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Message: 8875 - Contents - Hide Contents Date: Sat, 27 Dec 2003 03:03:25 Subject: Re: 5-limit, 12-note Fokker blocks From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> (The scales can be obtained by letting i range from 1 to > 12.)I forgot to put in that you must round the exponents to integers; if you don't you'll merely get 12-equal.> (16/15)^i (2048/2025)^round(-5i/12) (81/80)^round(-2i/12) > No Scala name > > > (16/15)^i (81/80)^rounnd(8i/12) (648/625)^round(-5i/12) > No Scala name > > > (16/15)^i (128/125)^round(-8i/12) (648/625)^round(3i/12) > No Scala name, a mode of what I've called the thirds scale > > > (16/15)^i (2048/2025)^round(-3i/12) (128/125)^round(-2i/12) > Scala identifies it as lumma5r.scl > > ! lumma5r.scl > ! > Carl Lumma's scale, 5-limit just version, TL 19-2- > 99 ________________________________________________________________________ ________________________________________________________________________To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx ------------------------------------------------------------------------ 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: 8876 - Contents - Hide Contents Date: Sun, 28 Dec 2003 01:53:47 Subject: Re: 5-limit, 12-note Fokker blocks From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Scala has many others, including Marpurg's Monochord #1, Ramos/Ramis, > etc . . . see the Gentle Introduction to Fokker Periodicity Blocks."Many" is a little vague, though I see tamil_vi.scl is a mode of ramis.scl. Can you tell me what two commas give Marpurg #1 and Ramis? Do you know of any other specific examples?
Message: 8877 - Contents - Hide Contents Date: Sun, 28 Dec 2003 03:48:41 Subject: Re: 5-limit, 12-note Fokker blocks From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> "Many" is a little vague, though I see tamil_vi.scl is a mode of > ramis.scl. Can you tell me what two commas give Marpurg #1 and Ramis? > Do you know of any other specific examples?I'm going to answer part of my own question. Determination of whether or not something was a Fokker block might be considered as an addition to Scala. Consider the intervals of ramis.scl. These go 135/126, 256/243, 16/15, 135/128, 16/15, 135/128, 16/15, 256/243, 135/128, 16/15, 135/128, 16/15 There are only three sizes (in the 5-limit, there can be as many as four) and none is a singleton, which we should avoid using as a basis. Pick 16/15 as the "basis", and take ratios with the other two step sizes to find the commas: (16/15)/(135/128) = 2048/2025, and (16/15)/(256/243) = 81/80. Now invert the matrix whose rows are the monzos for 16/15, 81/80, 2048/2025 and get the matrix whose columns are the standard vals h12, -h2, -h5. If we take the triples [h12(q), h2(q), h5(q)] for q in the Ramis scale, we get 1 [0, 0, 0] 135/128 [1, 0, 1] 10/9 [2, 1, 1] 32/27 [3, 1, 1] 5/4 [4, 1, 2] 4/3 [5, 1, 2] 45/32 [6, 1, 3] 3/2 [7, 1, 3] 128/81 [8, 2, 3] 5/3 [9, 2, 4] 16/9 [10, 2, 4] 15/8 [11, 2, 5] 2 [12, 2, 5] The values for h2 and h5 are nondecreasing, and for h12 they go up one at a time (the scale is h12-epimorphic.) This tells us we have a Fokker block. Doing the same kind of analysis for Marpurg Monochord 1, we get 16/15, 128/125 and 81/80, and h12, h5, and h3 as the vals to check it with. This time we get 1 [0, 0, 0] 25/24 [1, 1, 0] 9/8 [2, 1, 1] 6/5 [3, 1, 1] 5/4 [4, 2, 1] 4/3 [5, 2, 1] 45/32 [6, 3, 2] 3/2 [7, 3, 2] 25/16 [8, 4, 2] 5/3 [9, 4, 2] 9/5 [10, 4, 3] 15/8 [11, 5, 3] 2 [12, 5, 3] Once again, we have the correct monotonic and epimorphic properties, and Marpurg Monochord 1 is a Fokker block. ________________________________________________________________________ ________________________________________________________________________ To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx ------------------------------------------------------------------------ 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: 8878 - Contents - Hide Contents Date: Mon, 29 Dec 2003 08:26:01 Subject: Re: 5-limit, 12-note Fokker blocks (attn Manuel) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Consider the intervals of ramis.scl. These go > > 135/126, 256/243, 16/15, 135/128, 16/15, 135/128, 16/15, 256/243, > 135/128, 16/15, 135/128, 16/15 > > There are only three sizes (in the 5-limit, there can be as many as > four) and none is a singleton, which we should avoid using as a basis. > Pick 16/15 as the "basis", and take ratios with the other two step > sizes to find the commas: (16/15)/(135/128) = 2048/2025, and > (16/15)/(256/243) = 81/80. Now invert the matrix whose rows are the > monzos for 16/15, 81/80, 2048/2025 and get the matrix whose columns > are the standard vals h12, -h2, -h5. If we take the triples > [h12(q), h2(q), h5(q)] for q in the Ramis scale, we get > > 1 [0, 0, 0] > 135/128 [1, 0, 1] > 10/9 [2, 1, 1] > 32/27 [3, 1, 1] > 5/4 [4, 1, 2] > 4/3 [5, 1, 2] > 45/32 [6, 1, 3] > 3/2 [7, 1, 3] > 128/81 [8, 2, 3] > 5/3 [9, 2, 4] > 16/9 [10, 2, 4] > 15/8 [11, 2, 5] > 2 [12, 2, 5]If now we take [i-h12(q), i/6-h2(q), 5i/12-h5(q)] we get 0 [0, 0, 0] 1 [0, 1/6, -7/12] 2 [0, -2/3, -1/6] 3 [0, -1/2, 1/4] 4 [0, -1/3, -1/3] 5 [0, -1/6, 1/12] 6 [0, 0, -1/2] 7 [0, 1/6, -1/12] 8 [0, -2/3, 1/3] 9 [0, -1/2, -1/4] 10 [0, -1/3, 1/6] 11 [0, -1/6, -5/12] The midpoint of values of the second column is (-2/3 +1/6)/2 = -1/4, and the midpoint of the third column is (-7/12 + 1/3)/2 = -1/8. Hence we can express the exponents in terms of round(-5i/12 - 1/8) and round(-i/6 - 1/4), and get ramis[i] = (16/15)^i * (2048/2025)^round(-5i/12-1/8) * (81/80)^round(-i/6-/14) We can automate this process with the following steps 1. Find the set of intervals between steps. If there are more of these than there are primes involved in the factorization of the scale degrees, return false. 2. Check if the scale is epimorphic. If not, return false. 3. Pick a "base" element from the scale steps, by any means (first step, smallest step, most common step, least Tenney height, etc.) Now take ratios of the basis element with the other scale steps, finding (n-1) other ratios (where n is the number of primes involved) which together with the basis element gives a matrix of row vectors whose determinant is +-1 (unimodular matrix.) If this is impossible, return false. 4. Invert the above unimodular matrix, and obtain vals from the columns of the inverted matrix. 5. For each of these vals aside from the first, take the maximum and minimum values of v(2)*i/N, where N is the number of scale steps and i goes from 0 to N-1, and find the average of the maximum and minimum (the midpoint value.) 6. Using the midpoint value as an offset, calculate a scale by the above method. If it gives the correct result, return true and the data defining the means of computing the scale (v(2) for the various vals, corresponding commas, and offsets. Just giving the formula would be good.) Can anyone improve on this?
Message: 8881 - Contents - Hide Contents Date: Tue, 30 Dec 2003 17:25:17 Subject: Re: Meantone reduced blocks From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> The >> meatone reduction therefore is >> >> 7i - 12(round(i/3) + round(i/4+1/8)) >> Please express this meantone scale in conventional letter-name-and- > accidental notation.Here is the scale in terms of meantone fifths: [0, 7, -10, -3, 4, -1, -6, 1, -4, 3, -2, -7] Using an "f" for my flat symbol, here it is in sharps and flats: [C, C#, Eff, Ef, E, F, Gf, G, Af, A, Bf, Cf] I'm not sure yet how uncommon this sort of thing is, but probably not very common. There is only one possible {128/125, 648/625} scale up to transposition, and one question is what other comma pairs give us something differing from Meantone[12] on reduction.
