Wed, 12 Nov 2003 05:26:52 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> In this message > Yahoo groups: /tuning-math/message/7528 * [with cont.] > you said the two could be identified (except for a question about > finiteness)?

I understood you to mean any kind of prime mapping, whether it could be called an et or not.

> Here is a possible Monz dictionary definition of an ET-prime-mapping > (improved from the one I already gave in the above message, that you > seem to have missed): > ---------------------------------------------------------------------- > ET-prime-mapping > > A list of the (whole) numbers of steps of some equal temperament (ET) > (not necessarily octave based) used to approximate each prime number > (considered as a frequency ratio). An "n-limit" ET-prime-mapping > (where n is a whole number) only lists numbers of steps for primes no > greater than n.

OK, but this isn't equivalent to val.

> So what would you estimate is the percentage of the vals posted to > tuning-math that could not be read as mapping ratios to steps of an > ET.

Who knows? Maybe 50-50. Please give some examples of these and explain what they _do_ mean

> in tuning terms.

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

> For example, in the application area of electrical theory we use > vectors to represent the magnitude and phase of sinusoidal voltages > and currents.

Unless you use complex numbers, of course.

> I assume that the val, strictly speaking, is the operation of taking > the dot product with a vector of step numbers, not the actual vector > of step numbers itself.

It's the mapping that generates, yes.

> So are you saying that you don't really care if only two or three > people on this list understand how the things you write about apply to > tuning?

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.

Message: 8139 Date: Wed, 12 Nov 2003 05:34:35 Subject: Gaps between Zeta function zeros and ets From: Gene Ward Smith I took a list of the first 10000 zeros of the Riemann zeta function, rescaled so that they read in terms of equal divisions of the octave. The top zero is then 1089.695, so we are looking at ets from 1 to 1089. The gaps between these zeros are on average 1/log2(n) in the vicinity of n, so I took the gaps between successive zeros and multiplied by log2 of the average of the two successive zeros. As I expected, for the largest gaps these are centered around equal divisions of the octave. The first fifty of these, in order, are as follows: 954, 1012, 311, 764, 422, 581, 270, 814, 742, 935, 718, 494, 882, 1041, 525, 908, 571, 1065, 342, 836, 653, 851, 1084, 460, 354, 1075, 692, 1029, 684, 566, 624, 472, 711, 400, 863, 639, 988, 243, 997, 441, 643, 597, 373, 1046, 795, 449, 224, 513, 328, 966 Some tendency for the ets in question to be the kind approximating a lot of primes (311) instead of a relatively few (411) seems to be in evidence, but it isn't really clear what is going on.

Message: 8140 Date: Wed, 12 Nov 2003 05:36:57 Subject: Re: Definition of microtemperament From: Gene Ward Smith --- 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.

> 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.

Message: 8141 Date: Wed, 12 Nov 2003 07:40:16 Subject: Re: Vals? 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: >

> > In this message > > Yahoo groups: /tuning-math/message/7528 * [with cont.] > > you said the two could be identified (except for a question about > > finiteness)?

> > I understood you to mean any kind of prime mapping, whether it could > be called an et or not.

I wrote:

> > How is a val different from an ET-mapping? i.e. a list of the > > numbers of steps approximating each prime in some ET.

So I'm rather surprised you didn't know I was talking about ETs? But that's good, because now it looks like "val", as applied to tuning, can be replaced by "prime-mapping", which is even simpler than "ET-prime-mapping".

> <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)? 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)? But assuming that there's a good reason, I'd simply call them "prime-mappings" or "1D-prime mappings". But I'd prefer to be more specific and call one an ET-mapping and the other an LT-generator-mapping. I'd call the missing row the LT-period-mapping. Together the LT-generator-mapping and the LT-period-mapping make up the LT-mapping. The word "prime" can be inserted before the word "mapping" whenever this is not clear from the context.

> > So are you saying that you don't really care if only two or three > > people on this list understand how the things you write about

apply > > to tuning?

> > 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.

