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Message: 8925 - Contents - Hide Contents Date: Sat, 03 Jan 2004 16:58:15 Subject: Re: name? From: Carl Lumma>>> >h so you're thinking about 13-equal. >>>> Yup. And I think only multiples of 13 do the job? >> >> -Carl >>don't get your question . . .Which ETs temper this comma out? -Carl
Message: 8926 - Contents - Hide Contents Date: Sat, 03 Jan 2004 16:59:12 Subject: Re: name? From: Carl Lumma>> >hat's funny, it's supposed to be 222 cents. Crap, so sorry, >> it should have been [-30 0 13]. >>Here we see the advantage of leaving the 2s in.Or in my case, not failing to leave in the 3s! -Carl
Message: 8927 - Contents - Hide Contents Date: Sat, 03 Jan 2004 17:21:27 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>>>> >ene and you had an exchange. Gene suggested what I was >>>> thinking, you said that each band represented only a >>>> denominator, not a comma. >>>> Or are you talking about something else? >>>>>> Yes, a later pair of graphs. >>>> D'oh! Link? (I even went to tuning-math to find it myself, but >> this thread is not connected to anything). > >Yahoo groups: /tuning-math/files/Paul/com5monz... * [with cont.] > >and > >Yahoo groups: /tuning-math/files/Paul/com5rat.gif * [with cont.]These are the graphs I thought Gene replied about -- do they look very similar to those?>>>>> No need to specify a consonance limit? -- wow that's hot. >>>>>>>> How do you mean? Isn't this implicit in the dimensionality of >>>> the Tenney lattice you're using? >>>>>> No, no consonance limit (aka odd-limit) is implicity in the >>> dimensionality of the Tenney lattice you're using -- and even the >>> latter, aka prime-limit, doesn't need to be specified -- for >>> example the results for the pythagorean comma will be valid in >>> both lower and higher prime limits. >>>> I must not be tracking you -- the pythagorean comma is of course >> a 3-limit comma. >I don't see anything new in any of this, I'm afraid. Let's back up. We're talking about the badness of commas? Typically that's been a function of their size in cents and the number of notes you can be expected to search to find them (which depends on their distance on the lattice and the dimensionality of the lattice). That's Gene's logflat badness, which I've implemented in Scheme. Now, what exactly are you up to here? -Carl
Message: 8928 - Contents - Hide Contents Date: Sun, 04 Jan 2004 21:22:32 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Oh, BTW, I've been looking at commas of the form (n*n):(n+1)(n-1). Do > they have a name?In post 849 on this list I called them B(1, n), which isn't much of a name, nor is Square(n) which I think I've also used. What about squarejack as a name? Then we could also have trianglejacks and so forth, if anyone feels inspired. They always come from setting two n-integer limit> intervals to be equivalent. They're also more likely than arbitrary > superparticulars to belong to a low prime limit.. The 11-prime limit > subset gives us > > 4:3, 9:8, 16:15, 25:24, 36:35, 49:48, 64:63, 81:80, > 100:99, 225:224, 441:440, 2401:2400Note 16/15, 81/80 and 2401/2400 have fourth powers in the numerator and 81/80 = (9/8)/(10/9), 2401/2400 = (49/48)/(50/49) are jumping jacks. We aslo have 225/224, where the numerator is a square of a triangular number and is a jumping jack by 225/224 = (15/14)/(16/15). 36/35 has a numerator which is both square and triangular, making it a high jack.
Message: 8929 - Contents - Hide Contents Date: Sun, 04 Jan 2004 22:18:01 Subject: Square triangular numbers, etc From: Gene Ward Smith Square Triangular Number -- from MathWorld * [with cont.] Reply from On-Line Encyclopedia * [with cont.] (Wayb.) Reply from On-Line Encyclopedia * [with cont.] (Wayb.)
