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"Ever since his
own age a great controversy has raged
about the teachings of Aristoxenus.
Instead of using ratios, he divided the
tetrachord into 30 parts, of which, in
his diatonic syntonon, each tone has 12
parts, each semitone 6. . . ." [J.
Murray Barbour, TUNING AND TEMPERAMENT A
HISTORICAL SURVEY, Da Capo Press, New
York, 1972, p22.]
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72 TONE EQUAL TEMPERAMENT (72-tET)
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Temperament [G. Temperatur]. The term denotes those systems of tuning in which the intervals deviate from the "pure," i.e., acoustically correct intervals as used in the Pythagorean system and in Just intonation. . . . compromise[s] . . . which, instead of being perfect in the simple keys and intolerably wrong in the others, spread the . . . inaccuracy over all the tones and keys. [Apel, Willi, Harvard Dictionary of Music, Harvard University Press, Cambridge, MA 1964, p734]
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The following is a selection from John H. Chalmers, Jr.book Divisions of the
Tetrachord
. . . Martin Vogel in Bonn and Franz
Richter Herf in Salzburg have been active
in various microtonal systems, the latter
especially in 72-tone equal temperament.
[p2]
Aristoxenos described his genera in
units
of twelfths of a tone (Macran, Henry S.
1902. The harmonics of Aristoxenos.
Oxford: The Clarendon Press), but later
theorists, notably Cleonides, translated
these units into a cipher consisting of
30 parts (moria) to the fourth (Barbera,
Charles Andre. 1978. "Arithmetic and
geometric divisions of the tetrachord," J.
Music Theory 22:294-323.). . . . Two such
30-part tetrachords and a whole tone of
twelve parts completed an octave of 72
parts. [pp18-19]
. . . neo-Aristoxenian tetrachords may
be
approximated in just intonation or
realized in equal temperaments whose
cardinalities are zero modulo 12. The
zero modulo 12 temperaments provide
opportunities to simulate many of the
other genera in the Catalogs as their
fourths are only two cents from 4/3 and
other intervals of just intonation are
often closely approximated. One may also
use them to discover or invent new neo-
Aristoxenian tetrachords.
To articulate a single part
difference, a temperament of 72 tones per
octave is required. [p37]
Chalmers, Jr., John H. Divisions of the
Tetrachord, Frog Peak Music, 1993
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The following is a section devoted to the music of Julian Carrillo and Augusto Novaro.
"Several experimentalists have managed to
survive in the predominantly conservative
musical culture of Mexico. Julian
Carrillo (1875-1965), an early western-
world experimenter with microtonal
scales, is perhaps best known. The
practically unknown Augusto Novaro (1891-
1960) designed and built string
instruments, guitars and pianos with
experimental intonational arrangements,
and published his treatise on a "Natural
System of Music" in Mexico City in 1951."
[Garland, Peter, "Selected Studies for
Player Piano," Soundings Book 4, spring-
summer 1977, Berkeley, California, p1]
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. . . [Regarding] 72ET as representing
the harmonic
series, from a practical standpoint, you
could consider 72ET to be a pitch
continuum, where ANY interval is
approximated well (with a maximum error
of
8 1/3 cents), and this viewpoint is
particularly appropriate to strings and
such instruments, where the variable
tuning is all by ear anyway. For fixed
pitch instruments, it may be more
reasonable to think of the 72 tones as
only representing intervals that are
"nearby" (the max error leads to
ambiguity); in that view, the harmonic
series is represented very well up to
harmonic 12 (worst error 3.9c), but with
13 the error jumps to 7.2c, so from
there on, the approximations are not
clear representations of the harmonic
relations... IMHO anyway. So one may need
to be content with "only" pure
octaves, fifths, thirds, sevenths,
ninths, and elevenths! Not bad!
From: Canright, David, Date: Sat, 21 Apr
2001 09:39:02 -0700
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72-tET Approximations of Just Intervals
Below is a table of [the first 23]. . . intervals found in the harmonic series . . ., and their closest approximations by 72-tET. These are ordered by first appearance in the harmonic series (by numerator, then by denominator; . . .); Not all of the "octave complements" appear; of course, their tempered approximations have equal but opposite errors. Each entry includes the ratio, cents error relative to nearest tempered (so, for example, 3/2 is +2.0c relative to note #42), tempered note # (0-71), keyboard # (0-5), key # (0-11), where keyboards 1-5 are assumed progressively sharper relative to keyboard 0.
