This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).
- Contents - Hide Contents - Home - Section 109000 9050 9100 9150 9200 9250 9300 9350 9400 9450 9500 9550 9600 9650 9700 9750 9800 9850 9900 9950
9850 - 9875 -
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.