Message: 8883 - Contents - Hide Contents Date: Tue, 30 Dec 2003 21:55:14 Subject: The Six Syndia Scales From: Gene Ward Smith These are the six possible Fokker blocks, up to transpositional equivalence, which can be obtained from the 81/80 (SYNtonic) and 2048/2045 (DIAschismic) commas. Since midpoints between numbers of the form i/12 are numbers of the form n/24, I took all offsets n/24 for n ranging from -12 to 12 to obtain these, though in fact taking only odd n should suffice. The first two are self-dual, or whatever the word is (and if there isn't one, there should be) for a scale transpositionally equivalent to its inverse. Then 3 and 4, 5 and 6 are inversionally related pairs. When a name already existed in the Scala archives, I used that form of the scale, otherwise I just picked one of the 12 which looked nice. All of these on reduction by meantone lead to Meantone[12]. ! syndia1.scl First 81/80 2048/2025 Fokker block = ramis.scl 12 ! 135/128 10/9 32/27 5/4 4/3 45/32 3/2 128/81 5/3 16/9 15/8 2 ! syndia2.scl Second 81/80 2048/2025 Fokker block 12 ! 16/15 256/225 6/5 32/25 4/3 64/45 3/2 8/5 128/75 9/5 256/135 2 ! syndia3.scl Third 81/80 2048/2025 Fokker block 12 ! 135/128 9/8 1215/1024 5/4 675/512 45/32 3/2 405/256 27/16 225/128 15/8 2 ! syndia4.scl Fourth 81/80 2048/2025 Fokker block 12 ! 135/128 9/8 6/5 5/4 4/3 45/32 3/2 8/5 27/16 16/9 15/8 2 ! syndia5.scl Fifth 81/80 2048/2025 Fokker block = pipedum_12.scl 12 ! 135/128 9/8 75/64 5/4 4/3 45/32 3/2 405/256 5/3 16/9 15/8 2 ! syndia6.scl Sixth 81/80 2048/2025 Fokker block 12 ! 135/128 9/8 6/5 5/4 4/3 45/32 3/2 405/256 27/16 16/9 15/8 2
Message: 8884 - Contents - Hide Contents Date: Tue, 30 Dec 2003 02:08:04 Subject: Meantone reduced blocks From: Gene Ward Smith It would be nice to classify 12-note, 5-limit Fokker blocks at least up to meantone reduction. While pondering that, I thought I'd see how an example which does not reduce to Meantone[12] worked out. The "thirds" scale, the genus derived from 6/5 and 5/4, can be analyzed as a Fokker block using the method I've given as thirds[i] = (25/24)^i (128/125)^round(i/3) (648/625)^round(i/4+1/8) In meantone, 25/24 maps to 7, and 128/125 and 648/625 to -12. The meatone reduction therefore is 7i - 12(round(i/3) + round(i/4+1/8)) I don't see a better way offhand of writing this function. Graphing it gives a result which looks somewhat erratic.
Message: 8885 - Contents - Hide Contents Date: Tue, 30 Dec 2003 02:09:26 Subject: Re: 5-limit, 12-note Fokker blocks (attn Manuel) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> We can automate this process with the following steps > > 1. Find the set of intervals between steps. If there are more of these > than there are primes involved in the factorization of the scale > degrees, return false.Sorry, I forgot my own analysis. Skip this step.> 2. Check if the scale is epimorphic. If not, return false. > > 3. Pick a "base" element from the scale steps, by any means (first > step, smallest step, most common step, least Tenney height, etc.) Now > take ratios of the basis element with the other scale steps, finding > (n-1) other ratios (where n is the number of primes involved) which > together with the basis element gives a matrix of row vectors whose > determinant is +-1 (unimodular matrix.) If this is impossible, return > false. > > 4. Invert the above unimodular matrix, and obtain vals from the > columns of the inverted matrix. > > 5. For each of these vals aside from the first, take the maximum and > minimum values of v(2)*i/N, where N is the number of scale steps and i > goes from 0 to N-1, and find the average of the maximum and minimum > (the midpoint value.) > > 6. Using the midpoint value as an offset, calculate a scale by the > above method. If it gives the correct result, return true and the data > defining the means of computing the scale (v(2) for the various vals, > corresponding commas, and offsets. Just giving the formula would be good.) > > Can anyone improve on this?
