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Message: 3725 - Contents - Hide Contents Date: Sat, 2 Feb 2002 12:06:40 Subject: rational approximations to logarithms (was: simple math question) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, February 02, 2002 11:44 AM > Subject: Re: [tuning-math] simple math question > > > there are some very good low-integer-ratio approximations to > the log_10 of the lowest primes > > > Examples: > > log_10 ~fractional value > of: of logarithm > > more accurate less accurate > > 2 3/10 > 3 10/21 = ~1/2 > 5 7/10 > 7 11/13 = ~5/6 > 11 25/24 > 13 39/35 = ~10/9 > 17 16/13 = ~11/9 > 19 23/18 = ~14/11 = ~5/4in fact, i observed before that 3/10 showed up in a sumerian calculation which i think relates to tuning -- see the middle of Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) i had a hunch there that this had something to do with tempering, and now i realize why i had that hunch 3/10 is so close to the log_10(2) that it would serve just fine as an approximation in doing calculations by hand hmmm .... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3726 - Contents - Hide Contents Date: Sat, 2 Feb 2002 14:45 +00 Subject: Re: interval of equivalence, unison-vector, period From: graham@xxxxxxxxxx.xx.xx genewardsmith wrote:> Because a "temperament" which sends > > 1-9/8--5/4--4/3--3/2--5/3--15/8 to > > 1--9--1/9--1/3--3--1/27--1/3 > > hardly seems worthy of the name. In any case, 2 isn't represented!Why not? Why is 2 special? Graham

Message: 3727 - Contents - Hide Contents Date: Sat, 2 Feb 2002 14:45 +00 Subject: Re: interval of equivalence, unison-vector, period From: graham@xxxxxxxxxx.xx.xx paulerlich wrote:> You'd have to invoke "tritone-equivalence", which is clearly not a > recognized psychoacoustical phenomenon!You certainly would. Psychoacoustics will have to make its own mind up. Graham

Message: 3728 - Contents - Hide Contents Date: Sat, 02 Feb 2002 20:28:23 Subject: Re: simple math question From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> particularly in tuning math, logs are often taken to base 2 > since 2 is the ratio of the "octave", and the author often > assumes that the reader will know that and assume 2 as the baseAnother base often found in these parts is 2^(1/1200); the log base the 1200 root of two defines interval size in cents.

Message: 3729 - Contents - Hide Contents Date: Sat, 2 Feb 2002 14:45 +00 Subject: Re: interval of equivalence, unison-vector, period From: graham@xxxxxxxxxx.xx.xx genewardsmith wrote:> Is pajara the new official name? I'd like to get this settled. As for > this val, which defines only one of two required generator mappings > being a temperament, that's only if you layer on some interpretation > and perform the extra calculations to find a good choice for the second > generator; taken by itself, it isn't one. It's telling us to send the > octave to a unison, and 5 and 7 both to 1/9; it's only after you stick > in half-octaves and send 7 to some tuning of 64/9 and 5 to a > half-octave below that that pajara emerges. Read literally as a > temperament, it sends 2 to 1 and 5 and 7 to 1/9, and I don't think that > qualifies.It defines one of the required mappings for an octave-specific linear temperament. On it's own it is an equal temperament, but a somewhat strange one. It could also be an octave-equivalent mapping for pajara. An equal temperament with no steps to the octave is an octave-equivalent linear temperament. I don't see what other sense making an octave an identity vector could make. Yes, it's telling us to map the octave to a unison which we could call "imposing octave equivalence" or making the octave an identity vector. As it has torsion, that makes a tritone the real identity vector. Yes, it sends 2 to 1. They only differ by identity vectors. That's what octave equivalent temperaments are like. Also 5, 7 and 1/9 differ only by identity vectors. So it's a tritone equivalent linear temperament.>> The octave is acting as a unison, but it's more complicated than >> that. As it has torsion, it's actually half an octave that's acting >> as a commatic unison vector. >> I would say it's acting as a generator, but if you make 2 a unison it > becomes a torsion element, since its square is an octave.It only acts as a generator if it isn't tempered out. In this case it is being tempered out, so it must be an identity vector, which means it's like a unison vector. Torsion is certainly involved, but I don't understand what you're saying there.>> (BTW, in an octave-equivalent system, half a unison is a half-octave >> as well as a unison. This is obvious if you think of >> octave-equivalent frequency space as a Hilbert space, and remember >> that half the pitch is the same as the square root of the frequency.) >> You get a real Hilbert space if you allow anything of the form > 3^e3 5^e5 ... which can have an infinite number of prime exponents so > long as e3^2 + e5^2 + ... converges. Is this what you mean? The result > isn't even guaranteed to be a real number, and I don't know what it > would be good for.3 and 5 don't enter into it. I mean octave equivalent frequency ratios behave like the complex numbers with modulus 1. Partly because it's a circular system, and also because nth roots have n values.>>>>> i0 = i2-h2 >>>>> i0.basis>> [0, 1, -2, -2] >> >> Hey, that's the same as g0 above! >> And which I think hardly counts as a temperament. As I said, it's not > one I want to listen to.There are all kinds of temperaments I wouldn't want to listen to. Why single this one out for opprobrium? Graham