| ratio |
cents error |
note |
key |
| 1/1 |
+0.0¢ |
0 |
0:0 |
| 3/2 |
+2.0¢ |
42 |
0:7 |
| 4/3 |
-2.0¢ |
30 |
0:5 |
| 5/3 |
+1.0¢ |
53 |
5:8 |
| 5/4 |
+3.0¢ |
23 |
5:3 |
| 6/5 |
-1.0¢ |
19 |
1:3 |
| 7/4 |
+2.2¢ |
58 |
4:9 |
| 7/5 |
-0.8¢ |
35 |
5:5 |
| 7/6 |
+0.2¢ |
16 |
4:2 |
| 8/5 |
-3.0¢ |
49 |
1:8 |
| 8/7 |
-2.2¢ |
14 |
2:2 |
| 9/5 |
+0.9¢ |
61 |
1:10 |
| 9/7 |
+1.8¢ |
26 |
2:4 |
| 9/8 |
+3.9¢ |
12 |
0:2 |
| 10/7 |
+0.8¢ |
37 |
1:6 |
| 10/9 |
-0.9¢ |
11 |
5:1 |
| 11/6 |
-0.6¢ |
63 |
3:10 |
| 11/7 |
-0.8¢ |
47 |
5:7 |
| 11/8 |
+1.3¢ |
33 |
3:5 |
| 11/9 |
-2.6¢ |
21 |
3:3 |
| 11/10 |
-1.7¢ |
10 |
4:1 |
| 12/7 |
-0.2¢ |
56 |
2:9 |
| 12/11 |
+0.6¢ |
9 |
3:1 |
Subject: table ordered by harmonic limit/Date: Wed, 2 May 2001 11:56:53 -0700/From: Canright, David
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I'd advise anyone wanting an ET to
approximate some intervals with 11 and 13
as well as the lower primes to look at 41
or 43, if said person doesn't want 72. I
also think of ETs as ETs, with all the
usual properties, because if you don't
need those, you might as well do JI to
get JI.
Ezra Sims has written several articles
besides the one or two noted
recently. I'm not wild about his
notation, although it's not hard to
figure out. Try Perspectives of New
Music, vol. 29 no. 1 (1991).
Paul Rapoport
rapoport@mcmaster.ca
Date: Sun, 12 Jun 94
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Ezra Sims has been using 72ET since the
early 70s. He uses it to
approximate harmonics up into the 20s and
even 30s. His main subset scale
approximates the pitches
1/1 (33/32) 25/24 (17/16) 13/12 (35/32)
9/8 (37/32) 7/6 (19/16) 29/24
(39/32) 5/4 21/16 11/8 23/16 13/8 27/16
7/4 29/16 15/8 31/16 2/1
He says it's very similar to Franz
Richter Herf's _Ecmelic Scale_. This
info is out of the book _Mikrotoene III_,
editor Horst-Peter Hesse, published
Edition Helbling, Innsbruck, (c) 1990.
Herf seems to have something of a
following, this being the proceedings of
their third annual conference, but the
rest of the book is hard for me to follow
because it's in German, and my German
nearly isn't. But I hope this gives you
some leads to follow anyway.
--
pH <manynote@library.wustl.edu>
Date: Fri, 10 Jun 94 13:53:25 -0700
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Didn't Tenney compose a piece for 72-TET
called "6 Harps" or
something? If I remember correctly, each
harp was in in 12-TET, but
all were tuned 1/6 semitone apart from
each other. He used a very
small subset of the 72 pitches, I
believe, to get close approximations
of various just intervals.
-greg higgs
Date: Fri, 10 Jun 94 14:31:23 -0700
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I have been using 72 equal quite a bit.
One thing that has helped me
immensely is a 72TET 'Bingo' card I made
up.
| 58 |
9 |
32 |
55 |
6 |
29 |
52 |
3 |
26 |
| 16 |
39 |
62 |
13 |
36 |
59 |
10 |
33 |
56 |
| 46 |
69 |
20 |
43 |
66 |
17 |
40 |
63 |
14 |
| 4 |
27 |
50 |
1 |
24 |
47 |
70 |
21 |
44 |
| 34 |
57 |
8 |
31 |
54 |
5 |
28 |
51 |
2 |
| 64 |
15 |
38 |
61 |
12 |
35 |
58 |
9 |
32 |
| 22 |
45 |
68 |
19 |
42 |
65 |
16 |
39 |
62 |
| 52 |
3 |
26 |
49 |
*0* |
23 |
46 |
69 |
20 |
| 10 |
33 |
56 |
7 |
30 |
53 |
4 |
27 |
50 |
| 40 |
63 |
14 |
37 |
60 |
11 |
34 |
57 |
8 |
| 70 |
21 |
44 |
67 |
18 |
41 |
64 |
15 |
38 |
| 28 |
51 |
2 |
25 |
48 |
71 |
22 |
45 |
68 |
| 58 |
9 |
32 |
55 |
6 |
29 |
52 |
3 |
26 |
| 16 |
39 |
62 |
13 |
36 |
59 |
10 |
33 |
56 |
| 46 |
69 |
20 |
43 |
66 |
17 |
40 |
63 |
14 |
~3/2's go up and ~5/4's go right.
On the card all intervals are uniformly
spaced, that is a 7/6 for
example is two right and one up.
