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Message: 11275 - Contents - Hide Contents Date: Sun, 04 Jul 2004 03:12:39 Subject: Re: from linear to equal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> 9-limit should also be considered when you're going "poptimal".True enough. Alas, even though we have the same wedgie, commas and tuning map, the poptimal range need not even overlap. Orwell is a typical example--there seems to be no overlap from 7 to 9, and none between 11 and 9, but the others are OK. So, 5 and 9 overlap, and have 43/190 as a common generator, but 7 and 9, no.
Message: 11276 - Contents - Hide Contents Date: Sun, 04 Jul 2004 23:10:23 Subject: Re: from linear to equal From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >> wrote: >>>>> For pure octave tunings, a system I sometimes use is to close at a >>> "poptimal" generator. A generator is "poptimal" for a certain set of >>> octave-eqivalent consonances if there is some exponent p, 2 <= p <= >>> infinity, such that the sum of the pth powers of the absolute value >> of>>> the errors over the set of consonances is minimal. >>>> This is quite an interesting approach. What makes poptimal generators >> good? And why can't p be 1? >> It could be 1. In fact, Paul thinks it should be 1.No I don't -- TOP actually uses p=inf, in a sense -- I just think 1 should be included in the range if infinity is.> What makes them good is that they approximate a > given list of target consonances in an optimal way, for some sense of > optimal.It's not a sense which gives a unique answer, which seems to have tripped everyone up so far. All your logic is right there in your math, but you have to understand that this is essentially a foreign language for most of us.
Message: 11277 - Contents - Hide Contents Date: Sun, 04 Jul 2004 03:21:03 Subject: Re: NOT tuning From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> except where TOP already had pure octaves, in >> which case it would actually change! >> That's impossible given the criterion of NOT. > > Maybe I don't comprehend you.Some examples of this method of tuning would be nice, and a definition even better.
Message: 11278 - Contents - Hide Contents Date: Sun, 04 Jul 2004 23:12:47 Subject: Re: from linear to equal From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> By the way, Gene, how does poptimal relate to TOP? >>>> Not very well, apparently. >> >> If the>>> commas dictate the TOP tuning, is there necessarily a >>> generator/period pair that give it? >>>> You've lost me. >> I meant, for a given TOP-tuned linear temperament, does it not > stand to reason that there is at least one generator/period > pair (in cents) that produces scales in said tuning?Yes, and there's exactly one if you fix that the octave is multiple of the period, and the generator is within a prescribed 1/2-period range (say, between 0 or 1/2 period, or between 1/2 and 1 period).
Message: 11279 - Contents - Hide Contents Date: Sun, 04 Jul 2004 23:17:50 Subject: Re: 9&11 poptimal secor From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> The 9 and 11 poptimal range intersect only at the minimax for 9 and > 11, which is the (18/5)^(1/19) secor. The continued fraction for this > gives > 10, 31, 41, 72, 329, 2046 ... as the et convergents. The 7/72 secor is > between the poptimal range for 9 and and 11 and the range for 5 and 7, > which makes it an all-purpose utility choice, and it's actually > possible that the 11-limit poptimal range includes it, since it at > least gets quite close. > > The 5 and 7 limit minimax tuning is (12/5)^(1/13), which defines the > upper part of their range. Is either of these the official George > Secor secor?The first one -- (18/5)^(1/19) -- is. He chose it because it's the 11- (odd-)limit minimax.