Message: 3730 - Contents - Hide Contents Date: Sat, 02 Feb 2002 20:42:38 Subject: Re: simple math question From: jpehrson2 --- In tuning-math@y..., "monz" <joemonz@y...> wrote: Yahoo groups: /tuning-math/message/3185 * [with cont.]> > hi joe, > >>> From: jpehrson2 <jpehrson@r...> >> To: <tuning-math@y...> >> Sent: Saturday, February 02, 2002 10:56 AM >> Subject: [tuning-math] simple math question >> >> >> Could it possibly be said that a logarithm is a way to find >> the "exponent" of a number?? >> >> I mean, in the most simple case... >> >> ?? > > >> the logarithm is t h e way to find the exponent > > by definition, that's exactly its purpose >***Thanks, everybody, for the answers... It looks like I was "kinda right" and "kinda wrong..." It seems to hinge mostly on *definitions...* Thanks! JP

Message: 3731 - Contents - Hide Contents Date: Sat, 02 Feb 2002 20:58:21 Subject: 7-limit MT reduced bases for meantone ets From: genewardsmith For all you meantone fans out there (and you know who you are), here are some meantone systems not already discussed. Rational implications are staring us in the face here. 24: [49/48, 81/80, 128/125] 26: [50/49, 81/80, 525/512] 38: [50/49, 81/80, 3125/3072] 45: [81/80, 525/512, 2401/2400] 50: [81/80, 126/125, 16807/16384] 55: [81/80, 686/675, 6144/6125] 67: [81/80, 1029/1024, 9604/9375] 74: [81/80, 126/125, 4194304/4117715] 81: [81/80, 126/125, 17294403/16777216]

Message: 3732 - Contents - Hide Contents Date: Sat, 02 Feb 2002 07:12:12 Subject: Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets) From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> 171: [2401/2400, 4375/4374, 32805/32768] >> >> Wouldn't want to do that--look at those three high-powered commas!> (note that the title is an homage to Helmholtz)Helmholtz liked the schismic temperament, and Vogel goes him one better by combining schismic with ennealimmal, which the above reduced basis shows isone way of thiking about 171-et. You could temper either 53 tones or 72 tones with it, among other things. Since I am now writing a piece in 46-et and just finished one in 53-et, I'll also add these: 46: [126/125, 245/243, 1029/1024] 53: [225/224, 1728/1715, 3125/3087] I'm finding the 43-et set of commas quite useful.

Message: 3733 - Contents - Hide Contents Date: Sat, 02 Feb 2002 07:21:55 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Well I confused the two things, which is completely my fault, but was >> not helped by Graham's opinion that the thing you declined to call a >> temperament was in fact pajara. >> If you ever get around to trying 222223 with a period of 3/2 in the >22-et, tell us about it.I think I will!