Message: 8886 - Contents - Hide Contents Date: Tue, 30 Dec 2003 08:56:52 Subject: Re: Meantone reduced blocks From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> It would be nice to classify 12-note, 5-limit Fokker blocks at least > up to meantone reduction. While pondering that, I thought I'd see how > an example which does not reduce to Meantone[12] worked out. > > The "thirds" scale, the genus derived from 6/5 and 5/4, can be > analyzed as a Fokker block using the method I've given as > > thirds[i] = (25/24)^i (128/125)^round(i/3) (648/625)^round(i/4+1/8) > > In meantone, 25/24 maps to 7, and 128/125 and 648/625 to -12. > > The > meatone reduction therefore is > > 7i - 12(round(i/3) + round(i/4+1/8))Please express this meantone scale in conventional letter-name-and- accidental notation.
Message: 8887 - Contents - Hide Contents Date: Tue, 30 Dec 2003 16:54:18 Subject: Re: 5-limit, 12-note Fokker blocks (attn Manuel) From: Manuel Op de Coul I've paid attention. It will be a useful addition, so I've put it on the todo list. Manuel ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 8888 - Contents - Hide Contents Date: Wed, 31 Dec 2003 00:44:18 Subject: The Four Syndie Scaless From: Gene Ward Smith The name this time comes from SYNtonic and DIEesis, or 81/80 and 128/125. These, unsurprisingly, turn out to be already known; what is new is that there are only four of them. This time I did things systematically and made the meantone reduction run from -3 to 8. The first two are inversions of each other, and 3 and 4 are self-dual or inversionally similar. ! syndie1.scl First Syndie scale ~ sauveur_ji.scl 12 ! 135/128 9/8 6/5 5/4 27/20 45/32 3/2 25/16 27/16 9/5 15/8 2 ! syndie2.scl Second Syndie scale = fogliano1.scl 12 ! 25/24 10/9 6/5 5/4 4/3 25/18 3/2 25/16 5/3 16/9 15/8 2 ! syndie3.scl Third Syndie scale ~ duodene.scl = efg33355.scl 12 ! 25/24 10/9 32/27 5/4 4/3 25/18 40/27 25/16 5/3 16/9 50/27 2 ! syndie.scl Fourth Syndie scale = marpurg1.scl 12 ! 25/24 9/8 6/5 5/4 4/3 45/32 3/2 25/16 5/3 9/5 15/8 2
Message: 8889 - Contents - Hide Contents Date: Wed, 31 Dec 2003 22:52:19 Subject: Re: The Two Diadie Scales From: Carl Lumma>The two scales using the DIAschisma and the DIEsis of 128/125 are >both known, and this seems like a Carl Lumma speciality. They don't >reduce to Meantone[12], but 22-et, pajara or orwell seem more to the >point. Reduction by 22-et or pajara leads to Pajara[12], but >reduction by orwell leads to two interesting new scales. Or at least >one is new, reducing the first diadie scale gives us something quite >close to lumma.scl, which Carl presented back in 1999.I did a non-thorough by-hand search for 12-tone 5- and 7-limit 'Fokker blocks' (before I knew the term, and before the subject had been explored by the list -- I certainly wasn't checking for epimorphism or monotonicity). Some of this was done before I joined the list, on paper with the rectangular lattices I'd learned about from Doty's JI Primer. -Carl ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 8890 - Contents - Hide Contents Date: Wed, 31 Dec 2003 02:15:38 Subject: The Three Majsyn Scales From: Gene Ward Smith These are the three possible Fokker blocks arising from the MAJor diesis of 648 and the SYNtonic comma of 81/80. The first two are inversional pairs, the third is self-dual. ! majsyn1.scl First Majsyn 648/625 81/80 scale 12 ! 250/243 10/9 6/5 100/81 4/3 25/18 40/27 125/81 5/3 16/9 50/27 2 ! majsyn2.scl Second Majsyn 648/625 81/80 scale 12 ! 25/24 10/9 6/5 5/4 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2 ! majsyn3.scl Third Majsyn 648/625 81/80 scale ~ duodene_skew.scl 12 ! 25/24 9/8 6/5 5/4 27/20 25/18 3/2 25/16 5/3 9/5 15/8 2