Message: 11280 - Contents - Hide Contents Date: Sun, 04 Jul 2004 23:20:11 Subject: Re: bimonzos, and naming tunings (was: Gene's mail server)) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:>>> Are you expecting people to read the comma values >>> off of the bimonzo? >>>> No. But as long as we're on the subject here, it might >> be worth reviewing here for list memmbers how you do that. >> Not in the paper.> yes, please do review it!If you have a bimonzo ||a1 a2 a3 a4 a5 a6>> then you can read off the odd comma of the temperament (the comma which is a ratio of odd integers) by taking out the common factor if needed of a1, a2, a3 to get b1, b2, b3, and then the comma is 3^b1 5^b2 7^b3, or its reciprocal if you need to make it bigger than 1. In general, however, reading the commas from a bimonzo is no easier than reading them from a bival, and in fact probably harder, and I don't think this makes much of a reason for using bimonzos. I really would like to know why Paul insists on this so stubbornly. If you have a bival <<a1 a2 a3 a4 a5 a6||, then 2^a4 3^(-a2) 5^a1 gives the 5-limit comma. That's 2 to the power of the (3,5) coefficient, 3 to minus the power of the (2,5) coefficent, and 5 to the power of the (2,3) coefficient. You can figure out the signs by putting the primes in the circular order, 235; then 35, 52 and 23 have the same sign, so we give 25 a sign opposite to 35 and 23, and we have the comma. Of course the prime we attach the exponent to is the prime not in the (a,b) of "(a,b) coefficient", and the coefficients are ordered <<(2,3), (2,5), (2,7), (3,5), (3,7), (5,7)||. The rest of these work similarly. 2^(a5) 3^(-a3) 7^(a1) gives the {2,3,7}-comma; that's 2 to the power of the (3,7) coefficient, 3 to minus the power of the (2,7) coefficient, and 7 to the power of the (2,3) coefficient; 237 in order tells you that (2,3) and (3,7) have a sign opposite from (2, 7). 2^a6 5^(-a3) 7^a2 gives the {2,5,7}-comma 3^a6 5^(-a5) 7^a4 gives the {3,5,7}-comma (the odd comma.) You get similar stuff in higher limits, but with a lot more commas. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 11281 - Contents - Hide Contents
Date: Mon, 05 Jul 2004 19:31:47
Subject: 24 7-limit temperaments with 245/243 as a comma
From: Gene Ward Smith
"Clyde" comes from the kleisma; I was thinking bi and clyde was sort
of like Bonnie and Clyde, but it isn't a bi, it's a semi. "Sidi" is
short for "semidicot", and "semiaug" for "semiaugmented". "Bohpier",
of course, comes from Bohlen and Pierce.
The numbers on the third line are Graham complexity, TOP error, TOP
badness and Graham-TOP badness. I used TOP l-infinity badness to sort
since that is more intrinsic than Graham, but it seems to make no
difference.
magic
[5, 1, 12, -10, 5, 25]
[[1, 0, 2, -1], [0, 5, 1, 12]]
12 1.276744 183.851136 23.327687
father
[1, -1, 3, -4, 2, 10]
[[1, 2, 2, 4], [0, -1, 1, -3]]
4 14.130875 226.094010 33.256527
sensi/semisixths
[7, 9, 13, -2, 1, 5]
[[1, -1, -1, -2], [0, 7, 9, 13]]
13 1.610469 272.169318 34.533812
godzilla/hemifourths
[2, 8, 1, 8, -4, -20]
[[1, 2, 4, 3], [0, -2, -8, -1]]
8 3.668842 234.805888 43.552336
superpythagorean
[1, 9, -2, 12, -6, -30]
[[1, 2, 6, 2], [0, -1, -9, 2]]
11 2.403879 290.869402 50.917023
rodan/supersupermajor
[3, 17, -1, 20, -10, -50]
[[1, 1, -1, 3], [0, 3, 17, -1]]
18 .894655 289.868296 52.638504
hedgehog
[6, 10, 10, 2, -1, -5]
[[2, 4, 6, 7], [0, -3, -5, -5]]
10 3.106578 310.657834 57.621529
octacot
[8, 18, 11, 10, -5, -25]
[[1, 1, 1, 2], [0, 8, 18, 11]]
18 .968741 313.872084 58.217715
shrutar
[4, -8, 14, -22, 11, 55]
[[2, 3, 5, 5], [0, 2, -4, 7]]