Message: 3734 - Contents - Hide Contents Date: Sat, 02 Feb 2002 07:24:05 Subject: Re: 7-limit MT reduced bases for ets From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>> 9: [21/20, 27/25, 128/125] >> 10: [25/24, 28/27, 49/48] >> 12: [36/35, 50/49, 64/63] >> 15: [28/27, 49/48, 126/125] >> 19: [49/48, 81/80, 126/125] >> 22: [50/49, 64/63, 245/243] >> 27: [64/63, 126/125, 245/243] >> 31: [81/80, 126/125, 1029/1024] >> 41: [225/224, 245/243, 1029/1024] >> 68: [245/243, 2048/2025, 2401/2400] >> 72: [225/224, 1029/1024, 4375/4374] >> 99: [2401/2400, 3136/3125, 4375/4374] >> 130: [2401/2400, 3136/3125, 19683/19600] >> 140: [2401/2400, 5120/5103, 15625/15552] >> This is seriously cool. >>> For any prime limit, we could consider the most characteristic >> linear temperament of a particular et to be the one leaving off >> the last member of the MT reduced basis. >> Does it have to be prime (not odd) limit? > > -CarlYeah, Carl, these are _tuning system building_ considerations and not _simultaneous consonance_ considerations.

Message: 3735 - Contents - Hide Contents Date: Sat, 02 Feb 2002 07:26:06 Subject: Re: 7-limit MT reduced bases for ets From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "clumma" <carl@l...> wrote: >>> Does it have to be prime (not odd) limit? >> Fraid so. It occurs to me another fun game to play with these is to >find the corresponding Fokker blocks.Or better yet, the most compact blocks where ratio odd-limit measures distance from a central 1/1 (this is what I refer to as the van Prooijen metric). Remember, these are constrained to be periodicity blocks, so will _not_ be ellipsoids.

Message: 3736 - Contents - Hide Contents Date: Sat, 02 Feb 2002 07:27:47 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Friday, February 01, 2002 3:56 PM >> Subject: [tuning-math] Re: interval of equivalence, unison- vector, period >> >>>>> I would say it's acting as a generator, but if you make 2 a unison >>> it becomes a torsion element, since its square is an octave. >>>> This, along with my message to Monzo this morning, seems to show the >> very real problems with considering 2 a unison! > >> How is 2^2 an octave? By definition, it's simply 2. > Now you guys have really lost me.Dude, what exactly are you referring to? I thought this was amazingly clear, but I guess I'm wrong!

Message: 3737 - Contents - Hide Contents Date: Sat, 02 Feb 2002 07:29:19 Subject: Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets) From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> 171: [2401/2400, 4375/4374, 32805/32768] >>> >>> Wouldn't want to do that--look at those three high-powered commas! >>> (note that the title is an homage to Helmholtz) >> Helmholtz liked the schismic temperament, and Vogel goes him onebetter by combining schismic with ennealimmal, which the above reduced basis shows is one way of thiking about 171-et. You could temper either 53 tones or 72 tones with it, among other things.> > Since I am now writing a piece in 46-et and just finished one in > 53-et, I'll also add these: > > 46: [126/125, 245/243, 1029/1024] > 53: [225/224, 1728/1715, 3125/3087] > > I'm finding the 43-et set of commas quite useful.Do you really mean 43, or one of the above?

Message: 3738 - Contents - Hide Contents Date: Sat, 02 Feb 2002 08:05:50 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Friday, February 01, 2002 11:27 PM >> Subject: [tuning-math] Re: interval of equivalence, unison- vector, period >> >>>>>>> I would say it's acting as a generator, but if you >>>>> make 2 a unison it becomes a torsion element, since >>>>> its square is an octave. >>>>>>>> This, along with my message to Monzo this morning, >>>> seems to show the very real problems with considering >>>> 2 a unison! >>> >>>>>> How is 2^2 an octave? By definition, it's simply 2. >>> Now you guys have really lost me. >>>> Dude, what exactly are you referring to? I thought this >> was amazingly clear, but I guess I'm wrong! > >> Oh, OK ... I think I get it. > > If 2 = a unison, then 2^2 = an octave. Yes?Umm . . . not exactly. 2 is an octave, and 2^2 is a double octave.