If all your explanations were similar to this one Definitions of tuning terms: val, (c) 2001 by ... * [with cont.] (Wayb.) I'm afraid it wouldn't have made any difference if I'd read them all. But I can't hold it against anyone that they are not good at explaining things. I'm pretty sure I did read an early one, and said to myself, "I have no idea what that means. I guess I need a bit more mathematical background. I'll look into it later." But it appears that little or no explanation would have been necessary if you had simply called them prime mappings. So is a val, as applied to tuning theory, simply a prime-mapping, or a 1D-prime-mapping?

Message: 8142 Date: Thu, 13 Nov 2003 02:16:33 Subject: Re: Vals? From: Dave Keenan 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?

Message: 8144 Date: Thu, 13 Nov 2003 22:13:53 Subject: Re: Definition of microtemperament From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"

<gdsecor@y...>

> wrote: >

> > I have never thought of any tuning as anything less than 11-limit

if

> > it contains 11-limit intervals.

> > You mean 11-prime limit? Miracle is of course a temperament, and it > can be derived easily and naturally from the 7-limit lattice. You > simply temper our 225:224 and 2401:2400 (or 225:224 and 1029:1024). > > You mean 11-odd limit? Well, meantone contained excellent > approximations to ratios of 7, but practically no one considered

them

> consonant historically. So i see no problem with considering

miracle

> a 7-limit temperament if someone uses it in a style where ratios of > 11 or their approximation are used as dissonances. In fact miracle

is

> one of the very best choices for a 7-limit temperament. >

> > Even from an historical perspective > > miracle has always been an 11-limit tuning.

> > Not true at all!! I think the 11-limit was an unexpected bonus,

that

> mostly Dave Keenan was keeping track of at the time.^

Okay, you win! --George

Message: 8145 Date: Thu, 13 Nov 2003 17:23:09 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 "Gene Ward Smith" <gwsmith@s...> > wrote:

> > 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.

> > Well this looks like a good system to me. But I'm afraid it will

make

> too many people unhappy to try to align the definition of > "microtemperament" with it. >

> > Miracle is a third magnitude temperament by this.

> > Hopefully this will make George happy.

Maybe and maybe not, because it doesn't take into account whether there might be better boundaries that would be suggested by the gaps indicated in a tabulated list of microtemperaments, which could also suggest that a factor of magnitude other than 2 might be more appropriate. Then again, after doing this you might find that whatever boundaries and factor you chose would still be rather arbitrary and there would not be any reason to use anything different from what Gene gave above. I'm not trying to give you guys a hard time about this, but just offering some constructive criticism. And considering what I have to say below, it might be best to forget about boundaries altogether and express the magnitude to one decimal place (as with stellar magnitude in astronomy).

> Paul, > > I wasn't considering the 72-EDO incarnation. When I gave 2.4 c and

3.3

> c I was considering the minimax optima for 7-limit and 11-limit > respectively (9-limit is the same as 11-limit). > > George, > > I think most of us find it quite natural to speak of "7-limit

miracle"

> even though some may consider the "true" miracle temperament to be > 11-limit.

I have never thought of any tuning as anything less than 11-limit if it contains 11-limit intervals. Even from an historical perspective miracle has always been an 11-limit tuning.

> We have enough trouble agreeing on names as it is. I'd hate > to have to find different names for different lower-limit subsets

(or

> whatever the right term is) of the same mapping.

My primary concern in engaging you in this discussion was to determine whether I should be calling my high-tolerance temperament (which I described here:) Yahoo groups: /tuning-math/message/7574 * [with cont.] a microtemperament, or whether some other category-name would be more appropriate. (See my comments on your definition, below.)

> The most important thing is for Monz to get rid of the current false > definition of microtemperament. > > Monz, > > You could just change every ocurrence of "microtemperament" in it to > "planar temperament" and change its name to "planartemp.htm" and > there's your definition for planar temperament. > > I'm hoping that the following will make everyone happy. I've changed > "would always be less than 3 cents" to "would typically be less than > 2.8 cents". I've also changed "JI scale" to "JI tuning" throughout. > > I'm sure this definition could be improved, but can we just get > something in place of that bad definition in Monz's dictionary? > > --------------------------------------------------------------------

--

> Microtemperament > > A microtemperament is a temperament where the consonances sound

justly

> intoned to most listeners in ordinary musical use. The allowed

errors

> in the approximated ratios are therefore somewhat context-dependent > but would typically be less than 2.8 cents.