22 1.079127 522.297520 76.825572
clyde Number 78 {245/243, 3136/3125}
[12, 10, 25, -12, 6, 30]
[[1, 6, 6, 12], [0, -12, -10, -25]]
25 .971298 607.061287 77.026097
sidi Number 93 {25/24, 245/243}
[4, 2, 9, -6, 3, 15]
[[1, 3, 3, 6], [0, -4, -2, -9]]
9 8.170435 661.805266 83.972208
semiaug Number 95 {128/125, 245/243}
[6, 0, 15, -14, 7, 35]
[[3, 5, 7, 9], [0, -2, 0, -5]]
15 2.939961 661.491318 84.758945
bohpier Number 106 {245/243, 3125/3087}
[13, 19, 23, 0, 0, 0]
[[1, 0, 0, 0], [0, 13, 19, 23]]
23 1.408527 745.110689 94.757554
[2, -16, 13, -30, 15, 75]
[[1, 1, 7, -1], [0, 2, -16, 13]]
29 .894655 752.405053 118.436634
[3, -7, 11, -18, 9, 45]
[[1, 3, -1, 8], [0, -3, 7, -11]]
18 2.563758 830.657592 122.182745
[9, -7, 26, -32, 16, 80]
[[1, 2, 2, 4], [0, -9, 7, -26]]
33 .894655 974.279552 134.754571
[15, 27, 24, 8, -4, -20]
[[3, 7, 11, 12], [0, -5, -9, -8]]
27 1.015676 740.428092 137.336304
[9, 17, 14, 6, -3, -15]
[[1, 3, 5, 5], [0, -9, -17, -14]]
17 2.747484 794.022876 147.277188
[17, 11, 37, -22, 11, 55]
[[1, -2, 0, -5], [0, 17, 11, 37]]
37 .894655 1224.783018 155.404830
[8, 8, 16, -6, 3, 15]
[[8, 13, 19, 23], [0, -1, -1, -2]]
16 4.953617 1268.126009 160.904343
[5, 11, 7, 6, -3, -15]
[[1, 1, 1, 2], [0, 5, 11, 7]]
11 7.195870 870.700224 161.499478
[2, -6, 8, -14, 7, 35]
[[2, 3, 5, 5], [0, 1, -3, 4]]
14 7.149508 1401.303568 206.119969
[0, 10, -5, 16, -8, -40]
[[5, 8, 12, 14], [0, 0, -2, 1]]
15 5.665687 1274.779496 213.343963
[22, 36, 37, 6, -3, -15]
[[1, -9, -15, -15], [0, 22, 36, 37]]
37 1.149341 1573.448359 276.284496
Message: 11282 - Contents - Hide Contents Date: Mon, 05 Jul 2004 22:09:04 Subject: Re: bimonzos, and naming tunings (was: Gene's mail server)) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> Why harder? Can you show this? >> From my explanation, you can see it is easy to read off the prime > factors in a frontward way and figure out the signs in a bival. With > the 7-limit bimonzo, you don't have the same regular pattern where two > signs are the same, and one opposite; sometimes they all have the same > sign, and sometimes they don't.When don't they? I must have missed something.> I think bivals are clearly easier. > Moreover, for higher limit linear temperaments, they are still bivals > and you can use the same rule, whereas trimonzos are what you get in > the 11-limit, etc. A big fat mess by comparison.I already answered this, so I won't repeat myself.>>> and I >>> don't think this makes much of a reason for using bimonzos. I really >>> would like to know why Paul insists on this so stubbornly. >>>> I welcome constructive suggestions for making the paper go bival, >> without adding to its math-heaviness. >> It's really, really, really easy. Simply replace the bimonzos you list > with the corresponding bivals, and you are done.You've got to be kidding me.> You need explain > nothing, nor define anything.I'd like to do better by my readers.>>> If you have a bival <<a1 a2 a3 a4 a5 a6||, then >>> >>> 2^a4 3^(-a2) 5^a1 gives the 5-limit comma. >>>> You don't need to remove common factors? >> I said "gives the comma", not "is the comma".But when you were talking about the bimonzo, you made it seem a whole lot more complicated by introducing b1, b2, b3.>>> 2^(a5) 3^(-a3) 7^(a1) gives the {2,3,7}-comma; that's 2 to the power >>> of the (3,7) coefficient, 3 to minus the power of the (2,7) >>> coefficient, and 7 to the power of the (2,3) coefficient; >>>> How is this easier than the bimonzo case? >> Because there is a simple rule I can explain which works for any prime > limit.Show me how a bimonzo gets so bad in a higher prime limit.
