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Message: 5050 - Contents - Hide Contents Date: Wed, 26 Jun 2002 20:57:35 Subject: Moving From: Gene W Smith What do people think of moving the list to Carl's freelist? I'd especially like Paul's take on this. Columbia would be another possibility, of course.
Message: 5051 - Contents - Hide Contents Date: Thu, 27 Jun 2002 15:17 +0 Subject: Re: Detempering miracle From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <aff3jh+vet5@xxxxxxx.xxx> genewardsmith wrote:> If we detemper miracle by h270, we get > > [1 0 0] > [1 6 0] > [3 -7 4] > [3 -2 2] > [2 15 -3] > > This actually doesn't go very well with 270, but it is well covered by > 954, which has its generators as 93.005/954 (the secor) and > 1.003/954. The temperament is 2401/2400^3025/3024, and is therefore a > detempering of hemiennealimmal, which is 2401/2400^3025/3024^9801/9800. > Since it both covers hemiennealimmal and has its own version of > miracle, it might be a good "universal" temperament, and in particular > a way of notating miracle in hemiennealimmal and vice-versa.Now there's a funny thing. I was looking for the miracle-plus temperament with 2401:2400^3025:3024 and got the mapping [1 0 0] [1 6 2] [3 -7 -1] [3 -2 0] [2 15 4] It looks like my generator's three times the size of yours. Does it still work, or is this a torsion problem? Graham
Message: 5052 - Contents - Hide Contents Date: Thu, 27 Jun 2002 20:30:39 Subject: Re: Detempering miracle From: Gene W Smith On Thu, 27 Jun 2002 15:17 +0100 (BST) graham@xxxxxxxxxx.xx.xx writes:> Now there's a funny thing. I was looking for the miracle-plus > temperament > with 2401:2400^3025:3024 and got the mapping > > [1 0 0] > [1 6 2] > [3 -7 -1] > [3 -2 0] > [2 15 4] > > It looks like my generator's three times the size of yours. Does it > still > work, or is this a torsion problem?The two matricies are equivalent; three times the third column minus the second column gives you my matrix.
Message: 5055 - Contents - Hide Contents Date: Thu, 27 Jun 2002 08:24:17 Subject: Re: Another approach to notating JI From: genewardsmith --- In tuning-math@y..., "jonszanto" <jonszanto@y...> wrote:> --- In tuning-math@y..., Gene W Smith <genewardsmith@j...> wrote:>> Always the rub. I suppose I could notate some Partch, claim it was >> JI, and see the fur fly over on the main group.> Like we don't read this list? :)Only on your best behavior. :)
Message: 5056 - Contents - Hide Contents Date: Thu, 27 Jun 2002 08:37:05 Subject: 494 From: genewardsmith This seems overdue for a deeper look. The MT reduced basis for the 11-limit h494 is <3025/3024, 9801/9800, 131072/130977, 234375/234256>. That immediately suggests the planar temperament defined by <3025/3024, 9801/9800> as something of interest. The (planar, not linear!) wedgie is 3025/3024^9801/9800 = h72^h270^h494 = [2,8,-6,-14,10,-2,-2,5,14,-25], and the Hermite reduced mapping is [2 0 0] [0 1 0] [0 0 1] [2 7 -4] [5 5 -3] This is an excellent microtemperament, but not the one Graham used; which is 3025/3024^825000/823543 = h31^h41^h494 = [13,-8,-9,-5,22,-17,-10,31,-26,5]. This is both more complex and less accurate than the previous, but it has a very nice feature--it is the detempering of Miralce, which is h31^h41, by h494. This suggests we look at this proceedure in general--take a linear temperament, and detemper it by wedging with a suitable JI (one which does not cover the orginal temperament.) Then relate it to the original temperament by keeping the two columns of that temperament as the first two columns of the mapping matrix for the new temperament. I'm off to take a look at this.
