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Message: 9850 - Contents - Hide Contents Date: Fri, 06 Feb 2004 06:52:43 Subject: [tuning] Re: question about 24-tET From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Can we get generators for 5-limit meantone, 7-limit schismic, > and 11-limit miracle for each of: > > (1) TOP > (2) odd-limit TOP > (3) rms TOP (or can you only do integer-limit rms TOP?) > (4) rms odd-limit TOPI can do the TOP. What's the definition for the others? If I'm doing rms analogs of TOP, don't I need a list of intervals and maybe weights for them in order to cook up a Euclidean metric? I think Paul wanted something like that, and I could do it if I could remember exactly what it was.

Message: 9851 - Contents - Hide Contents Date: Fri, 06 Feb 2004 22:22:17 Subject: Re: Ennealimmal[45] as a chord block From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> The major chord with root 21/20 is [0,2,0] in the 7-limit chord > lattice, that with root 2401/2400 is [2,-3,-3], and that with > 4375/4374 is [6,-6,-3]. If I form the block in the lattice centered at > [0,0,0] and with the inverse matrix coordinates running -1 < coordinat > <= 1, I get a block with 127 chords, consisting of 191 notes.Arrgh, as Albert the Alligator once said. Blocks are supposed to run from -1/2 to 1/2. That gives me something more reasonable, 18 chords, leading to 48 notes, which reduces to 45 notes after tempering by 2401/2400 and 4375/4374. The 18 chords are [[-2, 3, 1], [1, 0, -1], [-1, 0, 1], [1, 1, -1], [-2, 4, 1], [2, -2, -1], [-1, 4, 0], [-1, 1, 1], [1, -3, 0], [0, 0, 0], [-2, 0, 2], [0, 1, 0], [1, -2, 0], [2, -3, -1], [-2, 1, 2], [2, 0, -2], [-1, 3, 0], [2, 1, -2]] The 48 notes as follows; as a scale this has two highly cheesy 2401/2400 steps, and one even cheesier 4375/4374, all of which ennealimmal exterminates. [1, 49/48, 25/24, 21/20, 15/14, 27/25, 54/49, 441/400, 9/8, 567/500, 125/108, 7/6, 25/21, 343/288, 175/144, 49/40, 5/4, 63/50, 9/7, 21/16, 1323/1000,27/20, 49/36, 25/18, 567/400, 343/240, 35/24, 72/49, 3/2, 49/32, 54/35, 63/40,100/63, 81/50, 175/108, 1323/800, 5/3, 245/144, 12/7, 7/4, 343/192, 9/5, 90/49, 50/27, 189/100, 27/14, 35/18, 125/63]

Message: 9852 - Contents - Hide Contents Date: Fri, 06 Feb 2004 07:18:21 Subject: Re: 126 7-limit linears From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: After all the complaints, no response. :( Here are some high-error temperaments:> 25 [2, 1, 6, -3, 4, 11] 2392.139586 9.396316 7.231437commas: {21/20, 25/24} mapping: [<1 1 2 3|, <0 2 1 -2|]> 28 [1, -3, -4, -7, -9, -1] 2669.323351 9.734056 7.425960commas: {21/20, 135/128} mapping: [<1 2 1 1|, <0 -1 3 4|]> 39 [2, 1, -4, -3, -12, -12] 3625.480387 9.146173 8.470366commas: {25/24, 64/63} "Number 56" mapping: [<1 1 2 4|, <0 2 1 -4|]> 62 [6, 0, 0, -14, -17, 0] 5045.450988 5.526647 11.410361commas: {50/49, 128/125} "Number 85" Has a 12 tone DE mapping: [<6 10 14 17|, <0 -1 0 0|]