Message: 11283 - Contents - Hide Contents Date: Mon, 05 Jul 2004 22:16:36 Subject: Re: from linear to equal From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 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: >>>>>>>> This is quite an interesting approach. What makes poptimal >>>>> generators >>>>> good? >>>>>>>> Not much, IMHO -- the "true" value of p in any situation will be >> some>>>> number, not an infinite range of numbers. >>>>>> What in the world does this mean? What allegedly "true" value? >>>> If you're using this p-norm model in the first place, it's probably >> because you think it's true for some value of p. >> I have no idea how a norm can possibly be either true or false. My > assumption is that it is a valid definition of optimum for any p in > the range 2 to infinity, and that is simply because if you assume the > endpoints define a valid sense of optimum, so should all the > intermediate values.OK. And a lot of other values may define a valid sense of optimum, as well, for example weightings, etc.>>> As for 1, I think a lot of people would find the supposedly optimal >>> tunings not really very optimal in some cases. >>>> And yet there is a significant bunch of composers who refuse to >> temper, clinging to their JI scales. Might they be modelled too? (no >> offense to them, of course.) >> If you refuse to temper at all, what in the world are you doing trying > to decide which tuning is best, on the assumption that a given > temperament will be used?Exactly. This corresponds to the p<1 area, or at least to some of it.
Message: 11284 - Contents - Hide Contents Date: Mon, 05 Jul 2004 22:19:08 Subject: Re: NOT tuning From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> except where TOP already had pure octaves, in >>>> which case it would actually change! >>>>>> That's impossible given the criterion of NOT. >>> >>> Maybe I don't comprehend you. >>>> I didn't say NOT, I said "Graham" and "pure-octaves TOP". >> Ok, it would seem to violate the definition of > "pure-octaves TOP".It doesn't. It still has pure octaves.
Message: 11285 - Contents - Hide Contents Date: Mon, 05 Jul 2004 22:22:57 Subject: Re: from linear to equal From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hi Gene and Paul, > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>>>>> You'd maybe prefer Fokker? >>>>>> Did Fokker have a particular route along the circle of fifths that he >>> preferred to get 11? >>>> I doubt it. These two temperaments should both probably be melted down >> into 31 equal, however, which of course makes them the same; hence >> huygens or fokker might be good names. > >> huygens or fokker are indeed the two most appropriate names > for 31edo. > > but if your main criteria in naming is to honor someone > who advocated 11-limit,It would have to be a particular mapping of the 11-limit along a particular chain of fifths that would, for example, correspond to a particular path in the 31-equal circle of fifths. Huygens never went beyond 7-limit, and Fokker didn't prescribe one method of generating (by fifths) 31-equal's approximation of 11 over another.
Message: 11286 - Contents - Hide Contents Date: Mon, 05 Jul 2004 22:25:13 Subject: Re: 24 7-limit temperaments with 245/243 as a comma From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> godzilla/hemifourths > [2, 8, 1, 8, -4, -20] > [[1, 2, 4, 3], [0, -2, -8, -1]]Herman's chart calls this "mothra". My paper calls it "semaphore". Let's call the whole thing off :)
Message: 11287 - Contents - Hide Contents Date: Mon, 05 Jul 2004 00:21:18 Subject: Re: A chart of syntonic comma temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> What you call Mothra, I call Cynder, since it's basically the same as > the Slendric or Wonder temperament, but with 5 thrown into the primal > mix. > What you call Hemifourths, I call Semaphore.Actually, I had proposed Godzilla for that, but unless we use the rest of the 8/7 generator lineup that doesn't suggest as much as Semaphore. There's something attractive about giving similar names to similar generators, though, since we have more than one comma, but just the one generator. In the case of Mothra, I hardly think that a temperament with a comma of 81/80 is "basically the same" as Wonder, so I don't think Cynder as a name has a lot to recommend it on that basis. Were you thinking of Rodan=Supersupermajor? Godzilla has (4/3)/(8/7)^2 = 49/48 going for it, as well as 2^(-20) 7^8 5^(-1). Mothra has (3/2)/(8/7)^3 = 1029/1024, but we can hardly ignore the 81/80. Rodan has 1029/1024 also, as well as 2^(-50) 5 7^17, and so it would seem to be a lot more like something which could be called Wonder with 5 tossed in. Gamera is a microtemperament, with 4375/4374 its most familiar comma.