This language is very good in that it acknowledges that the term relates to one's (subjective) *perception* of a tuning rather than a hard-and-fast (objective) error limit, so that classification as a microtemperament for a particular tuning with approximations approaching or exceeding 2.8-cents is left open for debate. (And given the subjectivity of this definition, I would agree with Aaron's recommendation that this be changed to 3 cents.) The orders of magnitude that Gene gave could be useful for predicting the *probability* of a particular tuning being perceived as a microtemperament.

> A JI tuning might be microtempered to increase the number of

available

> consonances or to regularise the scale for some purpose such as > allowing more full-width continuous frets on a stringed instrument. > Microtemperament may also be used to introduce deliberate slight > mistunings to avoid phase-locking when a JI tuning is implemented

on an

> electronic instrument. > > A microtemperament may be equal, linear, planar or of any dimension > less than that of the JI tuning being approximated. > --------------------------------------------------------------------

-- Excellent! --George

Message: 8146 Date: Thu, 13 Nov 2003 23:24:11 Subject: Re: Definition of microtemperament From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

> > i changed it a couple of days ago when you proposed the > > earlier version of the part i snipped here. now it's as > > per your latest definition: > > > > Definitions of tuning terms: microtemperament,... * [with cont.] (Wayb.)

> > Could you change this back to "always less than three cents"? 2.8 > cents seems an absurd line to draw, and "usually" means it isn't even > a line.

This whole thread is hilarious. :-) I haven't had such a good laugh from tuning-math in a long time, but I admit I've been taking myself too seriously lately. Back when it said "always", and it had 3 cents (because I thought, as a lot of people apparently do) that 2 significant digits of cents was a bit too precise, George said "if you mean 2.8 cents then say 2.8 cents or you'll just encourage further creepage" or words to that effect. I thought he had a good point. But now that we've changed it to "typically" (which I understand most people support) then even George agrees 2.8 is too finicky. So "typically less than 3 cents" is ok with me. Paul, I assume you were merely arguing that "typically less than 2.8 cents" is as about as good as any other nearby number as a just-noticeble-difference, and you wouldn't really mind if the microtemperament definition was changed to "typically less than 3 cents". Gene, I wanted an actual cutoff too - an "always" rather than a "typically" - but it looks like we're outvoted. Or to put it another way, I can live with a "typically" for the sake of consensus.

Message: 8147 Date: Thu, 13 Nov 2003 05:41:58 Subject: Re: 7-limit optimal et vals From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > > wrote:

> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > > > wrote:

> > > > what is the optimality criterion?

> > > > > > Minimax error in the 7-limit.

> > > > any differences if you use rms?

> > and are you allowing the octaves to be tempered? i.e. Do they apply > strictly to EDOs or to ET's generally?

EDOs only, but I didn't know you were calling non=integer mappings ets.

Message: 8148 Date: Thu, 13 Nov 2003 18:51:27 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> > > Where did 3 come from?

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

> > excuse me, but i think the answer to carl's question is "the

complete

> 5-limit otonal chord has *3* notes". right?

Maybe I misunderstood the question.

Message: 8149 Date: Thu, 13 Nov 2003 23:30:01 Subject: Re: Vals? 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: >

> > I think we have quite complementary skills. You come up with the the > > math tools and methods and I may _eventually_ be able to understand > > them enough to put them into terms that others on this list can more > > easily understand. But I don't think you should worry too much if my > > explanations or recasting of terminology misses some of the more > > subtle points as far as the pure mathematician is concerned, at

> least

> > on a first pass.

> > Sounds reasonable, but I don't think you should worry to much if I > want to make precise mathematical definitions for things, or make the > definitions the way they are for reasons not immediately apparent to > you.

It's a deal. :-)

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