Message: 9854 - Contents - Hide Contents Date: Fri, 06 Feb 2004 08:10:20 Subject: Re: 126 7-limit linears From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> 1 [0, 0, 2, 0, 3, 5] 662.236987 77.285947 2.153690 > 2 [1, 1, 0, -1, -3, -3] 806.955502 64.326132 2.467788 > 3 [0, 0, 3, 0, 5, 7] 829.171704 30.152577 3.266201 ...Thanks for the list. I can certainly get it into a spreadsheet and plot it easily, but I have no idea what I'm plotting. I assume the last two columns are error and complexity but I have no idea which is which. Also, I'm not yet up to speed on reading wedgies directly so I have no idea of the identity of the temperaments. Can we please have generators or mappings or comma pairs, if not names (where they exist)? I suppose you figure it was difficult to generate so it should be difficult to interpret as well. ;-)

Message: 9855 - Contents - Hide Contents Date: Fri, 06 Feb 2004 10:44:30 Subject: Re: 126 7-limit linears From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> 1 [0, 0, 2, 0, 3, 5] 662.236987 77.285947 2.153690 >> 2 [1, 1, 0, -1, -3, -3] 806.955502 64.326132 2.467788 >> 3 [0, 0, 3, 0, 5, 7] 829.171704 30.152577 3.266201 > ... >> Thanks for the list. I can certainly get it into a spreadsheet and > plot it easily, but I have no idea what I'm plotting. I assume the > last two columns are error and complexity but I have no idea which is > which.It's easy to tell which is which, but it's badness, error, complexity.

Message: 9856 - Contents - Hide Contents Date: Fri, 06 Feb 2004 23:04:52 Subject: Some warped egresses From: Herman Miller 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/ in Wayback Machine I've been playing with retunings of _This Way to the Egress_ in different 7-limit temperaments. One thing I'm wondering is if it would be better to use the same JI mapping for all the retunings, or try to find a different one for each temperament. Since the original version is 14-ET, it could be mapped to JI in different ways. Here's the one I've been using for the Superpelog and Miracle warpings (in MIDICONV tuning format): ! 14 notes of 7-limit JI coords 4 two 1.0 three 1.584962501 five 2.321928095 seven 2.807354922 notes 14 0 0 0 0 0 ! A# 5/4 1 -2 1 -1 1 ! B 2 0 -2 -1 2 ! B# 3 2 0 -2 1 ! C 7/5 4 1 1 -1 0 ! C# 3/2 5 3 -2 -1 1 ! D 6 1 -1 -2 2 ! D# 7 0 0 -1 1 ! E 7/4 8 2 -3 -1 2 ! E# 9 4 -1 -2 1 ! F 10 3 0 -1 0 ! F# 1/1 11 5 -3 -1 1 ! G 12 3 -2 -2 2 ! G# 13 2 -1 -1 1 ! A 7/6 14 1 0 0 0 ! A# 5/4 The basic harmony of the piece is an alternation between F# A# C# E and A C E F# chords, which I'm interpreting as 4:5:6:7 and 1/(7:6:5:4) chords in all the retunings. But it's not obvious what mapping to use for the other notes, and I might want to adjust them to keep the number of generators down to a reasonable number. So for the Orwell version, I changed the tuning of B and G: 1 4 -1 -1 0 ! B 11 7 -1 -2 0 ! G and the Pajara version changed the tuning of E# and F: 8 -2 1 -2 2 ! E# 9 1 -2 0 1 ! F On the other hand, it might make it harder to directly compare temperaments if they're mapped differently. But it seems like it'd be easier to find mappings that work with each temperament individually than to try to find one mapping that will work for everything. -- see my music page ---> ---<The Music Page * [with cont.] (Wayb.)>-- hmiller (Herman Miller) "If all Printers were determin'd not to print any @io.com email password: thing till they were sure it would offend no body, \ "Subject: teamouse" / there would be very little printed." -Ben Franklin