Message: 11288 - Contents - Hide Contents Date: Mon, 05 Jul 2004 17:46:15 Subject: Re: NOT tuning From: Carl Lumma>>>>> >xcept where TOP already had pure octaves, in >>>>> which case it would actually change! >>>>>>>> That's impossible given the criterion of NOT. >>>> >>>> Maybe I don't comprehend you. >>>>>> I didn't say NOT, I said "Graham" and "pure-octaves TOP". >>>> Ok, it would seem to violate the definition of >> "pure-octaves TOP". >>It doesn't. It still has pure octaves.Oh, I thought you were saying the octaves changed. So in fact I have no clue what you were trying to say. -Carl
Message: 11289 - Contents - Hide Contents Date: Mon, 05 Jul 2004 00:32:12 Subject: Re: bimonzos, and naming tunings (was: Gene's mail server)) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Why harder? Can you show this?From my explanation, you can see it is easy to read off the prime factors in a frontward way and figure out the signs in a bival. With the 7-limit bimonzo, you don't have the same regular pattern where two signs are the same, and one opposite; sometimes they all have the same sign, and sometimes they don't. I think bivals are clearly easier. Moreover, for higher limit linear temperaments, they are still bivals and you can use the same rule, whereas trimonzos are what you get in the 11-limit, etc. A big fat mess by comparison.>> and I >> don't think this makes much of a reason for using bimonzos. I really >> would like to know why Paul insists on this so stubbornly. >> I welcome constructive suggestions for making the paper go bival, > without adding to its math-heaviness.It's really, really, really easy. Simply replace the bimonzos you list with the corresponding bivals, and you are done. You need explain nothing, nor define anything.>> If you have a bival <<a1 a2 a3 a4 a5 a6||, then >> >> 2^a4 3^(-a2) 5^a1 gives the 5-limit comma. >> You don't need to remove common factors?I said "gives the comma", not "is the comma".>> 2^(a5) 3^(-a3) 7^(a1) gives the {2,3,7}-comma; that's 2 to the power >> of the (3,7) coefficient, 3 to minus the power of the (2,7) >> coefficient, and 7 to the power of the (2,3) coefficient; >> How is this easier than the bimonzo case?Because there is a simple rule I can explain which works for any prime limit.
Message: 11290 - Contents - Hide Contents Date: Mon, 05 Jul 2004 19:44:40 Subject: Re: 24 7-limit temperaments with 245/243 as a comma From: Herman Miller Paul Erlich wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > > >> godzilla/hemifourths>> [2, 8, 1, 8, -4, -20] >> [[1, 2, 4, 3], [0, -2, -8, -1]] > >> Herman's chart calls this "mothra". My paper calls it "semaphore". > Let's call the whole thing off :)"Mothra" is <<3, 12, -1, 12, -10, -36|| (formerly known as "supermajor seconds"). I've added "Cynder" to my list of names, but I haven't decided yet which one I prefer. Actually, I think I prefer calling it "5&26" until I get more familiar with it. Godzilla/hemifourths/semaphore would be "5&19". In fact, almost all the useful temperaments can be named this way (if you specify the prime limit). Miracle is 10&31, porcupine is 7&8 in the 5-limit version, 15&22 in the 7-limit version (avoiding the 7-limit inconsistent ET's 7 and 8), negri is 9&10 (no surprise there), orwell is 9&22. This can be extended to naming planar temperaments, with starling as an example being 4&12&15; this immediately lets you know that 4&12 (diminished), 4&15 (kleismic) and 12&15 (tripletone) are starling temperaments. As an added bonus, if the ET's are consistent in a higher limit, there's no question which ET should get the name; in the case of porcupine, 15&22 in the 11-limit is <<3, 5, -6, 4, 1, -18, -4, -28, -8, 32|| (map [<1, 2, 3, 2, 4|, <0, -3, -5, 6, -4|]). By that logic, 7-limit 12&29 <<1, -8, -14, -15, -25, -10|| should have the same name as 5-limit 12&29 <<1, -8, -15|| (schismic). But once you get to the 11-limit, 12-ET is inconsistent.
Message: 11291 - Contents - Hide Contents Date: Mon, 05 Jul 2004 00:34:21 Subject: Re: from linear to equal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> You'd maybe prefer Fokker? >> Did Fokker have a particular route along the circle of fifths that he > preferred to get 11?I doubt it. These two temperaments should both probably be melted down into 31 equal, however, which of course makes them the same; hence huygens or fokker might be good names.