Message: 8930 - Contents - Hide Contents Date: Sun, 04 Jan 2004 22:55:29 Subject: Squarejacks From: Gene Ward Smith Here is a list n < 5000 such that n^2/(n^2-1) is within the 23 limit, together with the prime limit. 2 2 3 3 4 2 5 5 6 3 7 7 8 2 9 3 10 5 11 11 12 3 13 13 14 7 15 5 16 2 17 17 18 3 19 19 20 5 21 7 22 11 23 23 24 3 25 5 26 13 27 3 33 11 34 17 35 7 39 13 45 5 49 7 50 5 51 17 55 11 56 7 64 2 65 13 69 23 76 19 77 11 91 13 99 11 120 5 153 17 161 23 169 13 170 17 208 13 209 19 323 19 324 3 351 13 391 23 441 7 2024 23 2431 17
Message: 8931 - Contents - Hide Contents Date: Sun, 04 Jan 2004 22:58:43 Subject: Re: Squarejacks From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Here is a list n < 5000 such that n^2/(n^2-1) is within the 23 limit, > together with the prime limit.Sorry, my "prime limit" rountine is actually calculating the number of primes which appear in the factorization.
Message: 8932 - Contents - Hide Contents Date: Sun, 04 Jan 2004 23:08:16 Subject: Re: Squarejacks From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Here is a list n < 5000 such that n^2/(n^2-1) is within the 23 limit, > together with the prime limit. 2 3 3 3 4 5 5 5 6 7 7 7 8 7 9 5 10 11 11 11 12 13 13 13 14 13 15 7 16 17 17 17 18 19 19 19 20 19 21 11 22 23 23 23 24 23 25 13 26 13 27 13 33 17 34 17 35 17 39 19 45 23 49 7 50 17 51 17 55 11 56 19 64 13 65 13 69 23 76 19 77 19 91 23 99 11 120 17 153 19 161 23 169 17 170 19 208 23 209 19 323 23 324 19 351 13 391 23 441 17 2024 23 2431 19
Message: 8933 - Contents - Hide Contents Date: Sun, 04 Jan 2004 00:14:20 Subject: Re: name? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Oh so you're thinking about 13-equal. >> Yup. And I think only multiples of 13 do the job? > > -Carldon't get your question . . .
Message: 8934 - Contents - Hide Contents Date: Sun, 04 Jan 2004 00:28:05 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Gene and you had an exchange. Gene suggested what I was thinking, >>> you said that each band represented only a denominator, not a comma. >>> Or are you talking about something else? >>>> Yes, a later pair of graphs. >> D'oh! Link? (I even went to tuning-math to find it myself, but > this thread is not connected to anything). Yahoo groups: /tuning-math/files/Paul/com5monz... * [with cont.] and Yahoo groups: /tuning-math/files/Paul/com5rat.gif * [with cont.]>>>> No need to specify a consonance limit? -- wow that's hot. >>>>>> How do you mean? Isn't this implicit in the dimensionality of the >>> Tenney lattice you're using? >>>> No, no consonance limit (aka odd-limit) is implicity in the >> dimensionality of the Tenney lattice you're using -- and even the >> latter, aka prime-limit, doesn't need to be specified -- for example >> the results for the pythagorean comma will be valid in both lower and >> higher prime limits. >> I must not be tracking you -- the pythagorean comma is of course > a 3-limit comma.See the last paragraph of Yahoo groups: /tuning/message/50964 * [with cont.]
Message: 8935 - Contents - Hide Contents Date: Sun, 04 Jan 2004 00:41:58 Subject: Re: name? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> That's funny, it's supposed to be 222 cents. Crap, so sorry, > it should have been [-30 0 13].Here we see the advantage of leaving the 2s in.
Message: 8936 - Contents - Hide Contents Date: Sun, 04 Jan 2004 01:00:23 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> What would you call this kind of meantone? The 3:2 is flattened by > about 10/49-comma, while the octave is widened by about 3/38-comma.It's approximated by 19ED2.00196, very well approximated by 31ED2.001963, or if you need more accuracy, 205ED2.00196315 . . . 12 doesn't even seem to be a convergent here, it skips right from 7 to 19 . . .
Message: 8937 - Contents - Hide Contents Date: Sun, 04 Jan 2004 01:01:45 Subject: Re: name? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> Oh so you're thinking about 13-equal. >>>>>> Yup. And I think only multiples of 13 do the job? >>> >>> -Carl >>>> don't get your question . . . >> Which ETs temper this comma out? > > -CarlYes, as long as 4/13 oct. remains your operative (by being 'best' or whatever rule you base your mapping on) approximation of 5:4.
