Re: Miracle/Canasta 72-tet

I might add that Ezra Sims and Franz Richter Herf have also composed in
72-tet/EDO. Julian Carrillo had a 72-tet piano built, but the only use I
know that has been made of it is the scale recorded on an example tape
his daughter made some years ago.

--John

--- In tuning@y..., John Chalmers <JHCHALMERS@U...> wrote:
> I might add that Ezra Sims and Franz Richter Herf have also composed
in
> 72-tet/EDO.

It would be interesting to know how few notes per octave these
compositions would require in a chain of 7/72 oct generators.

Dave!
I understand that Ezra uses an 18 tone subset of 72

Dave Keenan wrote:

> --- In tuning@y..., John Chalmers <JHCHALMERS@U...> wrote:
> > I might add that Ezra Sims and Franz Richter Herf have also composed
> in
> > 72-tet/EDO.
>
> It would be interesting to know how few notes per octave these
> compositions would require in a chain of 7/72 oct generators.
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_22968.html#23068

> --- In tuning@y..., John Chalmers <JHCHALMERS@U...> wrote:
>
> > I might add that Ezra Sims and Franz Richter Herf have
> > also composed in 72-tet/EDO.
>
> It would be interesting to know how few notes per octave these
> compositions would require in a chain of 7/72 oct generators.

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:

/tuning/topicId_22968.html#23074

> Dave!
> I understand that Ezra uses an 18 tone subset of 72
>

Dave, I'm not sure I understand completely what you're
suggesting here. Kraig is partly correct, in that Sims
started out his microtonal investigations with an 18-tone
scale. But by now it's *much* more complicated than that.

I thought I'd post a little info on how Sims and Herf use
72-EDO, since I've studied their work a bit.

Sims devised an 18-tone microtonal scale by ear and intuition
years ago, which he referred to as "A structured, asymmetrical
set, founded on harmonic relations, like the diationic scale".

The primary basis of this intuitive scale was this
harmonic structure (usual triangular lattice, this time):

5/4 15/8
E-------B
/ \ /
/ \ /
/ \ /
1/1-----3/2
C G

Sims then posited the in-between pitches as equal divisions
of these basic ratios. So:

- the "major 3rd" from C to E was divided into 6 equal steps
so the step size was (5/4)^(1/6) = ~64.38561898 cents.

- the "minor 3rd" from E to G was divided into 4 equal steps
so the step size was (6/5)^(1/4) = ~78.91032175 cents.

- the "major 3rd" from G to B was exactly like that from C to E

- the "minor 2nd" from B to C was divided in half, giving
steps of (16/15)^(1/2) = ~55.86564263 cents.

Thus the total 18-tone scale, given in my preferred Semitones,
was:

scale
degree Semitones

0 12.00 == 0.00 C
17 11.44
16 10.88 B
15 10.24
14 9.59
13 8.95
12 8.31
11 7.66
10 7.02 G
9 6.23
8 5.44
7 4.65
6 3.86 E
5 3.22
4 2.57
3 1.93
2 1.29
1 0.64
0 0.00 C

After several intermediate tunings, he finally settled on
72-EDO to facilitate transposition, and this basic scale:

72-EDO 72-EDO approximate
degree notation Semitones proportions

0 C 0 32
68 B> 11&1/3 31
65 B- 10&5/6 30
62 Bb> 10&1/3 29
58 Bb< 9&2/3 28
54 A 9 27
50 Ab> 8&1/3 26
46 Ab< 7&2/3 25
42 G 7 24
38 F#> 6&1/3 23
33 F#v 5&1/2 22
28 F< 4&2/3 21
23 E- 3&5/6 20 30 40
20 Eb> 3&1/3 39
20 Eb> 3&1/3 29
Eb 3 19 38
16 Eb< 2&2/3 28
15 Ebv 2&1/2 37
12 D 2 18 27 36
Dv 1&1/2 35
8 Db> 1&1/3 26
6 Db 1 17 34
4 Db< 2/3 25
3 Dbv 1/2 33
0 C 0 16 24 32

(Sims sometimes uses 2^(69/72) B^ to represent 31/16.)

The usual scale is the one with proportions 24:25:26:27:28:29:30
("1/3-tones") at the bottom. Sometimes Sims feels the need
to have 1/4-tones instead, and uses the notes with proportions
32:33:34:35:36:37:38:39:40. The two sets are not used together.

Then Sims transposes this 37-prime-limit basic scale
successively to 3/2, 5/4, 9/8, 7/4, 11/8, 13/8 and 15/8,
fitting the resulting ratios into the nearest empty 72-EDO
spots. If a previous scale has a tone which already
occupies that spot, then that spot is unavailable for later
transpositions. This results in there being fewer and fewer
representatives of the transposed scales as one progresses
thru the series of transpositions.

This is essentially the JI conception of Sim's work, but he
has emphasized in letters to me that the precise tuning is
irrelevant: in comparing computer versions of his pieces
tuned in both 72-EDO and JI, he can't tell the difference.
So he pretty much considers 72-EDO to be a representation
of the virtual pitch continuum in the same way as Maneri.

Herf, too, thought of 72-EDO this way. His theories are
very interesting, based on the difference between what he
termed (1) structured, discernible _gestalt_ intervals,
and (2) amorphous, merged sound.

He devised an interesting method making use of proportions
from arithmetic series, which he considered gave "homogenous
harmonics (from bright to rough)". The method was to
repeatedly add difference d to initial term a, which would
be read "sequence d on a", and written d||a.