Message: 5057 - Contents - Hide Contents Date: Thu, 27 Jun 2002 13:22:57 Subject: Detempering miracle From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>This suggests we look at this proceedure in general--take a linear >temperament, and detemper it by wedging with a suitable JI (one >which does not cover the orginal temperament.) Then relate it to the >original temperament by keeping the two columns of that temperament >as the first two columns of the mapping matrix for the new >temperament. I'm off to take a look at this.If we detemper miracle by h270, we get [1 0 0] [1 6 0] [3 -7 4] [3 -2 2] [2 15 -3] This actually doesn't go very well with 270, but it is well covered by 954, which has its generators as 93.005/954 (the secor) and 1.003/954. The temperament is 2401/2400^3025/3024, and is therefore a detempering of hemiennealimmal, which is 2401/2400^3025/3024^9801/9800. Since it both covers hemiennealimmal and has its own version of miracle, it might be a good "universal" temperament, and in particular a way of notating miracle in hemiennealimmal and vice-versa.
Message: 5058 - Contents - Hide Contents
Date: Fri, 28 Jun 2002 13:39 +0
Subject: Detempering schismic
From: graham@xxxxxxxxxx.xx.xx
Okay, how about
( 1 0 0)
( 2 -1 0)
(-1 8 1)
( 3 14 0)
(-4 18 -2)
Tune schismic temperament so 9:7 is just, giving a 497.81 cent generator.
This leaves 9:8 and 7:4 both 0.48 cents out. Then add a (somewhat large)
schisma of 4.356 cents to get both 5:4 and 11:8 to about 0.50 cents. It
looks like everything adds up correctly, so no 11-limit intervals are more
than 0.5 cents out.
This is also very close to 270-equal. The schisma is one step. Although
that would only give 0.74 cent accuracy with 11:9 as the offending
interval.
Graham
Message: 5059 - Contents - Hide Contents Date: Sat, 29 Jun 2002 21:28:51 Subject: Re: Non octave over / under scale (continued from the main tuning list) From: Robert Walker HI there, for anyone following this, I've found the scale tree generating formula for the general case of r integer - probably. Also found interesting data for r/s rational but can't yet see the pattern there. In all this, it is entirely experimental, but to avoid continually saying "probably", I'll just present the (pro-tem) conclusions, without comment. Suppose one wants to find location for (k+m)/k in the under / over series scales with repeat at r/s. The general format seems to be: For x = (a+n)/a, (k+m)/k occurs at c(a+n)/(da+en) where c, d and e depend on k and m, and need to be determined. 4/3 occurs at x 1.1 1.2 1.3 r 2 11/13 12/16 .. c = 1, d = 1, e = 3 r 3 11/16 12/22 .. c = 1, d = 1, e = 6 r 4 11/19 12/28 .. c = 1, d = 1, e = 9 5/4 at r 2 11/14 12/18 .. c = 1, d = 1, e = 4 r 3 11/18 12/26 .. c = 1, d = 1, e = 8 r 4 11/22 12/34 .. c = 1, d = 1, e = 12 So we see: If scale repeat is r (integer), and the ratio is superparticular, e = (r-1)*k ................................... Now try a couple of others: 5/3 (k = 3, m = 2) at x 1.1 1.2 1.3 r 2 22/23 24/26 26/29 .. c = 2, d = 2, e = 3 r 3 22/26 24/32 24/38 .. c = 2, d = 2, e = 6 5/2 (k = 2, m = 3) at r 2 r 3 33/34 36/38 .. c = 3, d = 3, e = 4 r 4 33/36 .. c = 3, d = 3, e = 6 So we see: If scale repeat is r (integer), c = m, d = m, and e = (r-1)*k This is okay. ................................... What about general case of repeat at r/s rational: Try superparticular first r/s = 7/2, x = 1.1 1.2 3/2 at 22/30 24/40 .. c = 2, d = 2, e = 10 4/3 at 22/35 24/50 .. c = 2, d = 2, e = 15 r/s = 7/3, x = 1.1 3/2 at 33/38 .. c = 3, d = 3, e = 8 4/3 at 33/42 .. c = 3, d = 3, e = 12 5/4 at 33/46 .. c = 3, d = 3, e = 16 