Message: 8938 - Contents - Hide Contents Date: Sun, 04 Jan 2004 02:39:16 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> No one commented on the graphs I posted around Christmas, but I'll >>> keep going, if only for myself . . . >> Gene and you had an exchange. Gene suggested what I was thinking, > you said that each band represented only a denominator, not a comma. > Or are you talking about something else? >>> No need to specify a consonance limit? -- wow that's hot. >> How do you mean? Isn't this implicit in the dimensionality of the > Tenney lattice you're using? > > It would indeed be hot. > > -CarlLet's try to get a better grasp of what happens in this particular meantone, for a start. I could also do this for 7-limit intervals, treating the 81/80 temperament as a 'planar' temperament, but hopefully it's clear that the extra intervals won't have enough error to exceed the bound. Same for any higher limit. Interval...Approx....|Error|....Comp=log2(n*d)...|Error|/Comp 2:1........1201.70....1.70...........1..............1.70 3:1........1899.26....2.69..........1.58............1.70 4:1........2403.40....3.40...........2..............1.70 5:1........2790.26....3.94..........2.32............1.70 3:2.........697.56....4.39..........2.58............1.70 6:1........3100.96....0.99..........2.58............0.38 8:1........3605.10....5.10...........3..............1.70 9:1........3798.53....5.38..........3.17............1.70 10:1.......3991.96....5.64..........3.32............1.70 4:3.........504.13....6.09..........3.58............1.70 12:1.......4302.66....0.70..........3.58............0.20 5:3.........890.99....6.64..........3.91............1.70 15:1.......4689.52....1.25..........3.91............0.32 16:1.......4806.79....6.79...........4..............1.70 9:2........2596.83....7.08..........4.17............1.70 18:1.......5000.22....3.69..........4.17............0.88 5:4.........386.86....0.55..........4.32............0.13 20:1.......5193.65....7.34..........4.32............1.70 8:3........1705.83....7.79..........4.58............1.70 24:1.......5504.36....2.40..........4.58............0.52 25:1.......5580.52....7.89..........4.64............1.70 6:5.........310.70....4.94..........4.91............1.01 10:3.......2092.69....8.33..........4.91............1.70 30:1.......5891.22....2.95..........4.91............0.60 32:1.......6008.49....8.49...........5..............1.70 36:1.......6201.92....1.99..........5.17............0.38 8:5.........814.84....1.15..........5.32............0.22 40:1.......6395.35....9.04..........5.32............1.70 9:5........1008.27....9.33..........5.49............1.70 45:1.......6588.78....1.44..........5.49............0.26 16:3.......2907.53....9.49..........5.58............1.70 48:1.......6706.06....4.10..........5.58............0.73 25:2.......4378.82....6.19..........5.64............1.10 50:1.......6782.21....9.59..........5.64............1.70 27:2.......4496.09....9.77..........5.75............1.70 54:1.......6899.49....6.38..........5.75............1.11 12:5.......1512.40....3.24..........5.91............0.55 15:4.......2286.12....2.15..........5.91............0.36 20:3.......3294.39...10.03..........5.91............1.70 60:1.......7092.92....4.65..........5.91............0.79 1296:5..... and so on. Thinking about a few of these example spacially should help you see that the weighted error can never exceed cents(81/80)/log2(81*80) = 1.70 for ANY interval. Is there a just (RI) interval in this meantone? The idea of duality leads me to guess 81*80:1 = 6480:1 . . . 6480:1....15194.10....0.03 almost, but no cigar.
