Gene's transformation
- From: Carl Lumma <carl@xxxxxxxxx>
- To: tuning-math@xxxxxxxxxxxxx
- Date: Tue, 23 Jul 2002 14:02:33 -0700
I saw a bit on Robert Walker's tune's page...
http://robertinventor.com/tunes/index.htm
...on a technique developed by Gene 'for transforming the tuning of a
tune while keeping the melodic line intact'.
Robert (hello, Robert!) says everything he knows of the technique is
embodied in some posts of Gene's. I found the posts copied below,
but I cannot decipher the technique. Is there a post I'm missing,
Gene? Or a chance I could get a primer on how it works? The way I
read the first post, it seems like a trivial mapping of intervals
from one temperament to another? No?
-Carl
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From: Gene W Smith <genewardsmith@xxxxxxxx>
Date: Sun, 21 Apr 2002 20:13:08 -0700
Subject: [tuning-math] An amazing new kind of transformation
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Since I just discovered it, perhaps "amazing" is immodest, but that is
how it struck me. This is a transformation which transforms from the
64/63 planar temperament to the 126/125 planar temperament and back!
It works as follows: using the approximations 64/63~1 or 126/125~1
respectively, the 7-limit music can be pulled back to a 5-limit preimage
in a canonical way, since 7~64/9 in the first instance, and 7~125/18 in
the second instance. We then apply a standard triad preserving mapping
of order 2, namely
2-->2
3-->10/3
5-->5
While sending triads to triads, it also sends tetrads in 64/63~1 to
tetrads in 126/125~1, and tetrads in 126/125~1 to tetrads in 64/63~1;
for instance 1-5/4-3/2-7/4 in 64/63~1 becomes 1-5/4-5/3-10/7 in
126/25~1, and vice-versa.
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From: "genewardsmith" <genewardsmith@xxxxxxxx>
Date: Mon, 22 Apr 2002 03:22:48 -0000
Subject: [tuning-math] Bearings
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My new type of transformation was discovered in the course of looking at
the position of the minor triad 7/6-7/5-7/4 when reduced to the 5-limit
lattice by means of a 7-limit comma in which 7 appears only to a power
of +-1, such as 64/63, 126/125, 225/224 or 4375/4374. In this case we
can describe the position by a bearing (here in degrees on one or
another side of one of the six 5-limit consonances) and distance. One
way to produce scales suitable for these planar temperaments would be to
produce 5-limit scales running in the general direction of the bearings
below. Hence, 64/63 works well with chains of fifths, 126/125 with
chains of minor thirds, 225/224 with chains of 16/15s (secors), and
4375/4374 with chains of minor thirds.
I obtained the following:
Bearing for 64/63
7/4 ~ 16/9 distance = 2, bearing 4/3
7/6 ~ 32/27 distance = 3, bearing 4/3
7/5 ~ 64/45 distance sqrt(7) = 2.64575, bearing 19.10661 4/3 by 8/5
Triad distance sqrt(57)/3 = 2.51661, bearing 6.58678 4/3 by 8/5
Bearing for 126/125
7/4 ~ 125/72 distance sqrt(7) = 2.64575, bearing 19.10661 5/3 by 5/4
7/6 ~ 125/108 distance 3, bearing 5/3
7/5 ~ 25/18 distance 2, bearing 5/3
Triad distance sqrt(57)/3 = 2.51661, bearing 6.58678 5/3 by 5/4
Bearing for 225/224
7/4 ~ 225/128 distance 2sqrt(3) = 3.46410, bearing 30 3/2 by 5/4
7/6 ~ 75/64 distance sqrt(7) = 2.64575, bearing 19.10661 5/4 by 3/2
7/5 ~ 45/32 distance sqrt(7), bearing 19.10661 3/2 by 5/4
Triad distance 5sqrt(3)/3 = 2.88675, bearing 30 3/2 by 5/4
Straight at 15/8 = 2/secor
Bearing for 4375/4374
7/4~2187/1250 distance sqrt(37) = 6.08276, bearing 25.28500 6/5 by 3/2
7/6~729/625 distance 2sqrt(7) = 5.29150, bearing 19.10661 6/5 by 3/2
7/5~4374/3125 distance sqrt(39) = 6.24500, bearing 16.10211 6/5 by 3/2
Triad distance sqrt(309)/3 = 5.85947 bearing 20.17357 6/5 by 3/2
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From: "genewardsmith" <genewardsmith@xxxxxxxx>
Date: Tue, 23 Apr 2002 05:56:28 -0000
Subject: [tuning-math] Yet another new kind of transformation
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I did a search of possible transformations among the commas in the list
25/24, 28/27, 36/35, 49/48, 50/49, 64/63, 81/80, 2048/2025, 245/243,
126/125, 4000/3969, 1728/1715, 1029/1024, 225/224, 3136/3125, 5120/5103,
6144/6125, 2401/2400, 4375/4374. There is nothing special about this
list, it's just one I had handy. I checked to see which pairs had the
same full octahedral group invariants, where for 3^x 5^y 7^z the
invariants I used were
Degree 2 x^2+y^2+z^2+x*y+x*z+y*z
Degree 4 y*x^2*z+x*y*z^2+x*y^2*z
Degree 6
y^4*z^2+y^4*x^2+2*y^3*z^3+2*y^3*z^2*x+2*y^3*z*x^2+2*y^3*x^3+y^2*z^4+
2*y^2*z^3*x+4*y^2*z^2*x^2+2*y^2*z*x^3+y^2*x^4+2*y*z^3*x^2+2*y*z^2*x^3+
z^4*x^2+2*z^3*x^3+z^2*x^4
This gave me the following possibilites, of pairs of commas which had
the same values for all three invariants:
25/24 36/35 3 0 4
25/24 49/48 3 0 4
28/27 64/63 7 0 36
28/27 126/125 7 0 36
36/35 49/48 3 0 4
64/63 126/125 7 0 36
81/80 1029/1024 13 0 144
The last one was unexpected and particularly intriging; on checking it,
I found it associated to the 7-limit transformation (of order 4)
given by 2->2, 3->7/2, 5->14/3, 7->28/5. The orbit of 81/80 under this
transformation is 81/80->1029/512->2401/2000->378/625->81/80.
Hence a piece in meantone, as a 7-limit planar temperament, can be sent
to something in the 1029/1024 temperament. The comma, instead of
converting into another comma and then being tempered out, converts to
1029/512, which tempers to 1/2. Hence, 7-limit harmony *is* preserved!
We now have two new, albeit related, kinds of transformations.
We might also note the larger groups of transformations arising from
the {28/27, 64/63, 126/125} and {25/24, 36/25, 49/48} sets of commas.
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