Message: 3739 - Contents - Hide Contents Date: Sat, 02 Feb 2002 08:13:38 Subject: Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets) From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> I'm finding the 43-et set of commas quite useful. >> Do you really mean 43, or one of the above?I meant 46--just my usual terrific job of proof-reading. 43: [81/80, 126/125, 12288/12005]

Message: 3740 - Contents - Hide Contents Date: Sat, 2 Feb 2002 01:26:27 Subject: Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets) From: monz ------------> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, February 02, 2002 12:13 AM > Subject: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets) > > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> I'm finding the 43-et set of commas quite useful. >>>> Do you really mean 43, or one of the above? >> I meant 46--just my usual terrific job of proof-reading. > > 43: [81/80, 126/125, 12288/12005]thanks for sending t h a t now anyway, gene! 43-edo was advocated as a tuning by Sauveur file:///C:/interval/dict/meride.htm#43-1/5diff i've updated that page now thanks to the attention you caused me to give it, and included a link to your post above i'd like to change that link to point to something that really explains what "7-limit MT reduced bases" are, but i sure don't know to explain it, so if you do then i'll let the link point to that post thanks from manuel's page: Stichting Huygens-Fokker: Logarithmic Interval... * [with cont.] (Wayb.)>> méride: 1/43 part of an octave >> >> This name was chosen by Joseph Sauveur (1653-1716) >> in 1696. The méride and eptaméride were the first >> logarithmic interval measures proposed. Sauveur >> favoured 43-tone equal temperament because the small >> intervals are well represented in it. He had set the >> comma to one step, then found a range of 2, 3 or 4 >> steps for the chromatic semitone, corresponding to 31, >> 43 and 55 tones per octave. He found 43 to be optimal >> because 4 steps is almost exactly a 16/15 minor second >> and 7 steps almost exactly the geometric mean of >> three 9/8 and two 10/9 whole tones. The chromatic scale >> contained in 43-tET is virtually identical to 1/5-comma >> meantone tuning. -monz ------------ -monz _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3741 - Contents - Hide Contents Date: Sat, 2 Feb 2002 01:29:59 Subject: ---------- request for everyone to add line separators From: monz i have a request to make: i've begun compiling hard-copy volumes of tuning-math posts so that i can do serious studying of them the easiest way i've found to do this is to go to the "expand messages" version of the yahoo interface for the post on the 1st of each month, copy what's there, and keep clicking "next" until the month is finished when i print it, there's no easy way to see where one post ends and another begins would everyone please add something like this ---------- as a separator line at the beginning and end of each post? i'd really appreciate that -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3742 - Contents - Hide Contents Date: Sat, 02 Feb 2002 00:03:44 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> So why did you say "this was not a temperament"? >> Because a "temperament" which sends > > 1-9/8--5/4--4/3--3/2--5/3--15/8 to > > 1--9--1/9--1/3--3--1/27--1/3 > > hardly seems worthy of the name.You'd have to invoke "tritone-equivalence", which is clearly not a recognized psychoacoustical phenomenon!>> And isn't it true >> that, if you took it out to, say, 10 notes per approximate octave, >> and tuned the octaves pure, it would _not_ be an octave-repeating >> scale? This seems to be the point Graham is missing. >> We seem to be talking about different things--what is "it"? If you > mean pajara, it's a temperament, not a scale.What I thought we were talking about, and I thought you agreed, was the fact that you derived temperaments originally in terms of two generators, neither of which was guaranteed to be an octave, and then came up with a different basis for the temperament such that the octave was either a member of the basis or a power of a member of the basis. For example, didn't you originally state that a form of Blackjack which has a period of ~5/3 came out of your mechanism? I took that to mean that if there were no such phenomenon as octave equivalence, a Blackjack scale with a period of ~5/3 would better exploit the consonances than the standard Blackjack scale.