Message: 9857 - Contents - Hide Contents Date: Fri, 06 Feb 2004 12:18:16 Subject: Re: 126 7-limit linears From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> Also, I'm not yet up to speed on reading wedgies directly so I have no > idea of the identity of the temperaments.The first three numbers give the relevant information re the mapping, so this should not be a problem. The gcd is the period, and dividing by the period gives the mapping to primes of the "generator". There is no reason to think that [<1 2 4 7|, <0 -1 -4 -10|] is going to be any clearer or better than <<1 4 10 4 13 12|| that I can see; it's in the 13 limit and beyond that you end up using fewer numbers with the mapping. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9858 - Contents - Hide Contents Date: Sat, 07 Feb 2004 19:02:45 Subject: Val lattices From: Gene Ward Smith If we take the vals and put a norm on the vector space they live in, we get a lattice. A very useful one is what I've been calling the val lattice, with the L_inf norm the dual of the Tenney norm. However, many other norms are possible. One possibility is to take n independent vals, where n is the dimension of the space, and give them the same Euclidean length. In particular the vals in what I've called a "notation", which together form a unimodular matrix, can be give the same length. For example if I take 19,27,31, and 72, then the bilinear form 52*x2^2+8*x2*x3-14*x2*x5-30*x2*x7+58*x3^2-56*x3*x5- 22*x3*x7+33*x5^2-18*x5*x7+19*x7^2 has the same value of 1 for 19,27,31 and 72, where [x2,x3,x5,x7] is a val or point in the val space. We get a Euclidean norm by taking the square root. Vals which are sums or differences of any two of the above have a norm of sqrt(2), and +-v1+-v2+-v3, for three of the above vals, a norm of sqrt(3). The 4D cross polytope, or hyper-octahedron, with verticies + or - the above vals has edges corresponding to linear temperaments and faces corresponding to planar temperaments.

Message: 9859 - Contents - Hide Contents Date: Sat, 07 Feb 2004 19:54:01 Subject: Re: Val lattices From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: The 4D cross polytope, or> hyper-octahedron, with verticies + or - the above vals has edges > corresponding to linear temperaments and faces corresponding to planar > temperaments.It has 16 cells, and we may as well just pick the 19-27-31-72 cell and look at that. This is a tetrahedron, with faces planar temperaments, or 7-limit commas, and edges linear temperaments. Other regular tetrahedra can be found in the space which have similar properties, but the norm is a bit arbitary so I don't see too much importance in that.

Message: 9860 - Contents - Hide Contents Date: Sat, 07 Feb 2004 00:27:19 Subject: Re: Ennealimmal[45] as a chord block From: Gene Ward Smith It seems to work better to forget about 2401/2400 and 4375/4374 to start out with, and use generators of [0 1 0] and [1 0 -1]. The first changes major to minor and vice-versa, but two of them together are the same chord transposed 21/20 up; we have [0 0 0] ~ {1, 5/4, 3/2, 7/4} [0 1 0] ~ {21/20, 21/16, 3/2, 7/4} [0 2 0] ~ {21/20, 21/16, 63/40, 147/80} = (21/20){1, 5/4, 3/2, 7/4} [1 0 -1] is just transposing up a 7/6. Now we have two ennealimmal generators, since (7/6)^9 ~ 4 can work in place of (27/25)^9 ~ 2. If we take [i,j,-i] for i from -4 to 4, j from -1 to 0, we get Ennealimmal[36] on reducing. Note that [9 0 -9] is the same as [0 0 0] if we are tempering with ennealimmal. If we take j from -1 to 1, we get Ennealimmal[45] instead. We have [9, 0, -9] = 2[2 3 -3] + [5 -6 -3], where the last two are transposition by 2401/2400 and 4375/4374 respectively, so we may use the basis [9, 0, -9] and [2 3 -3] for lattice equivalencies. The determinant of the two generators [0 1 0] and [1 0 -1] together with [2 3 -3] is one, inverting the unimodular matrix with these rows gives a transformation from the lattice basis for tetrads I have been using to one in terms of the generators, plus the 2401/2400 transposition, given by the matrix with columns [3 1 3], [3 0 2] [-1 0 -1]. We may therefore use this to change 7-limit tetrads to a basis suitable to ennealimmal, and drop the last coordinate. Changing basis in this way we have for instance Major tetrad on 3/2: [4, 2, -1] Minor tetrad on 1: [-3, -3, 1] Major tetrad on 5/4: [6, 5, -2] and so forth. Dropping the last coodinate tells us where chord should be placed on the [0 1 0], [1 0 -1] plane, which gets translated as [1 0 0] and [0, 1, 0] respectively. We can then wrap the plane into a cylinder by [9 0 -9] ~ [0 0 0], which translates as [0 9 0] ~ [0 0 0]. We then see for instance that the major tetrad on 5/4 could be called [6, -4] after losing the last coordinate and reducing the second to the range -4 to 4. All of which, of course, could be put into Tonalsoft's 1.1 release, in theory, as a way of doing effective 7-limit JI chord noodling.