6/5 at 33/50 .. c = 3, d = 3, e = 20 r/s = 7/4, x = 1.1 3/2 at 44/46 .. c = 4, d = 4, e = 6 4/3 at 44/49 .. c = 4, d = 4, e = 9 So we see: c = s, d = s, e = k(r-s) Now the general rational case: r/s 7/3 x = 1.1 1.2 1.3 5/3 at 33/36 36/42 39/48 .. c = 3, d = 3, e = 6 7/5 at 33/40 36/50 39/60 .. c = 3, d = 3, e = 10 11/7 at 33/37 36/44 39/51 .. c = 3, d = 3, e = 7 11/5 at 99/100 108/110 117/120 .. c = 9, d = 9, e = 10 11/6 at 55/58 60/66 65/74 .. c = 5, d = 5, e = 8 9/5 at 33/35 9/10 13/15 21/25 r/s = 7/4 x = 1.1 1.2 1.3 8/5 at 44/45 48/50 52/55 .. c = 4, d = 4, e = 5 I can't see the pattern here. ................................... Constructing the scale tree lookup For r = 2, (k+m)/k occurs at m(a+n)/(a+kn) So 3/2 occurs at (a+n)/(a+2n) Then if 1/1 is at 0/v and 2/1 is at w/w 3/2 is at v/(v+w) so we can find v and w here. ................................... General case of r integer, To make the lookup tree, want to find (r+1)/2 Putting this in the form (k+m)/k k = 2, m = r-1 then putting that into m(a+n)/(ma+k(r-1)n) we get (r-1)(a+n)/((r-1)a+2(r-1)n) = (r-1)(a+n)/((r-1)(a+2n)) Then if 1/1 is at 0/v and r/1 is at w/w (r+1)/2 is at t/(s+t) so we can find v and w here. as before. ................................... General case: To construct the scale tree lookup, then you need to find a value for (r+1)/(s+1), from which you can get r/s and 1/1 by looking for them in the form 1/1 = 0/v and r/s = w/w as before, but here we need the formula for (r+1)/(s+1) which we don't know yet. However, to make the scale tree - the et one to use to look up the ratios one, we don't need the general case, we only need to know the formula for (r+1)/(s+1) so let's try that out. x = 1.1: r/s = 3/2, 4/3 at 22/23 c = d = 2, e = 3 r/s = 4/3, 5/4 at 33/34 c = d = 3, e = 4 ... r/s = 10/9 11/10 at 99/100 c = d = 9, e = 10 r/s = 7/4, 8/5 at 44/45 c = 4, e = 5 r/s = 7/2, 8/3 at 22/23 c = 2, e = 3 I don't get the pattern yet... Anyone see it? The page is now making Dan's scale tree for scales with integer repeats. Fractal Tune Smithy, Lissajous 3D, Virtual Flo... * [with cont.] (Wayb.) Robert
Message: 5062 - Contents - Hide Contents
Date: Sun, 30 Jun 2002 02:36:42
Subject: Geometry of 9-limit tetradic harmony
From: Gene W Smith
A 7-limit octave equivalence class is defined by the odd number 3^a 5^b
7^c, and so can be represented by the vector [a b c]. Consider the major
tetrad, 1-5/4-3/2-7/4, represented by
[0 0 0]-[0 1 0]-[1 0 0]-[0 0 1]. Taking the sum of this (which
corresponds to taking the product 3*5*7=105) gives us [1 1 1], which we
can use to represent the tetrad. The inverse
chord, 1-8/7-4/3-8/5 ~ 1-1/3-1/5-1/7, is a minor tetrad, represented by
[-1 -1 -1]. Multiplying
all four notes of a tetrad by any 7-limit interval will change the sum by
a power of four of that interval, or in terms of our vector notation,
will change it so as to leave it unchanged modulo four. Hence a major
tetrad is represented by a vector of three odd numbers, each of which is
of the form 4n+1, or 1 mod 4; and a minor tetrad by four odd numbers,
each of the form 4n-1, or -1 mod 4.
We've used up only a small proportion of the lattice of integral
3-vectors, so there is plenty of room to represent more types of tetrads.
The subminor tetrad, 1-7/6-3/2-5/3 adds up to
[-1 1 1], and so we can represent subminor tetrads by vectors equal
modulo 4 to [-1 1 1], or in other words, of the form [4n-1 4n+1 4n+1].
Inverting the subminor tetrad gives a supermajor tetrad 1-6/5-4/3-12/7
which adds up to [1 -1 -1], and we can represent all supermajor tetrads
by vectors of this form; for intense 1-9/7-3/2-9/5 is [5 -1 -1].