Message: 11292 - Contents - Hide Contents Date: Mon, 05 Jul 2004 00:38:23 Subject: Re: from linear to equal From: Gene Ward Smith --- 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: >>>>>> This is quite an interesting approach. What makes poptimal >>>> generators >>>> good? >>>>>> Not much, IMHO -- the "true" value of p in any situation will be > some>>> number, not an infinite range of numbers. >>>> What in the world does this mean? What allegedly "true" value? >> If you're using this p-norm model in the first place, it's probably > because you think it's true for some value of p.I have no idea how a norm can possibly be either true or false. My assumption is that it is a valid definition of optimum for any p in the range 2 to infinity, and that is simply because if you assume the endpoints define a valid sense of optimum, so should all the intermediate values.>> As for 1, I think a lot of people would find the supposedly optimal >> tunings not really very optimal in some cases. >> And yet there is a significant bunch of composers who refuse to > temper, clinging to their JI scales. Might they be modelled too? (no > offense to them, of course.)If you refuse to temper at all, what in the world are you doing trying to decide which tuning is best, on the assumption that a given temperament will be used?
Message: 11293 - Contents - Hide Contents Date: Mon, 05 Jul 2004 20:13:00 Subject: Re: NOT tuning From: Carl Lumma>Graham's "pure-octaves TOP" is just a uniform stretching or >compression of normal TOP -- except in those cases where normal TOP's >already got pure octaves, in which case the change is not a mere >uniform stretching or compression. Aha! Got'cha.That *is* interesting, and a bit unsettling I suppose. -Carl
Message: 11294 - Contents - Hide Contents Date: Mon, 05 Jul 2004 05:28:10 Subject: Beep, orwell, and schismic From: Gene Ward Smith Beep is the temperament with wedgie <<2 3 1 0 -4 -6||, the 5-limit comma of which is 27/25. This 5-limit temperament and 7-limit beep overlap, with 7/31 as a common poptimal generator. A 7-limit temperament with generator 7/31 representing a 7/6 sounds like orwell, however. In fact, if you run beep out very far it will run into the orwell range, and we have the phenomenon we've discussed where a temperament gets out of its applicable range, so to speak. Evidently this happens pretty soon for 7-limit beep. Meanwhile, the 9-limit version of beep is completely different so far as poptimal tuning goes, with 3/14 a poptimal generator. A very different example is provided by schismic. Here the 5-limit temperament has 169/289 as a poptimal generator; the fifth in question is very slightly flatter than the range for 7 or 9 limit schismic, by which I mean the temperament adding 4375/4374 to the schisma, which 171-et does well, but not so well as to give us a poptimal generator. To get one for 7 and 9 limit schismic, we need to go to 1031/1763, 1131/1934, etc. You could name it after famous people associated to the year in question, if you didn't like using "schismic" for a 7-limit temperament with a different TOP tuning, even if only slightly different, than its 5-limit reduction. In that case I would suggest using pontiac, for 1763, rather than hitler, for the year in which he became the German Fuehrer. But that's just me. Meanwhile, garibaldi is quite happy, for both the 7 and 9 limits, with 94-equal.
Message: 11295 - Contents - Hide Contents Date: Mon, 05 Jul 2004 00:25:37 Subject: Re: NOT tuning From: Carl Lumma>>> >xcept where TOP already had pure octaves, in >>> which case it would actually change! >>>> That's impossible given the criterion of NOT. >> >> Maybe I don't comprehend you. >>I didn't say NOT, I said "Graham" and "pure-octaves TOP".Ok, it would seem to violate the definition of "pure-octaves TOP". -Carl
Message: 11296 - Contents - Hide Contents Date: Mon, 05 Jul 2004 00:27:36 Subject: Re: from linear to equal From: Carl Lumma>> > meant, for a given TOP-tuned linear temperament, does it not >> stand to reason that there is at least one generator/period >> pair (in cents) that produces scales in said tuning? >>Yes, and there's exactly one if you fix that the octave is multiple >of the period, and the generator is within a prescribed 1/2-period >range (say, between 0 or 1/2 period, or between 1/2 and 1 period).Great, thanks. That's as I thought, then. -Carl
