Partch lattice in 144-eq notation

As an illustration of the 144-eq notation on which
I just posted:

Here's that lattice I made of Harry Partch's system,

pitches notated by Stearns/Monzo 144-eq
'ASCII cross-hatch Sims' notation instead of ratios

(Harry's rolling over in his grave).

1/1, or in this case 2^(0/144), = G (of course).
(so transpose my previous chart up a '5th')

G^
/| G>~
/ | /
/ | E+~ /
D#- ------------ C^ | / '-._ /
| /|'-._ | / C+
| / | A< | / /
| / B>|---|---|------A /
| F^ / | | | / |'-._ /
| /| / | F#-~ | / | F+
| / |/ |/ '|._ |/ | /
| / E>|------/|---|--D ------------- C<
Bv / | / | |/ |'-._ / | / '-._
C#+ --||-------|- B- | /| | Bb+ ------/----- Ab-
/'-._ ||/ |/ |'-._|/ | | / | / /
/ A>|------/|------G ---|---------- F< /
/ / | / | | / |'-._|/ | | |'-._ /
F#+ ----------|- E- | / | Eb+ ------/-|-----Db-
'-._ / |/ '-._ |/ | | / | / Ev
D> ------/|----- C --------------Bb< / |
/ | / |'-._ / | /| / |
A- | / | Ab+~ | / |/ |
/ '-._ |/ | | | / Av |
/ F ------------ Eb< / |
/ / | F> | / |
D- / | '-._|/ |
/ '-._ / | Dv ------------ B+
/ Bb-~ | /
/ | /
G<~ |/
Gv

The largest error for this notation from any ratio in the
system is ~4.1 cents for 21/16 and 32/21.

-monzo

Joseph L. Monzo....................monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html

|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |

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[MONZO:]
>pitches notated by Stearns/Monzo 144-eq 'ASCII cross-hatch Sims' notation
instead of ratios (Harry's rolling over in his grave).

It would seem to me that the successful application of Ezra Simms' 72-tET
notation at least points towards the _possibility_ of a (144-tET*)
UTILITARIAN 'mastering' of the pitch continuum... This somewhat hinges on
the caveat that the best chance for this type of a utilitarian 'mastering'
of the pitch continuum lies in the particular approach of navigating
successively complex deviations/approximations from the common target pitch
of standard notation, i.e. 12-tET. While I do think this type of an
approach stands a good chance of enabling standard notation to embrace the
widest possible range of intonational systems -- I think that any JI/ratio
examples would be more 'prudently' presented in this context without a
reference to Partch... (!) Unless perhaps it were to show (or debate) the
ideological shortcomings and benefits of this type of an approach.

Thanks Joe,
Dan

*While I believe that this ASCII 144-tET ('swung dash Simms') format
exaggerates the visual complexity of the actual glyphs and system... It
does maintains a strict fidelity to it's conceptual design: Adding ONE
'minus 24th tone tag' to the existing set of THREE symbols... [Where if you
think of the 'arrow,' the 'half-arrow,' and the upside-down 'square root' @
16&2/3rds, 33&1/3rd, and 50 cents sharp, as the main set of three symbols
i.e. -- the MS... then you could call the 50, 33&1/3rd, and 16&2/3rds cent
flat symbols the iMS -- the inverted main set... If you then add a minus
24th tone tag to the existing 12th tone symbols (rather than an independent
and additional 24th tone glyph) to the MS, it would yield the 24th, 8th,
and 5/24ths of a tone modified main set symbols -- the mMS... then adding
the crosshatch to the iMS would yield the 7/24ths, 3/8ths, and 11/24ths of
a tone imMS -- the inverted modified main set.] The score/transcription I
posted ("Don't Leave Me Here") uses/mixes 12, 19, and 20-tET. The full sets
of 20 and 19 would use the following 144-tET glyphs ["Don't Leave Me Here"
does not make use of them all.]:

19-tET

C, (<C#...Db), >~Db, -~D, (vD#...Eb), +Eb, <~E, (^~Fb...E), +~F, <F#, >Gb,
-~G, (v~G#...Ab), >~Ab, -A, (^Bb...A#), +~Bb, <~B, (>Cb...B#), C

C
8 +~F -~G 11
16 +~Bb -~D 3
5 +Eb -A 14
13 >~Ab <~E 6
2 >~Db <~B 17
10 >Gb <F# 9
18 >Cb <C# 1
7 ^~Fb v~G# 12
15 ^Bb vD# 4
4 vEb ^A# 15
12 v~Ab ^~E... 7
1 <Db >B# 18

[...As opposed to v-~E#.]

20-tET

C, (v~C#...Db), +Db, -D, ^~Ebb, (Eb...D#), v~E, +Fb, (-F...E#), ^~Gbb,
(F#...Gb), v~Fx, (+G...Abb), -G#, ^~Ab, (A...Bbb), v~A#, +Bb, -B,
(^~Cb...B#), C

C
8 -F +G 12
17 +Bb -D 3
5 Eb A 15
14 ^~Ab v~E 6
2 +Db -B 18
10 Gb F# 10
19 ^~Cb v~C# 1
7 +Fb -G# 13
15 Bbb D# 5
4 ^~Ebb v~A# 16
12 +Abb -E# 8
1... v~Db ^~B# 19
9 ^~Gbb v~Fx 11

[...As opposed to ^+~Dbb.]

Where:

16&2/3c# 33&1/3c# {50c# 50cb} 33&1/3cb 16&2/3cb
8&1/3c# 25c# 41&2/3c# 41&2/3cb 25cb 8&1/3cb

is:

+ > ^ v < -
+~ >~ ^~ v~ <~ -~

The peculiar spelling/array of 20-tET given here (where a clockwise fifth
is d/O*F, and a counterclockwise fourth is d/O*f, and equal "d"ivision of
the octave=20, "O"ctave=31, "F"ifth=18, and "f"ourth=13), was an attempt to
recast 20-tET in a faux (31-tET 1/4 comma) meantone array... While hardly
ideal - I thought it might match up a bit better (conceptually anyway...)
with the 19-tET of this particular example.

Joe,

I strongly disagree with the way you "transcribed" Harry Partch's system into 144-tET. Would Harry have
liked the 708.33-cent fifth between A and E+~? Of course not! There are probably more egregious errors in
your transcription; I don't have time to look right now. You would have achieved *much* better results had
you kept the approximation of each prime consistent throughout the system, and as a bonus you would have
ended up staying within 72-tET.

> The largest error for this notation from any ratio in the
> system is ~4.1 cents for 21/16 and 32/21.

The errors in intervals such as 21:16 are not very important, as Partch considered ratios of 15, 21, etc. to
be dissonant. More important are intervals such as 3:2, which your transcription renders with an error of
~6.4 cents betweem A and E+~.

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 lass 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. What is important in these transpositions is that the harmonic integrity of the each of the diamonds be
preserved, not that the complex, secondary ratios be represented well with respect to the 1/1 of the overall
system (G).

This is another case of what I have been criticizing you and Marion of: considering pitches as ratios, it is
easy to fall into the trap of viewing these ratios as harmonic entities in themselves, but the resulting
analyses are only valid when the ratio forms a harmony with the 1/1, in which case the ratio and the
harmonic interval are the same. Otherwise, the ratio must be considered in terms of the harmonic intervals
it will form with other, more closely related ratios, and it is these harmonic intervals (which I think
should be expressed with a colon rather than a slash) which must closely approximate simple-integer
proportions.

To sum up, even if a school of musicians appeared who could play perfect 144-tET, correct transcriptions of
Harry Partch's music would never require them to leave 72-tET, i.e., the ~ sign would never be needed.

What's more, 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.

Paul E. wrote
> 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.

Not true, counterexample is 87-tET.

> 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.

