Yahoo Tuning Groups Ultimate Backup tuning 144-ET approximations

[me, monz:]
>> in addition, because 11 * 13 = 143,
>> 144-eq approximates 11-eq and 13-eq extremely well:

[Erlich/Barbaro:]
> No it doesn't. It only approximates the smallest intervals
> of 11-eq and 13-eq extremely well, as you show:

<etc., quoting me>

> For larger intervals the approximation is nothing special.

Ahem...

11-eq: (approx.) 144-ET cents
2^x/11 (ratio) cents degree cents 'error'

0 1 0.00 0 0.00 0.00
1 1 8/123 1.09 13 1.08 -0.76
2 1 128/953 2.18 26 2.17 -1.52
3 1 36/173 3.27 39 3.25 -2.27
4 1 43/150 4.36 52 4.33 -3.03
5 1 10/27 5.45 65 5.42 -3.79
6 1 17/37 6.55 79 6.58 3.79
7 1 107/193 7.64 92 7.67 3.03
8 1 137/209 8.73 105 8.75 2.27
9 1 29/38 9.82 118 9.83 1.52
10 1 115/131 10.91 131 10.92 0.76

(Inconsistency alert! Between degrees 5 and 6!)

So a max error of ~3.79 cents isn't good enough
*for a notation* to call it 'special'?

OK, we're disagreeing over degree.
(say that 10 times real fast)

13-eq: (approx.) 144-ET cents
2^x/11 (ratio) cents degree cents 'error'

0 1 0.00 0 0.00 0.00
1 1 27/493 0.92 11 0.92 -0.64
2 1 44/391 1.85 22 1.83 -1.28
3 1 17/98 2.77 33 2.75 -1.92
4 1 92/387 3.69 44 3.67 -2.56
5 1 194/635 4.62 55 4.58 -3.21
6 1 305/809 5.54 66 5.50 -3.85
7 1 252/557 6.46 78 6.50 3.85
8 1 441/829 7.38 89 7.42 3.21
9 1 295/479 8.31 100 8.33 2.56
10 1 436/619 9.23 111 9.25 1.92
11 1 347/435 10.15 122 10.17 1.28
12 1 233/260 11.08 133 11.08 0.64

(there's the inconsistency, between degrees 6 and 7)

Ditto for max error = ~3.85 cents?

It's (kind of) like saying that the 'quarter-tone' system
offers 'nothing special' compared to 12-eq because its
approximations to the 'perfect 5th' and 'major 3rd' are
exactly as good and as bad as 12-eq's. No one seems to
be interested in the *terrific* approximations 24-eq gives
for 11/8 and, one of my all-time favs, 11/6, or in that
moody 950 cent '6th/7th'.

Here are the lowest 'errors' in 24-eq compared to
some of the ratios in the Partch system (only picked
because it offers a lot of small-integer ratios).
I purposely included 4/3, 3/2, and 5/4, otherwise
I only give 24-ET-degrees which are 'quarter-tones':

24-eq: 24-ET cents
ratio cents degree cents 'error'

11/6 10.49 21 10.50 0.64
12/11 1.51 3 1.50 -0.64
16/11 6.49 13 6.50 1.32
11/8 5.51 11 5.50 -1.32
4/3 4.98 10 5.00 1.96
3/2 7.02 14 7.00 -1.96
11/9 3.47 7 3.50 2.59
18/11 8.53 17 8.50 -2.59
64/33 11.47 23 11.50 3.27
33/32 0.53 1 0.50 -3.27
35/32 1.55 3 1.50 -5.14
39/32 3.42 7 3.50 7.52
13/8 8.41 17 8.50 9.47
13/12 1.39 3 1.50 11.43
5/4 3.86 8 4.00 13.69
9/7 4.35 9 4.50 14.92
14/9 7.65 15 7.50 -14.92
20/11 10.35 21 10.50 15.00
11/10 1.65 3 1.50 -15.00
12/7 9.33 19 9.50 16.87
7/6 2.67 5 2.50 -16.87
8/7 2.31 5 2.50 18.83
7/4 9.69 19 9.50 -18.83
25/24 0.71 1 0.50 -20.67
32/21 7.29 15 7.50 20.78
21/16 4.71 9 4.50 -20.78
25/16 7.73 15 7.50 -22.63
63/32 11.73 23 11.50 -22.74

Admittedly, some of the approximation towards the
bottom of the table are not that close, but they're
*all* (except 4/3 and 3/2) closer than 12-eq, and
*for notational purposes* I think that makes it useful.

And even as a tuning system, it gives excellent
approximations of several 11-limit, very good
approximations of 13-limit, and fair approximations
of 7-limit ratios.

The only reason I went so far off the 144-eq topic
was to give people a handy reference for some useful
possibilities for a scale that gets down-played a lot.

(BTW, [shameless plug:] it took me about 3 nano-seconds
to produce these tables using my 'PrimeET' spreadsheet,
a very handy scale calculator in Excel:)

http://www.ixpres.com/interval/monzo/micro/prime-et.htm

Along the same line of reasoning, 144-ET provides
a slew of great approximations of all kinds of ratios
and other ETs, with a minimal increase in the notational
complexity.

Don't get me wrong: I'm all for totally new notations.
I like the one you invented for 22-eq, and think it suits
the decatonic scales in that system extremely well.
In like manner, Erv Wilson's notations are uniquely
crafted for their individual systems, and they relate
to each other in families of notational systems ingeniously
and are perfect for representing the inventive keyboard
designs he's created.

But I also think there's a tremendous usefulness
and durability to 'traditional' music notation -
that's one reason why I came up with my own using
the prime-factors. I wanted to be able to comprehend
rational JI theory in terms of the music notation I was
already familiar with, in addition to having to learn
ratios and the various other mathematical representations
for these intonations.

Consistency is certainly something that I think should
be considered in an actual system of tuning, but in a
notation I think it's not as important. There is
almost always some kind of intonational approximation
involved in a notation, unless you're sticking strictly
with the piano.

I honestly don't think that these inconsistencies are
bothersome in actual practice. You're dealing here
with 144 steps to the 'octave'. I haven't played with
144-eq audibly (i.e., *as* a tuning system) to hear
what that sounds like, but we've been talking about
it strictly as an approximative notational representation
of other pitches, anyway.

With so many fine shadings of pitch available to the
notation, in my experience of using it *merely as a
notation*, those inconsistencies don't phase me.
I get used to the intonational meaning of the different
symbols, and my knowledge of the *target* pitch
(for me, generally a JI ratio) gets me to the correct
tuning. The notation is merely an aid, and I think
144's a pretty good one.

I think, at least so far (I'm still not off the fence),
that any disadvantages 144-eq notation has are far
outweighed by its usefulness in representing shadings
of intonation as subtle as ~8 cents, and covering
enough of the virtual pitch continuum to be able to
represent several (perhaps many) different tunings
successfully *with the same set of symbols*, which
is important for someone who composes polymicrotonally.

I mean really, if Partch's chromelodeon players
could get used to hearing those ratios while playing
from his keyboard-tablature notation, 144-eq's a
piece of cake! For that matter, if anyone can actually
use Blackwood's 13-to-24-eq notations, they can use this.
My point is, performers get used to notational quirks
quite readily if they're really interested in performing
the music.

So rest assured that we both agree with each other that
72-eq is an excellent tuning system *and* notational
choice for a variety of reasons and purposes.

But that one little extra symbol sure does a lot for it...
(IMO)

-monz

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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144-ET approximations

🔗monz@juno.com

🔗monz@juno.com