We still have plenty of space left, and can fill a small further amount
with the two so-called "asses" given by 1-3-5-5/3 and 1-3-7-7/3, which
are represented by [0 2 0] and [0 0 2] respectively. I'd be interested to
hear what else people think should be represented.
Let us call a tetrad interval adjacent if it shares an interval, or in
other words at least two notes, with another tetrad, and note adjacent if
it shares a single note. For major and minor tetrads, the difference
vectors defining interval adjacency are:
[2, 2, -2] [2, -2, 2] [-2, 2, -2] [-2, -2, 2] [2, -2, -2] [-2, 2,2]
[2, 0, 0] [-2, 0, 0] [0, -2, 2] [0, 2, -2] [4, -2, -2] [-4, 2, 2]
where the first row sends major to minor, and vice versa, and the second
sends to subminor/supermajor. If the tetrad is major, then
[2 0 0] attaches to the unique subminor tetrad with *three* notes in
common (for intense, 1-5/4-3/2-7/4 to 9/8-5/4-3/2-7/4), while if the
tetrad is minor, [-2 0 0] attaches to the unique supermajor tetrad with
three notes in common; for 1-6/5-3/2-12/7, represented by [3 -1 -1], this
would be 1-6/5-4/3-12/7 represented by [1 -1 -1].
For subminor/supermajor tetrads, the interval adjacency vectors are:
[2, 2, -2] [2, -2, 2] [-2, 2, -2] [-2, -2, 2] [6, -2, -2] [-6, 2, 2]
[2, 0, 0] [-2, 0, 0] [0, -2, 2], [0, 2, -2] [4, -2, -2] [-4, 2, 2]
By symmetry the second line, moving from sub/super to major/minor, is the
same for both, but the subminor/supermajor line involves more movement
along the chain of fifths.
For note adjacency, all four types of triads have different sets, though
with an approximate 2/3 overlap and 20 connections in common. These are,
in Maple readable form:
# note adjacency of major tetrad
majn := {[6, -2, -2], [6, 0, -4], [8, -2, -2], [4, 0, 0], [4, 2, -2], [6,
-4,
0], [2, 0, -4], [2, 4, -4], [4, -4, 0], [4, -2, 2], [4, 0, -4], [2, -4,
0],
[2, -4, 4], [0, -2, -2], [0, 0, -4], [0, 0, 4], [0, 4, -4], [0, 4, 0],
[-2,
6, -2], [0, -4, 0], [0, -4, 4], [-2, -2, 6], [-2, 0, 4], [-2, 4, 0], [-4,
0
, 4], [-4, 2, -2], [-4, 4, 0], [-4, 6, -2], [-2, -2, -2], [-6, 4, 0],
[-4, -
2, 2], [-4, -2, 6], [-4, 0, 0], [-6, 0, 0], [-6, 0, 4]}:
# note adjacency of minor tetrad
minn := {[-6, 2, 2], [-8, 2, 2], [6, 0, -4], [4, 0, 0], [4, 2, -2], [6,
-4, 0]
, [2, 0, -4], [4, -4, 0], [4, -2, 2], [4, 0, -4], [2, -4, 0], [0, 0, -4],
[0
, 0, 4], [0, 4, -4], [0, 4, 0], [0, -4, 0], [0, -4, 4], [-2, 0, 4], [-2,
4,
0], [-4, 0, 4], [-4, 2, -2], [-4, 4, 0], [-6, 4, 0], [-4, -2, 2], [-4, 0,
0]
, [-6, 0, 4], [6, 0, 0], [4, 2, -6], [2, 2, -6], [2, 2, 2], [4, -6, 2],
[2,
-6, 2], [0, 2, 2], [-2, 4, -4], [-2, -4, 4]}:
# note adjacency of subminor tetrad
subn := {[6, 0, -4], [8, -2, -2], [4, 0, 0], [4, 2, -2], [6, -4, 0], [2,
0, -4
], [4, -4, 0], [4, -2, 2], [4, 0, -4], [2, -4, 0], [0, -2, -2], [0, 4,
-4],
[0, -4, 4], [-2, 0, 4], [-2, 4, 0], [-4, 0, 4], [-4, 2, -2], [-4, 4, 0],
[-4
, 6, -2], [-6, 4, 0], [-4, -2, 2], [-4, -2, 6], [-4, 0, 0], [-6, 0, 4],
[6,
0, 0], [-2, 4, -4], [-2, -4, 4], [8, -4, 0], [8, 0, -4], [10, -2, -2],
[2, -
2, -2], [-8, 0, 4], [-8, 4, 0], [-6, -2, 6], [-6, 6, -2]}:
# note adjacency of supermajor tetrad
supn := {[-10, 2, 2], [-8, 2, 2], [6, 0, -4], [4, 0, 0], [4, 2, -2], [6,
-4, 0
], [2, 0, -4], [2, 4, -4], [4, -4, 0], [4, -2, 2], [4, 0, -4], [2, -4,
0], [
2, -4, 4], [0, 4, -4], [0, -4, 4], [-2, 0, 4], [-2, 4, 0], [-4, 0, 4],