Message: 9861 - Contents - Hide Contents Date: Sat, 07 Feb 2004 01:47:34 Subject: Re: Ennealimmal[45] as a chord block From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> If we take [i,j,-i] for i from -4 to 4, j from -1 to 0, we get > Ennealimmal[36] on reducing. Note that [9 0 -9] is the same as [0 0 0] > if we are tempering with ennealimmal.If we take j from -1 to 1, we> get Ennealimmal[45] instead.j from -1 to 2

Message: 9862 - Contents - Hide Contents Date: Sat, 07 Feb 2004 07:14:58 Subject: Basis change for monzos, vals and wedgies From: Gene Ward Smith For whatever insight it may bring, here is an example. Suppose instead of 2,3,5,7 as a basis for 7-limit, we use 27/25, 21/20, 2401/2400 and 4375/4374. Then the corresponding basis for vals is <19 13 19 24|, <0 2 3 2|, <-1 -2 -3 -3| and <4 6 9 11|. The definitions for bimonzo, bival and compliment are the same, giving a new basis there as well. We have 441-et: <49 31 0 0| 612-et: <68 43 0 0| ennealimmal: <<1 0 0 0 0 0|| This can aslo be computed from 2401/2400: |0 0 1 0> 4375/4374: |0 0 0 1> For miracle, we have 225/224: |-5 8 -2 -1> 1029/1024: |-5 8 -1 -1> miracle: <<1 0 8 0 5 0|| We could also have used, for instance 72-et: <8 5 0 0| 175-et: <19 12 0 1| I'm fond of 12-et in this system: 12-et: <1 1 1 1| ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 9863 - Contents - Hide Contents Date: Sun, 08 Feb 2004 12:30:09 Subject: Re: Some warped egresses From: Herman Miller I think I've found a couple of good JI approximations for retuning _Egress_. The first one is a nice symmetrical looking one with lots of consonances, which looks like it'd work nicely with any of the pelog-type approximations, and would also work as 14 consecutive steps of meantone, Fb-B. 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/egress-jimajor.mid in Wayback Machine 0 0 0 0 0 ! Bb 5/4 -2 0 1 0 1 -2 1 -1 1 ! B 2 4 -1 -1 0 ! Cb 3 2 0 -2 1 ! C 7/5 0 0 -1 1 Ab Eb Bb 4 1 1 -1 0 ! Db 3/2 -1 1 0 0 5 3 -2 -1 1 ! D D A E B 6 2 -1 0 0 ! Eb Fb Cb Gb Db 7 0 0 -1 1 ! E 7/4 -2 0 0 1 8 6 -2 -1 0 ! Fb F C G 9 4 -1 -2 1 ! F 10 3 0 -1 0 ! Gb 1/1 1 0 0 0 11 1 1 -2 1 ! G 12 4 -2 0 0 ! Ab 13 2 -1 -1 1 ! A 7/6 -1 -1 0 1 14 1 0 0 0 ! Bb 5/4 -2 0 1 0 Does anyone know of any better 14-note block of JI that might be useful for this purpose? I haven't done any kind of systematic searches, but I've tried a number of possibilities, and I haven't found a better one. I think I'll want to use this one as the foundation for the Warped Egress page, with slight modifications for particular temperaments if necessary. Since 14-ET is highly ambiguous, I've also been looking for a minor key version that's consistent with the _Egress_ harmony and melody. This is the best I've come up with thus far: 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/egress-jiminor.mid in Wayback Machine 0 0 0 0 0 ! A 6/5 1 1 -1 0 1 -1 1 1 -1 ! Bb 2 1 -2 1 0 ! B G# D# 3 0 -1 2 -1 ! C 10/7 1 0 1 -1 F C 4 -2 0 1 0 ! C# 3/2 -1 1 0 0 5 0 2 0 -1 ! Db B F# C# 6 -1 -2 2 0 ! D# Ab Eb Bb 7 1 0 1 -1 ! Eb 12/7 2 1 0 -1 8 -1 1 0 0 ! E A E 9 2 -2 2 -1 ! F Gb Db 10 0 -1 1 0 ! F# 1/1 1 0 0 0 11 2 1 0 -1 ! Gb 12 1 -3 2 0 ! G# 13 3 -1 1 -1 ! Ab 8/7 3 0 0 -1 14 1 0 0 0 ! A 6/5 1 1 -1 0