Message: 11297 - Contents - Hide Contents Date: Mon, 05 Jul 2004 07:50:51 Subject: 26 7-limit temperaments with 1029/1024 as a comma From: Gene Ward Smith I figured, correctly as it turned out, that this would be a fertile source of monster (8/7) generator temperaments miracle [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]] 13 .631014 106.641366 valentine [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]] 12 1.049791 151.169908 blackwood [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]] 5 7.239629 180.990733 hemithirds [15, -2, -5, -38, -50, -6] [[1, 4, 2, 2], [0, -15, 2, 5]] 20 .479706 191.882395 superkleismic [9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]] 13 1.371918 231.854073 lemba [6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]] 8 3.740932 239.419651 tritikleismic (number 87) [18, 15, -6, -18, -60, -56] [[3, 6, 8, 8], [0, -6, -5, 2]] 24 .448679 258.439259 gidorah [3, 2, -1, -4, -10, -8] [[1, 1, 2, 3], [0, 3, 2, -1]] 4 17.564918 281.038694 unidec [12, 22, -4, 7, -40, -71] [[2, 5, 8, 5], [0, -6, -11, 2]] 26 .421488 284.925888 mothra/cynder [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]] 13 1.698521 287.050049 rodan [3, 17, -1, 20, -10, -50] [[1, 1, -1, 3], [0, 3, 17, -1]] 18 .894655 289.868296 guiron [3, -24, -1, -45, -10, 65] [[1, 1, 7, 3], [0, 3, -24, -1]] 27 .486331 354.535421 [21, -9, -7, -63, -70, 9] [[1, 7, 0, 1], [0, -21, 9, 7]] 30 .421488 379.339200 [6, 29, -2, 32, -20, -86] [[1, 4, 14, 2], [0, -6, -29, 2]] 31 .422358 405.886038 gorgo (Number 90) [3, 7, -1, 4, -10, -22] [[1, 1, 1, 3], [0, 3, 7, -1]] 8 7.279064 465.860138 [9, 0, -3, -21, -30, -7] [[3, 6, 7, 8], [0, -3, 0, 1]] 12 3.361446 484.048237 [6, -12, -2, -33, -20, 29] [[2, 2, 7, 6], [0, 3, -6, -1]] 18 1.557920 504.766152 [15, 3, -5, -30, -50, -20] [[1, -5, 1, 5], [0, 15, 3, -5]] 20 1.276744 510.697600 [3, -19, -1, -37, -10, 51] [[1, 1, 6, 3], [0, 3, -19, -1]] 22 1.297058 627.776195 [12, 12, -4, -9, -40, -43] [[4, 7, 10, 11], [0, -3, -3, 1]] 16 3.494156 894.503841 [30, 37, -10, -11, -100, -127] [[1, 7, 9, 1], [0, -30, -37, 10]] 47 .431718 953.663965 [3, -14, -1, -29, -10, 37] [[1, 1, 5, 3], [0, 3, -14, -1]] 17 3.304346 954.955890 [9, -5, -3, -29, -30, 7] [[1, 4, 1, 2], [0, -9, 5, 3]] 14 4.889160 958.275451 [9, 15, -3, 3, -30, -49] [[3, 6, 9, 8], [0, -3, -5, 1]] 18 3.211020 1040.370387 [3, -9, -1, -21, -10, 23] [[1, 1, 4, 3], [0, 3, -9, -1]] 12 7.272811 1047.284848 [6, 8, -2, -1, -20, -27] [[2, 2, 3, 6], [0, 3, 4, -1]] 10 10.946338 1094.633800
Message: 11298 - Contents - Hide Contents Date: Mon, 05 Jul 2004 09:49:03 Subject: Re: from linear to equal From: monz hi Gene and Paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>>> You'd maybe prefer Fokker? >>>> Did Fokker have a particular route along the circle of fifths that he >> preferred to get 11? >> I doubt it. These two temperaments should both probably be melted down > into 31 equal, however, which of course makes them the same; hence > huygens or fokker might be good names.huygens or fokker are indeed the two most appropriate names for 31edo. but if your main criteria in naming is to honor someone who advocated 11-limit, a good choice might be Ptolemy. his "smooth (or 'equable' as quoted by Partch) diatonic" and "syntonic chromatic" genera both used ratios of 11. here are the tetrachord structures of those two tunings: Ptolemy - _genos homalon diatonon_ = "even diatonic genus" Tonalsoft Encyclopaedia of Tuning - diatonic, ... * [with cont.] (Wayb.)>> string-length proportions: 9 : 10 : 11 : 12 >> >> >> note ...... ratio ... ~ cents >> >> mese ...... 1/1 ....... 0 >> .................................... > . 9:10 . ~ 182.4037121 cents >> lichanos .. 9/10 .. - 182.4037121 >> .................................... >. 10:11 . ~ 165.0042285 cents >> parhypate . 9/11 .. - 347.4079406 >> .................................... >. 11:12 . ~ 150.6370585 cents >> hypate .... 3/4 ... - 498.0449991 >> >> The string-length proportions of Ptolemy's "even diatonic" >> have the smallest-number consecutive ratios which can describe >> a four-fold division of the 4:3 "perfect-4th". The top interval >> is thus the 5-limit 10:9 "lesser tone", the middle interval is >> the 11:10 "undecimal tone", and the bottom interval is the >> 12:11 "neutral 2nd" functioning as a very wide semitone.Ptolemy - _genos syntonon chromatikon_ = "tense chromatic genus" Tonalsoft Encyclopaedia of Tuning - chromatic,... * [with cont.] (Wayb.)