Message: 8891 - Contents - Hide Contents Date: Wed, 31 Dec 2003 08:31:08 Subject: The Twelve Ragisyn Scales From: Gene Ward Smith The interval 6561/6250 is shy of a minor semitone of 21/20 by a ragisma, so we may consider it a RAGIismic minor semitone; together with the SYNtonic comma of 81/80 we have the name "ragisyn" for this type of Fokker block. Below I list all twelve of them; the first two are self-dual, the rest are inversively related in successive pairs. As before, they all reduce to -3 to 8 under the meantone val. ! ragisyn1.scl Ragasyn 6561/6250 81/80 scale 12 ! 250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2 ! ragisyn2.scl Ragisyn 6561/6250 81/80 scale 12 ! 250/243 10/9 6/5 100/81 4/3 1000/729 40/27 10000/6561 400/243 16/9 4000/2187 2 ! ragisyn3.scl Ragasyn 6561/6250 81/80 scale 12 ! 25/24 9/8 243/200 5/4 27/20 25/18 3/2 125/81 5/3 9/5 50/27 2 ! ragisyn4.scl Ragisyn 6561/6250 81/80 scale 12 ! 250/243 10/9 6/5 100/81 4/3 1000/729 40/27 10000/6561 400/243 16/9 50/27 2 ! ragisyn5.scl Ragisyn 6561/6250 81/80 scale 12 ! 250/243 10/9 6/5 100/81 4/3 1000/729 40/27 125/81 5/3 9/5 50/27 2 ! ragisyn6.scl Ragisyn 6561/6250 81/80 scale 12 ! 250/243 10/9 6/5 5/4 27/20 25/18 3/2 125/81 5/3 9/5 50/27 2 ! ragisyn7.scl Ragisyn 6561/6250 81/80 scale 12 ! 250/243 10/9 6/5 100/81 4/3 1000/729 40/27 10000/6561 400/243 9/5 50/27 2 ! ragisyn8.scl Ragasyn 6561/6250 81/80 scale 12 ! 250/243 9/8 243/200 5/4 27/20 25/18 3/2 125/81 5/3 9/5 50/27 2 ! ragisyn9.scl Ragisyn 6561/6250 81/80 scale 12 ! 250/243 10/9 6/5 100/81 4/3 1000/729 3/2 125/81 5/3 9/5 50/27 2 ! ragisyn10.scl Ragisyn 6561/6250 81/80 scale 12 ! 250/243 10/9 6/5 100/81 27/20 25/18 3/2 125/81 5/3 9/5 50/27 2 ! ragisyn11.scl Ragisyn 6561/6250 81/80 scale 12 ! 250/243 10/9 6/5 100/81 4/3 1000/729 40/27 10000/6561 5/3 9/5 50/27 2 ! ragisyn12.scl Ragisyn 6561/6250 81/80 scale 12 ! 250/243 10/9 243/200 5/4 27/20 25/18 3/2 125/81 5/3 9/5 50/27 2 ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 8892 - Contents - Hide Contents Date: Thu, 01 Jan 2004 05:26:56 Subject: The Two Diadie Scales From: Gene Ward Smith Let me start out by saying that 81/80 with either the schisma or the pythagorean commas (or those two taken together) give us the 12-note Pythagorean scale, and that this completes the classification for scales using 81/80, unless you want to go past 0.75 in epimericity. The two scales using the DIAschisma and the DIEsis of 128/125 are both known, and this seems like a Carl Lumma speciality. They don't reduce to Meantone[12], but 22-et, pajara or orwell seem more to the point. Reduction by 22-et or pajara leads to Pajara[12], but reduction by orwell leads to two interesting new scales. Or at least one is new, reducing the first diadie scale gives us something quite close to lumma.scl, which Carl presented back in 1999. ! diadie1.scl First Diadie 2048/2025 128/125 scale = lumma5r.scl 12 ! 16/15 9/8 75/64 5/4 4/3 45/32 3/2 8/5 5/3 225/128 15/8 2 ! diadie2.scl Second Diadie 2048/2025 128/125 scale ~ pipedum_12a.scl 12 ! 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 128/75 225/128 15/8 2 ! diadieorw1.scl 84-et version of diadie1.scl (similar to lumma.scl) 12 ! 114.285714 200.000000 271.428571 385.714286 500.000000 585.714286 700.000000 814.285714 885.714286 971.428571 1085.714286 1200.000000 ! diadieorw2.scl 84-et version of diadie2.scl 12 ! 114.285714 200.000000 314.285714 385.714286 500.000000 585.714286 700.000000 814.285714 928.571429 971.428571 1085.714286 1200.000000