Message: 8939 - Contents - Hide Contents Date: Sun, 04 Jan 2004 02:45:22 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> Gene and you had an exchange. Gene suggested what I was >>>>> thinking, you said that each band represented only a >>>>> denominator, not a comma. >>>>> Or are you talking about something else? >>>>>>>> Yes, a later pair of graphs. >>>>>> D'oh! Link? (I even went to tuning-math to find it myself, but >>> this thread is not connected to anything). >> >> Yahoo groups: /tuning-math/files/Paul/com5monz... * [with cont.] >> >> and >> >> Yahoo groups: /tuning-math/files/Paul/com5rat.gif * [with cont.] >> These are the graphs I thought Gene replied about -- do they look > very similar to those? Pretty similar.>>>>>> No need to specify a consonance limit? -- wow that's hot. >>>>>>>>>> How do you mean? Isn't this implicit in the dimensionality of >>>>> the Tenney lattice you're using? >>>>>>>> No, no consonance limit (aka odd-limit) is implicity in the >>>> dimensionality of the Tenney lattice you're using -- and even the >>>> latter, aka prime-limit, doesn't need to be specified -- for >>>> example the results for the pythagorean comma will be valid in >>>> both lower and higher prime limits. >>>>>> I must not be tracking you -- the pythagorean comma is of course >>> a 3-limit comma. >>> I don't see anything new in any of this, I'm afraid. > > Let's back up. We're talking about the badness of commas? >Typically > that's been a function of their size in centsNo, not really. Rather, it's the error they induce when tempered out. I seem to have figured out a simple formula to get this when the tempering is Tenney-weighted-minimax optimal over ALL intervals. And a simple way to do such tempering.> and the number of notes > you can be expected to search to find them (which depends on their > distance on the lattice and the dimensionality of the lattice).I'm just looking at distance on the lattice, and dimensionality is irrelevant -- it can be assumed to be infinite.
Message: 8940 - Contents - Hide Contents Date: Sun, 04 Jan 2004 04:31:17 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: Thinking about a few of these example spacially should> help you see that the weighted error can never exceed > > cents(81/80)/log2(81*80) = 1.70 > > for ANY interval.The light dawns. Neat!
Message: 8941 - Contents - Hide Contents Date: Sun, 04 Jan 2004 01:18:00 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>No, not really. Rather, it's the error they induce when tempered out. >I seem to have figured out a simple formula to get this when the >tempering is Tenney-weighted-minimax optimal over ALL intervals. And >a simple way to do such tempering.What is that formula a way? Sorry, can we start at the top? It seems like your earlier message was either written in a white heat or addressed to someone who knows things I don't. -Carl
Message: 8942 - Contents - Hide Contents Date: Sun, 04 Jan 2004 12:28:04 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Graham Breed Paul Erlich wrote:> No, not really. Rather, it's the error they induce when tempered out. > I seem to have figured out a simple formula to get this when the > tempering is Tenney-weighted-minimax optimal over ALL intervals. And > a simple way to do such tempering.The odd-limit rule simplifies to interval size / complexity where "complexity" is the smallest number of intervals in the relevant limit that make up the comma. The result is the optimum minimax error by tempering out only this comma. So any temperament involving this comma must be at least this well tuned. What you've done is plugged in the Tenney complexity, and got a result that'll have something to do with the weighted minimax. That's not too surprising, and should generalize to any weighted complexity measure. At least if it gives a result in terms of octaves. I thought this was all assumed by your hypothesis anyway. We know that the Tenney complexity and odd limits are linked, depending on whether or not you enforce octave equialence. As geometric complexity looks like being an octave-specific weighted complexity measure, this may be the way to progress. The problem remains knowing how best to combine these commas to get a temperament of a specific dimension. For that we need a straightness measure, as always. And if you're not enforcing a prime limit to start with, you'll need to take a variable number of commas. Oh, BTW, I've been looking at commas of the form (n*n):(n+1)(n-1). Do they have a name? They always come from setting two n-integer limit intervals to be equivalent. They're also more likely than arbitrary superparticulars to belong to a low prime limit.. The 11-prime limit subset gives us 4:3, 9:8, 16:15, 25:24, 36:35, 49:48, 64:63, 81:80, 100:99, 225:224, 441:440, 2401:2400 Graham ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 8943 - Contents - Hide Contents Date: Mon, 05 Jan 2004 10:03:42 Subject: Re: Squarejacks From: Graham Breed Gene Ward Smith wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>> Here is a list n < 5000 such that n^2/(n^2-1) is within the 23 limit, >> together with the prime limit.That's more like it! I can't find any more with n<100000, so it might be complete. Graham
Message: 8944 - Contents - Hide Contents Date: Mon, 05 Jan 2004 10:54:40 Subject: Re: Squarejacks From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Gene Ward Smith wrote: > That's more like it! I can't find any more with n<100000, so it might > be complete.Speaking of complete, now that Catalan's conjecture (now Mihailescu's theorem) has finally been proven we know that 9/8 is the only superparticular ratio where the numerator and denominator are both powers. The proof uses a lot of deep algebraic number theory. These sorts of things are tough to prove!