Message: 3743 - Contents - Hide Contents Date: Sat, 2 Feb 2002 01:50:10 Subject: Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets) From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, February 02, 2002 1:26 AM > Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT reducedbases for ets)> > > from manuel's page: > Stichting Huygens-Fokker: Logarithmic Interval... * [with cont.] (Wayb.) >>>> ... Sauveur ... found 43 to be optimal >>> because 4 steps is almost exactly a 16/15 minor second >>> and 7 steps almost exactly the geometric mean of >>> three 9/8 and two 10/9 whole tones. The chromatic scale >>> contained in 43-tET is virtually identical to 1/5-comma >>> meantone tuning.[-9 6 0] = 3 * [-3 2 0] (= 9:8 whole tone) + [ 2 -4 2] = 2 * [ 1 -2 1] (= 10:9 whole tone) ---------- [-7 2 2] (= 225:128 "augmented 6th") [7 2 2]^(1/2) = [-7/2 1 1] = ~488.2687147 cents but what significance does that have? i don't get it manuel? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3744 - Contents - Hide Contents Date: Sat, 02 Feb 2002 01:04:01 Subject: Re: interval of equivalence, unison-vector, period From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> The thing he said wasn't a temperament has no notes to an octave, >> No notes? He said it was generated by a fifth and a fifth-tritone -- > so it seems like it could have plenty of notes, up to an infinite > number, in fact.One is a basis for pajara/twintone. We have octaves in it, since (15/14)^(-2) (3/2)^2 = 49/25 ~ 2. The other thing, which I declined to call a temperament, doesn't even represent octaves, so it depends on which thing you are talking about.

Message: 3745 - Contents - Hide Contents Date: Sat, 2 Feb 2002 02:01:31 Subject: 43-edo (was: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets)) From: monz ------------> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, February 02, 2002 1:50 AM > Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT reducedbases for ets)> > from manuel's page: > Stichting Huygens-Fokker: Logarithmic Interval... * [with cont.] (Wayb.) >>>> ... Sauveur ... found 43 to be optimal >>> because 4 steps is almost exactly a 16/15 minor second >>> and 7 steps almost exactly the geometric mean of >>> three 9/8 and two 10/9 whole tones. The chromatic scale >>> contained in 43-tET is virtually identical to 1/5-comma >>> meantone tuning. > > >> [-9 6 0] = 3 * [-3 2 0] (= 9:8 whole tone) > + > [ 2 -4 2] = 2 * [ 1 -2 1] (= 10:9 whole tone) > ---------- > [-7 2 2] (= 225:128 "augmented 6th") > > > [7 2 2]^(1/2) = [-7/2 1 1] = ~488.2687147 cents > > > but what significance does that have? i don't get it > > manuel?the only thing that i think i can see is some kind of tritone-equivalence in action, because if you ignore prime-factor 2 you get a mean for the 225:128 of 15:8, which is 2^(1/2) higher than the above interval, and which is the interval that is given exactly by 5 generators of 1/5-comma meantone but i really don't understand what's going on -monz ------------- _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3746 - Contents - Hide Contents Date: Sat, 02 Feb 2002 01:11:57 Subject: Re: interval of equivalence, unison-vector, period From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> The thing he said wasn't a temperament has no notes to an octave, >>>> No notes? He said it was generated by a fifth and a fifth- tritone -- >> so it seems like it could have plenty of notes, up to an infinite >> number, in fact. >> One is a basis for pajara/twintone. We have octaves in it, since > (15/14)^(-2) (3/2)^2 = 49/25 ~ 2.This is the thing I was trying to call Graham's attention to earlier.>The other thing, which I declined >to call a temperament, doesn't >even represent octaves, so it depends >on which thing you are >talking about.Well I confused the two things, which is completely my fault, but was not helped by Graham's opinion that the thing you declined to call a temperament was in fact pajara.