Message: 9864 - Contents - Hide Contents Date: Sun, 08 Feb 2004 20:04:48 Subject: Re: Some warped egresses From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> I think I've found a couple of good JI approximations for retuning > _Egress_. The first one is a nice symmetrical looking one with lots of > consonances, which looks like it'd work nicely with any of the > pelog-type approximations, and would also work as 14 consecutive steps > of meantone, Fb-B. > > 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/egress-jimajor.mid in Wayback Machine > > 0 0 0 0 0 ! Bb 5/4 -2 0 1 0 > 1 -2 1 -1 1 ! B > 2 4 -1 -1 0 ! Cb > 3 2 0 -2 1 ! C 7/5 0 0 -1 1 Ab Eb Bb > 4 1 1 -1 0 ! Db 3/2 -1 1 0 0 > 5 3 -2 -1 1 ! D D A E B > 6 2 -1 0 0 ! Eb Fb Cb Gb Db > 7 0 0 -1 1 ! E 7/4 -2 0 0 1 > 8 6 -2 -1 0 ! Fb F C G > 9 4 -1 -2 1 ! F > 10 3 0 -1 0 ! Gb 1/1 1 0 0 0 > 11 1 1 -2 1 ! G > 12 4 -2 0 0 ! Ab > 13 2 -1 -1 1 ! A 7/6 -1 -1 0 1 > 14 1 0 0 0 ! Bb 5/4 -2 0 1 0 > > Does anyone know of any better 14-note block of JI that might be useful > for this purpose?Maybe not, but I did post one or two 14-note blocks once -- they're in Scala. I would search the tuning list for "What's your favorite number?", but it seems the yahoogroups search engine has suddenly become near-useless as it only searches a very small number of posts at a time.

Message: 9865 - Contents - Hide Contents Date: Sun, 08 Feb 2004 08:50:19 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Since there's a huge empty gap between complexity ~25+ and ~31, I was >> forced to look for a lower-complexity moat (probably a good thing >> anyway). I'll upload a graph showing the temperaments indicated by >> their ranking according to error/8.125 + complexity/25, since I saw a >> reasonable linear moat where this measure equals 1. Twenty >> temperaments make it in: >> Given that we normally relate error and complexity multiplicitively,normally . . .> I > think using log(err) and log(complexity) makes far more sense.I don't think they make more sense practically.> Can you > justify using them additively?Yes, or else some small power of them. Dave and I discussed this in depth. He initially proposed a*error^2 + b*complexity^2, partly because the local minima of harmonic entropy are parabolic.