[-4,
2, -2], [-4, 4, 0], [-6, 4, 0], [-4, -2, 2], [-4, 0, 0], [-6, 0, 0], [-6,
0
, 4], [6, 2, -6], [6, -6, 2], [-2, 2, 2], [4, 2, -6], [4, -6, 2], [0, 2,
2]
, [8, -4, 0], [8, 0, -4], [-8, 0, 4], [-8, 4, 0]}:
It would be interesting to see a three-dimensional diagram, with four
colors for the four different types of tetrads, and three colors for
lines connecting 3, 2, and 1 note adjacencies, of this geometry. I might
also point out that a harmony vector plus a 4-et interval uniquely
determines a 7-limit JI interval, but this is a large topic.
Message: 5063 - Contents - Hide Contents Date: Sun, 30 Jun 2002 15:54:52 Subject: Re: Dan's over / under scales From: Robert Walker Hi there, Just seen it for the case s>1, was obvious really but I was looking for something too compliciated For repeat r/s, and s > 1, (r+1)/(s+1) occurs at c = d = s, e = s+1. So for x = (a+n)/a, (r+1)/(s+1) occurs at (sa+sn)/(sa+(s+1)n) Testing several values in the tree confirms this It doesn't depend on r, and if you fix s, then the ratio you are looking for is always in the same place. e.g. trying s = 4, for x = 11/10 5/4: 6/5 occurs at 44/45, 3/2: 7/5 at 44/45 7/4: 8/5 at 44/45 2/1: 9/5 at 44/45 9/4: 2/1 at 44/45 ... For ratios of form r/1, the formula reduces to (a+n)/(a+2n) e.g. for x = 11/10, the required ratio is always found at 11/12 e.g. r = 2 2/1: 3/2 at 11/12 3/1: 2/1 at 11/12 4/1: 5/2 at 11/12 5/1: 3/1 at 11/12 etc. My previous observation was that for r integer, (r+1)/2 occurs at (r-1)(a+n)/((r-1)(a+2n)) - which reduces to this formula if you cancel the (r-1)s, as you can since the tree has the same ratio at all integer multiples of the degree, e.g. if it occurs in 11th position in 12th row, it will also occur at 22nd position in 24th row etc. Anyway I've written up the observations now on the page, and will try to prove it after the release of FTS. Fractal Tune Smithy, Lissajous 3D, Virtual Flo... * [with cont.] (Wayb.) Robert
Message: 5069 - Contents - Hide Contents Date: Thu, 4 Jul 2002 11:24:58 Subject: p-limit approximations From: Gene W Smith --- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:> I may be taking it out of context, but the thought I had is > that you could characterise the golden ratio say as 3 or 5 limit > etc. depending on whether it is more rapidly approximated by a sequence of > 3 limit or 5 limit ratios.Presumably, 5-limit beats 3-limit, unless you mean 5 with no 3.> I wrote a program a while back to look for ratio approimations > to another ratio, and just updated it to accept arbitrary decimals. > so that it can look for approximations to golden ratio etc. too.Did this use integer relations algorithms, brute force, or what?> Obviously these huge numbers aren't of immediate musical relevance, > but kind of interesting. It rather looks as if there is enough > of a trend there so that with some work one could define > a mathematically precise notion of the relative proprotions > of the various priimes needed to approximate an irrational, > which mightn't necessarily converge, so next thing would > be to see if one could prove it did converge, and if > every irrational has a definite flavour in the n-limit > or if only some do and so forth.This sounds more like a topic for the number theory list or sci.math, but in any event I'm skeptical.