Message: 3747 - Contents - Hide Contents Date: Sat, 02 Feb 2002 10:08:01 Subject: Re: 171-EDO, Vogel (was: 7-limit MT reduced bases for ets) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> From: monz <joemonz@y...> >> To: <tuning-math@y...> >> Sent: Saturday, February 02, 2002 1:26 AM >> Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT reduced> bases for ets) >> >>>> from manuel's page: >> Stichting Huygens-Fokker: Logarithmic Interval... * [with cont.] (Wayb.) >>>>>> ... Sauveur ... found 43 to be optimal >>>> because 4 steps is almost exactly a 16/15 minor second >>>> and 7 steps almost exactly the geometric mean of >>>> three 9/8 and two 10/9 whole tones. The chromatic scale >>>> contained in 43-tET is virtually identical to 1/5-comma >>>> meantone tuning. > > > >> [-9 6 0] = 3 * [-3 2 0] (= 9:8 whole tone) > + > [ 2 -4 2] = 2 * [ 1 -2 1] (= 10:9 whole tone) > ---------- > [-7 2 2] (= 225:128 "augmented 6th") > > > [7 2 2]^(1/2) = [-7/2 1 1] = ~488.2687147 cents > > > but what significance does that have? i don't get it > > manuel?what are you trying to do here, monz?

Message: 3748 - Contents - Hide Contents Date: Sat, 02 Feb 2002 01:12:36 Subject: Re: interval of equivalence, unison-vector, period From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> What I thought we were talking about, and I thought you agreed, was > the fact that you derived temperaments originally in terms of two > generators, neither of which was guaranteed to be an octave, and then > came up with a different basis for the temperament such that the > octave was either a member of the basis or a power of a member of the > basis.That's a trick you can do with linear temperaments, and since this form is both useful and commonly used by Graham, it seems like a good one. This thread in my mind is partly about the point that the octave does not have a special status, in that you can do exactly the same for other intervals, such as a fifth. For example, didn't you originally state that a form of> Blackjack which has a period of ~5/3 came out of your mechanism?That didn't come out of my mechanism, it came out of my misunderstanding of a comment you made. I> took that to mean that if there were no such phenomenon as octave > equivalence, a Blackjack scale with a period of ~5/3 would better > exploit the consonances than the standard Blackjack scale.It seems like an interesting plan, at any rate. As I say, I can take any linear temperament such as miracle, and use anything I like (and 5/3 would seem to be a particularly good choice) as one of the generators, or at least as a power of one of the generators.

Message: 3749 - Contents - Hide Contents Date: Sat, 02 Feb 2002 10:08:34 Subject: Re: 43-edo (was: 171-EDO, Vogel (was: 7-limit MT reduced bases forets)) From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> ------------ >>> From: monz <joemonz@y...> >> To: <tuning-math@y...> >> Sent: Saturday, February 02, 2002 1:50 AM >> Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT reduced> bases for ets) >>>> from manuel's page: >> Stichting Huygens-Fokker: Logarithmic Interval... * [with cont.] (Wayb.) >>>>>> ... Sauveur ... found 43 to be optimal >>>> because 4 steps is almost exactly a 16/15 minor second >>>> and 7 steps almost exactly the geometric mean of >>>> three 9/8 and two 10/9 whole tones. The chromatic scale >>>> contained in 43-tET is virtually identical to 1/5-comma >>>> meantone tuning. > > > > > >> >> >>>> [-9 6 0] = 3 * [-3 2 0] (= 9:8 whole tone) >> + >> [ 2 -4 2] = 2 * [ 1 -2 1] (= 10:9 whole tone) >> ---------- >> [-7 2 2] (= 225:128 "augmented 6th") >> >> >> [7 2 2]^(1/2) = [-7/2 1 1] = ~488.2687147 cents >> >> >> but what significance does that have? i don't get it >> >> manuel? > > >> the only thing that i think i can see is some kind of > tritone-equivalence in action, because if you ignore > prime-factor 2 you get a mean for the 225:128 of 15:8, > which is 2^(1/2) higher than the above interval, and > which is the interval that is given exactly by 5 generators > of 1/5-comma meantone > > but i really don't understand what's going oni don't understand what you're trying to do

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