Message: 8945 - Contents - Hide Contents Date: Mon, 05 Jan 2004 20:59:18 Subject: More on Jacks From: Gene Ward Smith Calling my former A and B "upjack" and "downjack" I give some types beyond square and triangular. The two upjack types are in the Sloane list. A045944 n*(3*n+2) rhombic matchstick numbers upjack(2/3) A033954 n*(4*n+3) second decagonal numbers upjack(3/4) (2*n+3)*(3*n+2) downjack(2/3) (3*n+5)*(4*n+3) downjack(3/4) Here is some Maple code: farey:=proc(n) # nth row of farey sequence local p,q,s,t; s:=[]; for q from 1 to n do for p from 1 to q do if igcd(p,q)=1 then s:=[p/q,op(s)] fi od od; RETURN(sort(s)) end: nex := proc(q, n) # next in row n of farey sequence from q local r,s; s := n - modp(n+1/numer(q), denom(q)); r := modp(1/denom(q), s); r/s end: pre := proc(q, n) # previous in row n of farey sequence from q local r,s; s := n - modp(n-1/numer(q), denom(q)); r := modp(-1/denom(q), s); r/s end: med := proc(p, q) # mediant (numer(p)+numer(q))/(denom(p)+denom(q)) end: comma := proc(q) # comma induced by q local p,d,w; p := q; if q>1 then p := 1/q fi; d := denom(p); w := pre(p,2*d)*nex(p,2*d)/p^2; if w<1 then w := 1/w fi; w end: comr := proc(q,n) # recursive comma induction local p,d,w; p := q; if q>1 then p := 1/q fi; d := denom(p); if n=1 then w := comma(q) fi; if n>1 then w := comr(pre(p,2*d), n-1)/comma(nex(p,2*d), n-1) fi; if w<1 then w := 1/w fi; w end: downjack := proc(r, n) # B(r, n) in tuning-math message 849 local u, v, a, b, s; u := numer(r); v := denom(r); s := pre(r, v); a := numer(s); b := denom(s); ((n*v-v+b)*(n*u+a))/((n*u-u+a)*(n*v+b)) end: upjack := proc(r, n) # A(r, n) in tuning-math message 849 local u, v, a, b, s; u := numer(r); v := denom(r); s := nex(r, v); a := numer(s); b := denom(s); ((n*u-u-a)*(n*v-b))/((n*v-v-b)*(n*u-a)) end:
Message: 8946 - Contents - Hide Contents Date: Mon, 05 Jan 2004 21:25:32 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> No, not really. Rather, it's the error they induce when tempered out. >> I seem to have figured out a simple formula to get this when the >> tempering is Tenney-weighted-minimax optimal over ALL intervals. And >> a simple way to do such tempering. >> What is that formula a way? Sorry, can we start at the top?For the comma p/q, p>q, the number of cents you need to temper out is cents(p/q) = log2(p/q)*1200. The distance in the Tenney lattice (taxicab, by definition) is log2 (p*q). So the tempering per unit length in the direction of the comma is cents(p/q)/log2(p*q). This was (aside from a factor of 1200) the vertical axis in my graphs. Now, for all primes r, If p contains any factors of r, the r-rungs in the lattice (which have length log2(r)) are shrunk from cents(r) to cents(r) - log2(r)*cents(p/q)/log2(p*q). If q contains any factors of 2, they are instead stretched to cents(r) + log2(r)*cents(p/q)/log2(p*q).