Message: 9866 - Contents - Hide Contents Date: Sun, 08 Feb 2004 12:11:14 Subject: Re: Some warped egresses From: Carl Lumma>Maybe not, but I did post one or two 14-note blocks once -- they're >in Scala. I would search the tuning list for "What's your favorite >number?", but it seems the yahoogroups search engine has suddenly >become near-useless as it only searches a very small number of posts >at a time.Also I remember it not enforcing "". I don't find any tuning stuff with this on google. -C.

Message: 9867 - Contents - Hide Contents Date: Sun, 08 Feb 2004 08:52:11 Subject: Re: Comma reduction? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" >> >>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >>> wrote:>>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" >>>> >>>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" >>>>>>>> wrote: >>>>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" >>>>>>>>>>> >>>>>>> Thanks. Are they called 2-val and 2-monzo because they >>>>> are "linear">>>>>>> or is there some other reason? >>>>>>>>>>>> 2-vals are two vals wedged, 2-monzos are two monzos wedged. >> The >>>>> former>>>>>> is linear unless it reduces to the zero wedgie, the latter > is >>>> linear>>>>>> only in the 7-limit. >>>>>>>>>> Thanks! So the latter is linear in the 7-limit because the 7- >>> limit >>>> is>>>>> formed from two commas...I see. >>>>>>>> The 7-limit is 4-dimensional, so if you temper out 2 commas >> you're>>>> left with a 2-dimensional system, which is what we usually > refer >> to>>>> as "linear". Is that what you meant? >>>>>> Yes, I guess so. Why does tempering out two commas in a 4- >> dimensional>>> system leave a 2-dimensional system? >>>> Roughly: the two commas in addition to two other basis vectors will >> span the 4-dimensional system (only if the four vectors are > linearly>> independent). If you temper out the two commas, the remaining two >> basis vectors will form a basis for the entire resulting system of >> pitches, which we therefore regard as two-dimensional. >> Got it. How does one find the "remaining two basis vectors?" Is it > with Graham's matrix method?I suppose, or with Gene's algorithm, in which he uses Hermite reduction.

Message: 9868 - Contents - Hide Contents Date: Sun, 08 Feb 2004 18:55:02 Subject: Beep isn't useless.... From: Herman Miller 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/egress-beep.mid in Wayback Machine It doesn't sound as bad as I was imagining it would. It really warps the melody and harmony, but it could have its uses. Compare with the superpelog version, which I originally thought was beep before I figured out the mapping. 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/egress-superpelog.mid in Wayback Machine I still think the best use of extreme temperaments like beep and father is for their exotic melodic and harmonic properties, and not as approximations of JI. But the beep version doesn't seem as extreme as the father version: 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/egress-father.mid in Wayback Machine

Message: 9869 - Contents - Hide Contents Date: Sun, 08 Feb 2004 20:17:16 Subject: Jamesbond in 14-et From: Gene Ward Smith The jamesbond temperament, from the 007 in the wedgie <0 0 7 0 11 16|, has TM basis {25/24, 81/80}. If you look at the TOP tuning of its generator pair, you find one generator is almost exacly twice another; this strongly suggests we may effectively identify jamesbond with an et--but what et? The val for it needs to be of the form <7n 11n 16n m|, and if we set n=2 and m=39 we find our answer: <14 22 32 39|. The 7 limit TOP tuning for this has an octave of 1209.43 cents; 1/7 of this is g1 of the TOP generator pair <g1, g2> of jamesbond, which means 1/14 of it is g1/14 = (g2 + (g1-g2))/2, the average value of the two nearly equal generators which are both between 86 and 87 cents in size, g2 and g1-g2. This looks to me like something Herman might want to think about; you take TOP 14 with its stretched octaves and treat it in the 5-limit as two 7-ets, meaning that it is a meantone with neutral thirds; to this you add septimal harmony ad lib.