Message: 5070 - Contents - Hide Contents Date: Fri, 5 Jul 2002 04:53:59 Subject: Re: Digest Number 420 From: Robert Walker HI Gene,> Presumably, 5-limit beats 3-limit, unless you mean 5 with no 3.Well, kind of vague here. Two ideas in mind - the relative proportions of each prime if you leave them free and find the best one for a given value of product of denominator and denumerator, and then the idea that one could maybe leave some of them out altogether and see which such methods work well. With that other idea you could e.g. find out if the golden ratio is better approximated using say 2^a*3^b*7^c or using 2^a*3^b*5^c in the limit as the numbers tend to infinity - from the data so far, one would rather suspect that the 7 will win here. (while I suppose if you think in terms of density the 5 should win). Of course any finite sequence doesn't really give much indication of what will happen in the infinite without some proof backing it.> Did this use integer relations algorithms, brute force, or what?Pretty much just brute force search of the lattice. It's sparse enough so that is quite effective. I wanted to do something more sophisticated but couldn't find out how to do it at the time, anyway easily goes up to 10^50 to 10^100 kind of a range so not much incentive to look for a better method. I was originally interested in using it to search for xenharmonic bridges.> This sounds more like a topic for the number theory list or sci.math, but > in any event I'm skeptical.Well, just an intriguing idea at present. :-). Robert ----- Original Message ----- From: <tuning-math@xxxxxxxxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Friday, July 05, 2002 1:06 AM Subject: [tuning-math] Digest Number 420> > To unsubscribe from this group, send an email to: > tuning-math-unsubscribe@xxxxxxxxxxx.xxx > > > ------------------------------------------------------------------------ > > There is 1 message in this issue. > > Topics in this digest: > > 1. p-limit approximations > From: Gene W Smith <genewardsmith@xxxx.xxx> > > > ________________________________________________________________________ > ________________________________________________________________________ > > Message: 1 > Date: Thu, 4 Jul 2002 11:24:58 -0700 > From: Gene W Smith <genewardsmith@xxxx.xxx> > Subject: p-limit approximations > > --- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote: >>> I may be taking it out of context, but the thought I had is >> that you could characterise the golden ratio say as 3 or 5 limit >> etc. depending on whether it is more rapidly approximated by a sequence > of>> 3 limit or 5 limit ratios. >> Presumably, 5-limit beats 3-limit, unless you mean 5 with no 3. >>> I wrote a program a while back to look for ratio approimations >> to another ratio, and just updated it to accept arbitrary decimals. >> so that it can look for approximations to golden ratio etc. too. >> Did this use integer relations algorithms, brute force, or what? >>> Obviously these huge numbers aren't of immediate musical relevance, >> but kind of interesting. It rather looks as if there is enough >> of a trend there so that with some work one could define >> a mathematically precise notion of the relative proprotions >> of the various priimes needed to approximate an irrational, >> which mightn't necessarily converge, so next thing would >> be to see if one could prove it did converge, and if >> every irrational has a definite flavour in the n-limit >> or if only some do and so forth. >> This sounds more like a topic for the number theory list or sci.math, but > in any event I'm skeptical. > > > ________________________________________________________________________ > ________________________________________________________________________ > > > > Your use of Yahoo! Groups is subject to Yahoo! Terms of Service * [with cont.] (Wayb.) > > >
Message: 5072 - Contents - Hide Contents Date: Tue, 9 Jul 2002 16:54:12 Subject: n-limit chord search From: manuel.op.de.coul@xxxxxxxxxxx.xxx Gene has asked to implement an n-limit chord finding operation in Scala. So I'd like to know whether what I've done so far is what he or anyone else is looking for. The command will be called CHORDS/ODD_LIMIT. The first parameter is the odd limit, and the second one an optional scale degree for limiting the search to one degree. SET MAXDIFF is used to set the maximum deviation for any chord tone. Any octave multiples are not shown in the chords. Here's an example output, where the scale is 20-tET, limit = 15 and maxdiff = 8.0 cents. Scale is equal tempered, only showing first degree. 0-54 2:13 diff. 0.528 0-34 4:13 diff. 0.528 0-17 5:9 diff. -2.404 0-13-22 7:11:15 diff. 2.492,-0.557 0-14 8:13 diff. 0.528 0-3 9:10 diff. 2.404 0-7-9 11:14:15 diff. -2.492,-3.049 0-4-6 13:15:16 diff. 7.741,-0.528 0-2-13 14:15:22 diff. -0.557, 2.492 0-11-16-18 15:22:26:28 diff. 3.049,-7.741, 0.557 0-11-13 105:154:165 diff. 3.049, 2.492 0-2-9 154:165:210 diff. -0.557,-3.049 0-2-4 182:195:210 diff. -0.557, 7.741 Total of 13 Manuel
Message: 5073 - Contents - Hide Contents Date: Tue, 9 Jul 2002 08:34:33 Subject: Re: n-limit chord search From: Gene W Smith On Tue, 9 Jul 2002 16:54:12 +0200 manuel.op.de.coul@xxxxxxxxxxx.xxx writes:> > Gene has asked to implement an n-limit chord finding > operation in Scala. So I'd like to know whether what I've > done so far is what he or anyone else is looking for.It looks good to me!