Some interval equivalences:
3/2 = 42, 9/8 = 12, 4/3 = 30, 16/9 = 60,
15/8 = 65, 5/4 = 23, 5/3 = 53, 10/9 = 11,
9/5 = 61, 6/5 = 19, 8/5 = 49, 16/15 = 7,
7/5 = 35, 7/4 = 58, 7/6 = 16, 14/9 = 46,
11/8 = 33.. etc.
Many interesting symmetries and
enharmonicities can be found..
--- james mccartney
Date: Mon, 13 Jun 94 13:04:26 -0700
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72-tET approximates all of
Partch's
consonant intervals (i.e., all the ratios
in his 11-limit tonality
diamond) consistently and with a maximum
error of ~3.9 cents. This is much better
than any ET with less than
118 notes. 144-tET cannot improve upon
any of 72-tET's approximations.
Partch obtained his full 43-tone system
by transposing parts of the tonality
diamond to centers other than
1/1.
<snip>
correct transcriptions of
Harry Partch's music would never require
them to leave 72-tET <snip>
if 13- or higher-limit
intervals were included, using 144-tET to
approximate them could lead to
dangerous inconsistencies, since 144-tET
is only consistent through the 11-limit.
72-tET is consistent
through the 17-limit, although the ~7.2-
cent error in the 13:8 might cause one to
deem it insufficiently
accurate beyond the 11-limit. The
simplest ET to approximate the 13-limit
consistently with less than 4
cents error is 130-tET, and the simplest
to do so for the 17-limit is 149-tET.
Unfortunately, these are not
multiples of 12.
-Paul E.
Subject: [tuning] Re: Partch lattice in
144-eq notation
Date: Sun, 18 Apr 1999 18:36:28 -0400
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The following article was contributed by Paul Erlich.
This table shows which 72-tET intervals
best approximate the just ratios of
the 17-limit within half an octave. "17-
limit" here means the largest odd
factor of either of the numbers forming
the ratio does not exceed 17, as in
Harry Partch's Tonality Diamonds.
Intervals larger than half an octave are
easily calculated: for example, 7:4 is an
octave minus 8:7, so it's 72 - 14
= 58 steps.
The table shows that all 11-limit ratios
are represented uniquely; that is,
no 72-tET interval has to represent more
than one of them. What the table
does not show is that the maximum error
of the 72-tET approximations of the
11-limit ratios is 4 cents. The lowest ET
that improves on that is 118-tET
where the largest error is 3 cents.
72-tET is consistent through the 17-
limit. To explain what this means, let
me show why 72-tET is not consistent in
the 19-limit. In 72-tET, the best
approximation of 13:8 is 50 steps, the
best approximation of 19:8 is 90
steps, and the best approximation of
19:13 is 39 steps. That means that if
you try to play the chord 8:13:19 in 72-
tET, you cannot use the best
approximation of all the intervals
involved. That difficulty will never
arise in a chord all of whose intervals
are within the 17-limit, thus the
term "consistency". 58-tET is also
consistent through the 17-limit, but with
worse approximations; the lowest ET that
is consistent through the 19-limit
is 80-tET, and the smallest that improves
on 72-tET's 17-limit
approximations is 94-tET, which is also
consistent through the 23-limit.
| 72-tET steps |
Tones |
11-limit ratio |
Remaining 17-limit ratios |
| 0 |
0 |
1:1 |
| 6 |
1/2 |
|
18:17, 17:16 |
| 7 |
7/12 |
|
16:15, 15:14 |
| 8 |
2/3 |
|
14:13, 13:12 |
| 9 |
3/4 |
12:11 |
| 10 |
5/6 |
11:10 |
| 11 |
11/12 |
10:9 |
| 12 |
1 |
9:8 |
| 13 |
1-1/12 |
|
17:15 |
| 14 |
1-1/6 |
8:7 |
| 15 |
1-1/4 |
|
15:13 |
| 16 |
1-1/3 |
7:6 |
| 17 |
1-5/12 |
|
13:11, 20:17 |
| 19 |
1-7/12 |
6:5 |
| 20 |
1-2/3 |
|
17:14 |
| 21 |
1-3/4 |
11:9 |
| 22 |
1-5/6 |
|
16:13 |
| 23 |
1-11/12 |
5:4 |
| 25 |
2-1/12 |
14:11 |
| 26 |
2-1/6 |
9:7 |
| 27 |
2-1/4 |
|
13:10, 22:17 |
| 28 |
2-1/3 |
|
17:13 |
| 30 |
2-1/2 |
4:3 |
| 32 |
2-2/3 |
|
15:11 |
| 33 |
2-3/4 |
11:8 |
| 34 |
2-5/6 |
|
18:13 |
| 35 |
2-11/12 |
7:5 |
| 36 |
3 |
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17:12, 24:17 |
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Wed, 30 Jun 1999
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