Message: 8893 - Contents - Hide Contents Date: Thu, 01 Jan 2004 20:50:53 Subject: 225/224-planar equal temperaments From: Gene Ward Smith This is a kind of generic tuning to apply to large enough (and 12 seems to be large enough) 5-limit JI scales, since they always seem to have these relations. I ran a badness filter on the rms 225/224-planar temperaments, treating it just like a 5-limit JI computation but using the adjusted values for 3 and 5, and got the following list of temperaments, where the first column is the et (to 10000), the second the 225/224-planar val, the third column where the standard val maps 225/224 (if this is 0, the standard val and the 225/224 val are the same in the 7-limit) and the fourth the badness measure, set to be less than 1. From 72 and 175 we can conclude that 7-limit miracle is a good 225/224-planar tuning, but 156 and 228 give duodecimal (mapping [<12 19 28 34|, <0 0 -1 -2|]) with rms error 1.5 cents vs 1.64 for miracle. Beyond that, 403, 559 and 631 work; 559 gives us two vals, one an excellent 5-limit JI val <559 886 1298| and the other an excellent 225/224-planar val <559 885 1297|. Since there is nothing especially magical about the rms 225/224 tuning versus, eg, minimax the results beyond this point are pretty meaningless, I suspect. A kind of adaptive tuning, where the first 559 val was used for purely 5 limit chords and the second, with major thirds and fifths shifted down a schisma, for chords involving 7 would be possible. 1 [1, 2, 2] 0 .736966 2 [2, 3, 5] 0 .743974 3 [3, 5, 7] 1 .433823 4 [4, 6, 9] -1 .665419 5 [5, 8, 12] 1 .892828 7 [7, 11, 16] -1 .637629 12 [12, 19, 28] 0 .548311 15 [15, 24, 35] 1 .977303 19 [19, 30, 44] 0 .360557 31 [31, 49, 72] 0 .857806 53 [53, 84, 123] 0 .666832 72 [72, 114, 167] 0 .522038 84 [84, 133, 195] 0 .989711 103 [103, 163, 239] 0 .939217 156 [156, 247, 362] 0 .720534 175 [175, 277, 406] 0 .741466 228 [228, 361, 529] 0 .536746 403 [403, 638, 935] 4 .411588 559 [559, 885, 1297] 4 .946951 631 [631, 999, 1464] 4 .635276 1034 [1034, 1637, 2399] 7 .896596 1593 [1593, 2522, 3696] 11 .551130 1996 [1996, 3160, 4631] 15 .989239 2224 [2224, 3521, 5160] 14 .632155 2627 [2627, 4159, 6095] 18 .805402 3817 [3817, 6043, 8856] 25 .870642 4220 [4220, 6681, 9791] 29 .567603 6444 [6444, 10202, 14951] 41 .504831 8037 [8037, 12724, 18647] 50 .945557
Message: 8894 - Contents - Hide Contents Date: Thu, 01 Jan 2004 05:39:28 Subject: Re: The Two Diadie Scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: I made a triad circle (around the circle of fifths) for the two diadieorw scales, but neglected to include them. Here they are. Diadieorw1 triad circle -5 [385.714286, 314.285714, 700.0000000] -4 [385.714286, 271.4285714, 657.1428571] -3 [428.5714286, 271.4285714, 700.0000000] -2 [428.5714286, 300.0000000, 728.5714286] -1 [385.7142854, 314.2857143, 700.0000000] 0 [385.714286, 314.2857143, 700.0000000] 1 [385.714286, 314.2857143, 700.0000000] 2 [385.714286, 300.0000000, 685.7142857] 3 [428.5714286, 271.4285714, 700.0000000] 4 [428.5714286, 271.4285717, 700.0000000] 5 [385.714286, 314.2857143, 700.0000000] 6 [385.714286, 342.8571429, 728.5714286] I count five excellent 84-et major triads, one weird flat one, and three supermajor triads. Diadieorw2 triad circle -5 [385.714286, 314.285714, 700.000000] -4 [385.714286, 314.2857143, 700.0000000] -3 [385.714286, 271.4285714, 657.1428571] -2 [428.5714286, 300.0000000, 728.5714286] -1 [428.5714283, 271.4285714, 700.0000000] 0 [385.714286, 314.2857143, 700.0000000] 1 [385.714286, 314.2857143, 700.0000000] 2 [385.714286, 342.8571429, 728.5714286] 3 [385.714286, 271.4285714, 657.1428571] 4 [428.5714286, 271.4285717, 700.0000000] 5 [428.5714286, 271.4285714, 700.0000000] 6 [385.714286, 342.8571429, 728.5714286] Four 84-et major triads, three supermajor triads.