Message: 5074 - Contents - Hide Contents Date: Wed, 10 Jul 2002 22:15:09 Subject: More on chord geometry From: Gene W Smith I posted a while back on the geometric relations between 9-limit tetrads, using a system of coordinates defined by the vector sum of the exponents, or in other words, the product of the notes of the chord in exponent-vector form. This form of the coordinates allows one to find the vector form easily from the JI form, is generalizable to any p-limit, and works in the 5-limit also. However, there is a coordinate transformation which is special to the 7-limit which is more or less essential to understanding the how the 7-limit chord geometry works. The coordinates I've used may be related to the exponent coordinates of the notes by the observation that it is simply the centroid of the notes, times a scalar product of four. Since the notes themselves have a symmetric Euclidean metric given by the quadratic form Q(3^a 5^b 7^c) = a^2+b^2+c^2+ab+ac+bc, the chords inherit this metric. If we look at the interval adjacency vectors for major/minor, minor/major connections, namely [-2,2,2],[2,-2,2],[2,2,-2],[2,-2,-2],[-2,2,-2],[-2,-2,2] we find that these are either of opposite sign or are orthogonal. We may therefore select three of these to give us an orthogonal basis; if we also shift the coordinate center to the major tetrad [1,5/4,3/2,7/4] we may define a coordinate transformation which takes [a,b,c] to [(b+c-2)/4, (a+c-2)/4, (a+b-2)/4], the inverse of which takes [a,b,c] to [-2a+2b+2c+1, 2a-2b+2c+1, 2a+2b-2c+1]. This now makes the major and minor tetrads represented as the points of a cubic lattice, a nice feature which is unique to the 7-limit (the 5-limit gives a hexagonal tiling, but not a lattice; the lattice appears because of a special property of the 7-limit note lattice, which belongs to two different classes of lattices at once.) The pumps are particularly easy to find and understand in this coordinate system; for example [3 4 1][3 3 1][3 2 1][2 2 1][2 1 1][1 1 1][0 1 1][0 0 0] is a 1029/1024 pump I gave previously; in this form it is easy to see how we can obtain other such pumps. We again can represent subminor and supermajor tetrads, which transform to non-lattice points. In particular, 1-7/6-3/2-5/3 is represented by [0 -1/2 -1/2] and 1-9/7-3/2-9/5 by [-1 1/2 1/2] We can also represent complete 9-limit harmonies, basing ourselves on the major quintad 1-9/8-5/4-3/2-7/4 and its minor quintad inverse; the same coordinates serve for these. Formerly, given a 4-et value (that is, the value h4(q) for h4 = [4,6,9,11], which reduces mod 4 to [0 2 1 3], a complete set of representatives mod 4) and a chord, we could reconstruct the note. Now we may similarly use a 5-et value, based on h5 = [5,8,12,14], and because h5(1)-h5(9/8)-h5(5/4)-h5(3/2)-h5(7/4) is 0-1-2-3-4, a complete set of mod 5 residues, we can again recover the note from a chord and h5 value. (Similar comments apply to triadic harmony using h3 and 13-limit harmony using h7.) In terms of these chords, there are 12 interval adjacencies from major to minor, and 12 from minor to major.
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