Also not, 87-tET does 15-limit with 3.3 cents accuracy. And 94-tET even
23-limit with
less than 4 cents error.

Manuel Op de Coul coul@ezh.nl

On Tue, 20 Apr 1999 manuel.op.de.coul@ezh.nl wrote:
> Paul E. wrote
>> 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.
>
> Not true, counterexample is 87-tET.
>
>> 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.
>
> Also not, 87-tET does 15-limit with 3.3 cents accuracy. And 94-tET even
> 23-limit with less than 4 cents error.

Sorry, Manuel, but my calculations support Paul E.'s assertions. See
my consist3 table (http://library.wustl.edu/~manynote/consist3.txt),
which I have just redone to include another decimal of accuracy.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "You just ran nine racks but you won't give me a spot?"
-\-\-- o "I can't; I haven't seen you shoot yet."

NOTE: dehyphenate node to remove spamblock. <*>

Manuel wrote,

>Paul E. wrote
>> 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.
>
>Not true, counterexample is 87-tET.

You are wrong, Manuel. 87-tET has a 4.8 cent error in the 7/6 and a 6.3 cent
error in the 7/9.

>
>> 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.
>
>Also not, 87-tET does 15-limit with 3.3 cents accuracy.

False, see above.

And 94-tET even
>23-limit with
>less than 4 cents error.

Nope, the 7:5 is off by ~4.7 cents and 13:10 is off by ~5.4 cents.

Perhaps we have found an error in Scala?

-Paul E.

> You are wrong, Manuel. 87-tET has a 4.8 cent error in the 7/6 and a 6.3
cent
> error in the 7/9.

Oh, yes I see. My mistake was to only look at the primary harmonics. Sorry
about that.

> Perhaps we have found an error in Scala?

No, I just looked at the wrong numbers.

Manuel Op de Coul coul@ezh.nl

I was wondering If when all these mathematical cent errors are look at
if the notion as to whether they are off in the same direction or not
because if not these numbers mean little!
-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

On Wed, 21 Apr 1999, Kraig Grady wrote:
> I was wondering If when all these mathematical cent errors are look at
> if the notion as to whether they are off in the same direction or not
> because if not these numbers mean little!

The errors are absolute, i.e. unsigned, but I don't think it's as big a
problem as you may fear. For example, if the 5/4 and the 3/2 are both
off by 4 cents in the same direction, the 6/5 will be right on, but if
they are off by 4 cents in the opposite direction the 6/5 will be off by
8 cents. So although my consist3 table only shows the max (unsigned)
error, it still doesn't equate ETs like this.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "You just ran nine racks but you won't give me a spot?"
-\-\-- o "I can't; I haven't seen you shoot yet."

NOTE: dehyphenate node to remove spamblock. <*>

Paul Hahn wrote:

>
>
> The errors are absolute, i.e. unsigned, but I don't think it's as big a
> problem as you may fear. For example, if the 5/4 and the 3/2 are both
> off by 4 cents in the same direction, the 6/5 will be right on, but if
> they are off by 4 cents in the opposite direction the 6/5 will be off by
> 8 cents. .

This is exactly the problem I was mentioning as I so inarticulately said:)

-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

On Wed, 21 Apr 1999, Kraig Grady wrote:
> Paul Hahn wrote:
>> if the 5/4 and the 3/2 are both
>> off by 4 cents in the same direction, the 6/5 will be right on, but if
>> they are off by 4 cents in the opposite direction the 6/5 will be off by
>> 8 cents.
>
> This is exactly the problem I was mentioning as I so inarticulately said:)

Yes, but how much of a problem really is it? The table lists the worst
error for the first tuning as 4 cents, and the latter as 8 cents.
Clearly the first tuning is better, and the error values reflect that.
Where exactly is the problem?

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "You just ran nine racks but you won't give me a spot?"
-\-\-- o "I can't; I haven't seen you shoot yet."

NOTE: dehyphenate node to remove spamblock. <*>