Message: 8947 - Contents - Hide Contents Date: Mon, 05 Jan 2004 21:33:47 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Paul Erlich wrote: >>> No, not really. Rather, it's the error they induce when tempered out. >> I seem to have figured out a simple formula to get this when the >> tempering is Tenney-weighted-minimax optimal over ALL intervals. And >> a simple way to do such tempering. >> The odd-limit rule simplifies to > > interval size / complexity > > where "complexity" is the smallest number of intervals in the relevant > limit that make up the comma. The result is the optimum minimax error > by tempering out only this comma. So any temperament involving this > comma must be at least this well tuned. > > What you've done is plugged in the Tenney complexity, and got a result > that'll have something to do with the weighted minimax. That's not too > surprising, and should generalize to any weighted complexity measure. > At least if it gives a result in terms of octaves. > > I thought this was all assumed by your hypothesis anyway.I don't see the relationship.> We know that > the Tenney complexity and odd limits are linked, depending on whether or > not you enforce octave equialence.Yes, but for octave equivalence (pegged to 1200 cent octaves), I'd like to eventually be able to use Kees's expressibility measure instead of Tenney harmonic distance. Just as there was no finitistic 'limit' assumed for my 'optimization' in the Tenney lattice, no odd limit will have to be specified in the octave- equivalent case (if it can work).> As geometric complexity looks like > being an octave-specific weighted complexity measure, this may be the > way to progress.What do you mean?> The problem remains knowing how best to combine these commas to get a > temperament of a specific dimension. For that we need a straightness > measure, as always.That's why I was asking about heron's formula, etc. But if we have some way of acheiving this Tenney-weighted minimax for the relevant temperaments, we may be able to skip this step.
Message: 8948 - Contents - Hide Contents Date: Mon, 05 Jan 2004 14:45:25 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>For the comma p/q, p>q, the number of cents you need to temper out is >cents(p/q) = log2(p/q)*1200. > >The distance in the Tenney lattice (taxicab, by definition) is log2 >(p*q). > >So the tempering per unit length in the direction of the comma is >cents(p/q)/log2(p*q). This was (aside from a factor of 1200) the >vertical axis in my graphs. > >Now, for all primes r, > >If p contains any factors of r, the r-rungs in the lattice (which >have length log2(r)) are shrunk from >cents(r) >to >cents(r) - log2(r)*cents(p/q)/log2(p*q). >If q contains any factors of 2, they are instead stretched to >cents(r) + log2(r)*cents(p/q)/log2(p*q).Thanks. I understand this 100%. But I don't understand what's new. Perhaps it has something to do with using this to get optimum generators for a linear temperament? And I don't understand your 'limitless' claim -- since p/q contains the factors it does and no others, one wouldn't expect its vanishing to effect intervals different factors. -Carl
Message: 8949 - Contents - Hide Contents Date: Mon, 05 Jan 2004 23:35:07 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> For the comma p/q, p>q, the number of cents you need to temper out is >> cents(p/q) = log2(p/q)*1200. >> >> The distance in the Tenney lattice (taxicab, by definition) is log2 >> (p*q). >> >> So the tempering per unit length in the direction of the comma is >> cents(p/q)/log2(p*q). This was (aside from a factor of 1200) the >> vertical axis in my graphs. >> >> Now, for all primes r, >> >> If p contains any factors of r, the r-rungs in the lattice (which >> have length log2(r)) are shrunk from >> cents(r) >> to >> cents(r) - log2(r)*cents(p/q)/log2(p*q). >> If q contains any factors of 2, they are instead stretched to >> cents(r) + log2(r)*cents(p/q)/log2(p*q). >> Thanks. I understand this 100%. But I don't understand what's > new.Where have you seen this before?> Perhaps it has something to do with using this to get > optimum generators for a linear temperament?Well, that's exactly what this does (when the dimensionality is right), as I've illustrated already in a few cases. Here's something new -- Top meantone is, it seems, exactly 1/4-comma meantone (I get 0.24999999999997, but that's probably just rounding error) except a uniform (in cents, or log Hz) stretch of 1.00141543374547 is applied to all intervals . . .> And I don't understand your 'limitless' claim -- since p/q contains > the factors it does and no others, one wouldn't expect its vanishing > to effect affect? > intervals different factors.intervals with different factors? Well, 5:4 and 5:3 have *some* factors differing from those in the Pythagorean comma, yet both intervals are affected by its vanishing, in this scheme.
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