Message: 8895 - Contents - Hide Contents Date: Thu, 01 Jan 2004 20:54:55 Subject: Re: The Two Diadie Scales From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Let me start out by saying that 81/80 with either the schisma or the > pythagorean commas (or those two taken together) give us the 12- note > Pythagorean scale, and that this completes the classification for > scales using 81/80, unless you want to go past 0.75 in epimericity. > > The two scales using the DIAschisma and the DIEsis of 128/125 are > both known, and this seems like a Carl Lumma speciality. They don't > reduce to Meantone[12], but 22-et, pajara or orwell seem more to the > point. Reduction by 22-et or pajara leads to Pajara[12],Gene -- you keep saying Pajara but don't you mean Diaschismic?
Message: 8896 - Contents - Hide Contents Date: Thu, 01 Jan 2004 20:59:35 Subject: Re: The Two Diadie Scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> The two scales using the DIAschisma and the DIEsis of 128/125 are >> both known, and this seems like a Carl Lumma speciality. They don't >> reduce to Meantone[12], but 22-et, pajara or orwell seem more to > the>> point. Reduction by 22-et or pajara leads to Pajara[12], >> Gene -- you keep saying Pajara but don't you mean Diaschismic?I'm assuming that in 22-equal, it is more correctly called Pajara.
Message: 8897 - Contents - Hide Contents Date: Thu, 01 Jan 2004 21:04:02 Subject: The Four Tertiadie Scales From: Gene Ward Smith If we consider 262144/253125 to be a third of a tone or a tertiatone, these may be named TERTIAtone DIEesis scales. ! tertiadie1.scl First Tertiadie 262144/253125 128/125 scale 12 ! 16/15 256/225 75/64 5/4 4/3 64/45 375/256 25/16 5/3 2048/1125 15/8 2 ! tertiadie2.scl Second Tertiadie 262144/253125 128/125 scale 12 ! 16/15 1125/1024 75/64 5/4 4/3 5625/4096 3/2 8/5 128/75 225/128 15/8 2 ! tertiadie3.scl Third Tertiadie 262144/253125 128/125 scale 12 ! 16/15 1125/1024 75/64 5/4 4/3 45/32 3/2 8/5 128/75 225/128 15/8 2 ! tertiadie4.scl Fourth Tertiadie 262144/253125 128/125 scale 12 ! 2048/1875 32768/28125 6/5 32/25 512/375 8192/5625 3/2 8/5 128/75 2048/1125 15/8 2
Message: 8898 - Contents - Hide Contents Date: Thu, 01 Jan 2004 21:20:33 Subject: Re: The Two Diadie Scales From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>>> The two scales using the DIAschisma and the DIEsis of 128/125 are >>> both known, and this seems like a Carl Lumma speciality. They > don't>>> reduce to Meantone[12], but 22-et, pajara or orwell seem more to >> the>>> point. Reduction by 22-et or pajara leads to Pajara[12], >>>> Gene -- you keep saying Pajara but don't you mean Diaschismic? >> I'm assuming that in 22-equal, it is more correctly called Pajara.No, Pajara is simply the 7-limit extension of Diaschsimic that you do get in 22-equal (and pretty much in no other ET): GX Networks * [with cont.] (Wayb.) As long as you're talking 5-limit though, there's no reason to bring Pajara into it.
Message: 8899 - Contents - Hide Contents Date: Thu, 01 Jan 2004 21:56:21 Subject: Re: The Two Diadie Scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> No, Pajara is simply the 7-limit extension of Diaschsimic that you do > get in 22-equal (and pretty much in no other ET): > > GX Networks * [with cont.] (Wayb.) > > As long as you're talking 5-limit though, there's no reason to bring > Pajara into it.We are tuning Diaschismic so that the 7-limit works well, whether we want to acknowledge that fact or not. It's a Pajara tuning of Diaschismic, in other words, and so correctly called Pajara. Your method gives two precisely equal scales, one of which is called Diaschismic[12] and the other Pajara[12].
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