>> string-length proportions: 66 : 77 : 84 : 88 >> >> >> note ...... ratio ... ~ cents >> >> mese ....... 1/1 ...... 0 >> .................................... >. 6:7 .. ~ 266.8709056 cents >> lichanos ... 6/7 .. - 266.8709056 >> .................................... > 11:12 . ~ 150.6370585 cents >> parhypate . 11/14 . - 417.5079641 >> .................................... > 21:22 . ~ 80.53703503 cents >> hypate ..... 3/4 .. - 498.0449991 >> >> In his "tense" version of the chromatic genus, Ptolemy used >> the narrow 7:6 "septimal minor-3rd" as his "characteristic >> interval" at the top of the tetrachord, thus simultaneously >> placing his lichanos a wide 8:7 "septimal tone" above the >> bottom note hypate. The middle interval is the 12:11 >> "undecimal neutral 2nd", functioning here as a very wide semitone.in a later era, Partch might be the obvious choice ... but then again, he stressed JI so much that it would never be a good idea to honor him by naming a temperament after him. he was philosophically opposed to temperament, you could almost say on moral grounds. Ben Johnston speaks of temperament in much the same way. and of course, here on the list Jon Szanto, Kraig Grady, and David Beardsley and Pat Pagano would hold similar views. -monz ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 11299 - Contents - Hide Contents Date: Tue, 06 Jul 2004 19:36:20 Subject: Re: bimonzos, and naming tunings (was: Gene's mail server)) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: I'm a little late replying, because I've got a new computer now. I'm taking a break from things like config files.>> It's really, really, really easy. Simply replace the bimonzos you > list>> with the corresponding bivals, and you are done. >> You've got to be kidding me.Why? You said you were not introducing any multilinar algebra, I thought. Certainly you cannot do so and stay suitably nonmathematical.>> You need explain >> nothing, nor define anything. >> I'd like to do better by my readers.How does simply tossing a bimonzo in their face do better by them?> Show me how a bimonzo gets so bad in a higher prime limit.The bimonzo isn't even what you would use in a higher prime limit, for starters. To get a linear temperament, you need a monzo in the 5- limit, a bimonzo in the 7-limit, a trimonzo in the 11-limit, and so forth. It's bi*val*, not bimonzo, which gives linear temperaments always. (1) In any limit, the first pi(p)-1 entries of the bival give us the period, and the generator part of the period-generator map. For any limit above 5, the advantage goes on this point to bivals. (2) In all limits, we can read off the three-prime commas of the temperament by taking two of the exponents to have one sign, and the other to have another. This makes the rule for these commas an easier one in all limits above 5. For example, in the 7 limit, suppose we have the bival <<a23 a25 a27 a35 a37 a57|| Then the commas can be found from 2^a35 3^-a25 5^a23 2^a37 3^-a27 7^a23 2^a57 5^-a27 7^a25 3^a57 5^-a37 7^a35 All of the commas have two exponents of one sign and one of the opposite sign in terms of the components of the bival. If we take the complement of this, and so use the bimonzo signs, two of the commas (the odd comma, {3,5,7}, and the 5-limit comma, {2,3,5}) have all the exponents the same sign, and the other two have two the same and one the opposite. In the 11 and higher limits, it is similar. The bival commas are all of the ++- form, but the complementary multimonzo does not give all of them the same form. In the 11-limit, the {2,5,11}, {3,7,11} and {5,7,11} commas have the +++ form, and the rest the --- form. Again, the bival form on this basis is clearly superior. I would really like to know why you are so intransigent on this score; there seems to be no advantage in using multimonzos beyond the 5-limit as a canonical form, and you don't seem to have any argument beyond a dislike of vals for not using bivals. But this isn't a good argument, because vals are important and should be introduced before multilinear algebra is even mentioned.
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