Message: 9870 - Contents - Hide Contents Date: Sun, 08 Feb 2004 08:55:36 Subject: Re: Basis change for monzos, vals and wedgies From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> For whatever insight it may bring, here is an example. Suppose instead > of 2,3,5,7 as a basis for 7-limit, we use 27/25, 21/20, 2401/2400 and > 4375/4374.Paul Hj., this would interest you. Then the corresponding basis for vals is> <19 13 19 24|, <0 2 3 2|, <-1 -2 -3 -3| and <4 6 9 11|. The > definitions for bimonzo, bival and complimentWhy, thank you :)> are the same, giving a > new basis there as well. We have > > 441-et: <49 31 0 0| > 612-et: <68 43 0 0| > > ennealimmal: <<1 0 0 0 0 0|| > > This can aslo be computed from > > 2401/2400: |0 0 1 0> > 4375/4374: |0 0 0 1> > > For miracle, we have > > 225/224: |-5 8 -2 -1> > 1029/1024: |-5 8 -1 -1> > > miracle: <<1 0 8 0 5 0|| > > We could also have used, for instance > > 72-et: <8 5 0 0| > 175-et: <19 12 0 1| > > I'm fond of 12-et in this system: > > 12-et: <1 1 1 1| ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------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: 9871 - Contents - Hide Contents Date: Sun, 08 Feb 2004 20:22:48 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> I >> think using log(err) and log(complexity) makes far more sense. >> I don't think they make more sense practically.I think they probably will make more sense both practically and theoretically, but you've been ignoring this issue. Are you going to think about it, at least?>> Can you >> justify using them additively? >> Yes, or else some small power of them. Dave and I discussed this in > depth. He initially proposed a*error^2 + b*complexity^2, partly > because the local minima of harmonic entropy are parabolic.None of this has convinced me in the least.

Message: 9872 - Contents - Hide Contents Date: Sun, 08 Feb 2004 20:30:18 Subject: Re: Jamesbond in 14-et From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> The jamesbond temperament, from the 007 in the wedgie <0 0 7 0 11 16|, > has TM basis {25/24, 81/80}. If you look at the TOP tuning of its > generator pair, you find one generator is almost exacly twice another; > this strongly suggests we may effectively identify jamesbond with an > et--but what et?Did you see the horagram I posted?

Message: 9873 - Contents - Hide Contents Date: Sun, 08 Feb 2004 20:33:29 Subject: Re: Some warped egresses From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> I think I've found a couple of good JI approximations for retuning > _Egress_. The first one is a nice symmetrical looking one with lots of > consonances, which looks like it'd work nicely with any of the > pelog-type approximations, and would also work as 14 consecutive steps > of meantone, Fb-B. > > 404 Not Found * [with cont.] Search for http://www.io.com/~hmiller/midi/egress/egress-jimajor.mid in Wayback Machine > > 0 0 0 0 0 ! Bb 5/4 -2 0 1 0 > 1 -2 1 -1 1 ! BThis looks as if it should be 2 * 3^(-2) * 5 * 7^(-1) * 11 = 110/63, but instead it's |-2 1 -2 1> = 21/20.

Message: 9874 - Contents - Hide Contents Date: Sun, 08 Feb 2004 20:33:28 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > >>> I>>> think using log(err) and log(complexity) makes far more sense. >>>> I don't think they make more sense practically. >> I think they probably will make more sense both practically and > theoretically,As I see it, no way. Example: when you look at the graph with log (err) as one of the axes, the indication is that JI is infinitely far away. This is ridiculous. The JI line should be right there, with some temperaments many times more distant from it than others. Otherwise, you're operating in the realm of hopelessly impractical abstraction.> but you've been ignoring this issue. Are you going to > think about it, at least?Countless hours already spent thinking about it, and discussing it here.

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