A few examples:

a=1, d=1 (the simplest sequence) 1||1 = 1:2:3:4:5:6:7:8...

a=1, d=3 3||1 = 1:4:7:10:13:16:19:22...

A chord could be formed from an extract of this, for example:

(5)3||4 = 4:7:10:13:16 i.e., a 5-note chord starting on 4.

Herf felt that chords with a particular character could only
be used if the scale contained the tones needed. So the
structure of the scale made available certain types of chords.
This is exactly the opposite of Haba's theory, wherein the
chords one desired to use determined the construction of
the scale. (Alois Haba, _Neue Harmonielehre_ [1927])

Herf originally got the idea to use 72-EDO from Haba's
book, and he used it to represent the different kinds of
arithmetical proportions in his music. Incidentally, due
to the rather large numbers he sometimes used in his
proportions, I would call his conception RI (rational
intonation) rather than JI.

Herf's 72-EDO notation is, IMO, a bit better than the
Sims/Maneri/Boston notation, and *much* better than those
devised by Haba and Wyschnegradsky (_La Loi de Pantonalité_).
My ASCII adaptation is based on Herf's notation.

I'm not sure when Herf died; I believe it was 1971.

A bit of information on Herf's work is available at:
http://www.bmwf.gv.at/3uniwes/01uniprofile/sbgmoze.htm
<http://www.moz.ac.at/user/herf/index.html> (in German)

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_22968.html#23086

> I'm not sure when Herf died; I believe it was 1971.

Scratch that... I'm just not sure. Perhaps it was 1989.

>
> A bit of information on Herf's work is available at:
> http://www.bmwf.gv.at/3uniwes/01uniprofile/sbgmoze.htm
> <http://www.moz.ac.at/user/herf/index.html> (in German)

Duh! My bad. That last page is available in English also:
http://www.moz.ac.at/user/herf/index_gb.html

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Dave!
> I understand that Ezra uses an 18 tone subset of 72

Aha! Here's one from the Scala archive?

! sims.scl
!
Ezra Sims' 18-tone mode
18
!
25/24
13/12
9/8
7/6
29/24
5/4
21/16
11/8
23/16
3/2
25/16
13/8
27/16
7/4
29/16
15/8
31/16
2/1

Who can figure out how many miracle generators it would need?

There's also a 20 and a 24 note Sims scale.

-- Dave Keenan

Monz! Thanks for that tutorial on the scales of Sims and Richter-Hoef.
I'm sorry I hadn't read it before sending my previous message. I can
now announce that there is no significant correlation between Sims
scales and the miracle generator. It would take a chain of 67 to fit
Sims' 23 notes.

-- Dave Keenan

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_22968.html#23086

> After several intermediate tunings, he finally settled on
> 72-EDO to facilitate transposition, and this basic scale:

I mistakenly left out two of the 72-EDO degrees.
Here's the bottom section of the scale:

72-EDO 72-EDO approximate
degree notation Semitones proportions

<upper section snipped>

23 E- 3&5/6 20 30 40
20 Eb> 3&1/3 39
20 Eb> 3&1/3 29
18 Eb 3 19 38
16 Eb< 2&2/3 28
15 Ebv 2&1/2 37
12 D 2 18 27 36
9 Dv 1&1/2 35
8 Db> 1&1/3 26
6 Db 1 17 34
4 Db< 2/3 25
3 Dbv 1/2 33
0 C 0 16 24 32

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_22968.html#23094

> Monz! Thanks for that tutorial on the scales of Sims and
> Richter-Hoef [that's Herf]. I'm sorry I hadn't read it
> before sending my previous message. I can now announce that
> there is no significant correlation between Sims scales and
> the miracle generator. It would take a chain of 67 to fit
> Sims' 23 notes.
>
> -- Dave Keenan

Right - I should have recognized right away when I wrote
that tutorial last night that Sims's implied JI proportions
would not fit in MIRACLE because his prime-limit is 37!

... Which I've known for years. In fact, the lattice I
made of Sim's basic scale is one of the coolest-looking
ones in my book.

monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> Right - I should have recognized right away when I wrote
> that tutorial last night that Sims's implied JI proportions
> would not fit in MIRACLE because his prime-limit is 37!

That would have been incorrect reasoning, Monz. Remember, both are
being treated as subsets of 72-tET.

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_22968.html#23096

>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> /tuning/topicId_22968.html#23086
>
>
> > After several intermediate tunings, he finally settled on
> > 72-EDO to facilitate transposition, and this basic scale:
>
>
> I mistakenly left out two of the 72-EDO degrees.
> Here's the bottom section of the scale:
>
>
> 72-EDO 72-EDO approximate
> degree notation Semitones proportions
>
> <upper section snipped>
>
> 23 E- 3&5/6 20 30 40
> 20 Eb> 3&1/3 39
> 20 Eb> 3&1/3 29
> 18 Eb 3 19 38
> 16 Eb< 2&2/3 28
> 15 Ebv 2&1/2 37
> 12 D 2 18 27 36
> 9 Dv 1&1/2 35
> 8 Db> 1&1/3 26
> 6 Db 1 17 34
> 4 Db< 2/3 25
> 3 Dbv 1/2 33
> 0 C 0 16 24 32
>
>

Sorry, Joe, I hadn't read this stuff when asking about the way that
Sims notates. I WILL catch up. Es muss sein!

_________ ______ ______
Joseph Pehrson