Yahoo Tuning Groups Ultimate Backup tuning 7-limit pitch-bend lattice

I have created a 7-limit ASCII lattice that gives a consistent
letter-name notation, ratios, and pitch-bend values for use
in MIDI-files.

The full lattice is online at
http://www.ixpres.com/interval/monzo/lattices/pitch-bend-lattice.htm

The central portion is quoted below.

pitch-bend lattice
------------------

Extended accidentals are derived from my adaptation of the
Herf and Sims 72-tET notations:

sharper flatter - than 12-tET pitch

approximate or usual meaning

+ - ~1/12-tone ~ 17 cents 80:81 syntonic comma
> < ~1/6-tone ~ 33 cents 63:64 septimal comma
^ v ~1/4-tone ~ 50 cents 35:36 septimal diesis
# b ~1/2-tone ~100 cents 2048:2187 apotome

The cents adjustments are approximate because the actual pitches
are just-intonation ratios whose deviation from 12-tET varies
with every note.

Pitch-bend values show the deviation from the 12-tET pitch
(designated by the letter-name and any sharps or flats);
they assume 4096 units per 12-tET semitone.

F#< C#< G#< D#<
- 25/18 --------- 25/24 --------- 25/16 --------- 75/64 --------
-1281 ` '-1201 ` '-1121 ` '-1041 `
/|\ ` D# ' /|\ ` A# ' /|\ ` E# ' /|\ `B# '
/ | \ 25/21 / | \ 25/14 / | \ 75/56 / | \ 225/224
/ | \ +76 / | \ +156 / | \ +236 / | \ +316
/ Cv \ /|\ / Gv \ /|\ / Dv \ /|\ / Av \ /|\
35/18 -------- 35/24 --------- 35/32 --------- 105/64 --------
'-1998` /\ | /\'-1918 ` /\ | /\'-1837 ` /\ | /\'-1757 ` /\ | /\
/ \ /` \|/' \ / \ /` \|/' \ / \ /` \|/' \ / \ /` \|/'
/ \/ A- \/ \/ E- \/ \/ B- \/ \/ F#-
-/-----\-- 5/3 --/-----\-- 5/4 --/-----\- 15/8 --/-----\- 45/32
/ / \'-641 `/ \ / \'-561 `/ \ / \'-481 `/ \ / \'-400 `
` B+ ' \ /|\ / ` F#+ ' \ /|\ / ` C#+ ' \ /|\ / ` G#+ ' \ /|\ /
40/21 --\-|-/-- 10/7 ---\-|-/-- 15/14 --\-|-/-- 45/28 --\-|-/-
+636 / \|/ \ +716 / \|/ \ +796 / \|/ \ +876 / \|/ \
\ /|\ / Eb< \ /|\ / Bb< \ /|\ / F< \ /|\ / C<
---------- 7/6 ----------- 7/4 ---------- 21/16 --------- 63/32
/\ | /\ -1357 /\ | /\ -1277 /\ | /\ -1197 /\ | /\ -1117
/ \|/ ' / \` / \|/ ' / \` / \|/ ' / \ `/ \|/ ' / \ `
` F ' \/ \/ ` C ' \/ \/ ` G ' \/ \/ ` D ' \/ \/
-- 4/3 --/-----\-- 1/1 --/-----\-- 3/2 --/-----\-- 9/8 --/-----\
\'-80 ` / \ / \' 0 ` / \ / \' +80 `/ \ / \'+160 `/ \ /
\ /|\ / ` D> ' \ /|\ / ` A> ' \ /|\ / ` E> ' \ /|\ / ` B> '
--\-|-/--- 8/7 ---\-|-/-- 12/7 ---\-|-/--- 9/7 ---\-|-/-- 27/14
/ \|/ \ +1277 / \|/ \ +1357 / \|/ \ +1437 / \|/ \ +1517
/ Cb- \ | / Gb- \ | / Db- \ | / Ab- \ |
- 28/15 ---------- 7/5 ---------- 21/20 --------- 63/40 --------
-796 \ | / -716 \ | / -636 \ | / -556 \ | /
' ` \|/ ' ` \|/ ' ` \|/ ' ` \|/
` Ab+ ' ` Eb+ ' ` Bb+ ' ` F+ '
---------- 8/5 ----------- 6/5 ----------- 9/5 ---------- 27/20
+561 +641 +721 +801

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

Nice work, Monz! Glad you're comfortable with "our" way of drawing lattices.
72-tET has the fortunate circumstance of having the conventional notation
apply in the simplest possible way to a circle of fifths, while the
alterations required to produce ratios of 5 or of 7 (or of 9 (except 9:8) or
of 11) simply move one to a different circle of fifths:

Each factor of 5 moves you to the circle of fifths 1/72 octave (=1/12
tone)lower.
Each factor of 7 moves you to the circle of fifths 2/72 octave (=1/6
tone)lower.
Each factor of 11 moves you to the cirle of fifths 3/72 octave (=1/4
tone)lower.

This works for all 11-(odd-)limit ratios, since 72-tET is consistent in the
11-limit.

Giving each circle of fifths a "name" suffices to allow one to notate
11-(odd-)limit ratios unambiguously, since the 11-limit tonality diamond is
uniquely articulated in 72-tET (see the bottom of.

Even compounds of any two 11-(odd-)limit ratios (i.e. the set of ratios with
prime limit 11 and odd limit 121), form a consistent set relative to JI in
72-tET (by virtue of 72-tET's Hahn consistency level at the 11-limit being
2).

Hence 72-tET-based notation is an excellent notation for Partch's music.

144-tET, by contrast, is not consistent at all in the 11-limit. . . .

🔗Joe Monzo <monz@juno.com>

> [Paul Erlich, TD 528.18]
> Nice work, Monz! Glad you're comfortable with "our" way of
> drawing lattices.

Thanks, Paul. It took all day to draw *that* one - if I had
tried to use my lattice formula I'd probably still be working
on it! Besides, for a 3-D lattice the tetrahedral version
works fine for me; if I had included 11-limit ratios then
I probably would have tried to use my formula instead.

> [Paul]
> 72-tET has the fortunate circumstance of having the
> conventional notation apply in the simplest possible way
> to a circle of fifths, while the alterations required to
> produce ratios of 5 or of 7 (or of 9 (except 9:8) or
> of 11) simply move one to a different circle of fifths:
>
> Each factor of 5 moves you to the circle of fifths 1/72 octave
> (=1/12 tone)lower.
> Each factor of 7 moves you to the circle of fifths 2/72 octave
> (=1/6 tone)lower.
> Each factor of 11 moves you to the cirle of fifths 3/72 octave
> (=1/4 tone)lower.
>
> <snip>

Thanks for giving the detailed explanation. That's exactly
why I use the 72-tET notation.

> [Paul]
> Hence 72-tET-based notation is an excellent notation for
> Partch's music.
>
> 144-tET, by contrast, is not consistent at all in the
> 11-limit. . . .

Well, we've been thru *that* before... I'm not going to go
there, except to say that in some circumstances the greater
pitch discrimination provided by 144-tET, with only one
extra symbol as Dan Stearns and I use it, [see TD 144.16]
http://www.onelist.com/messages/tuning?archive=144
is useful, even if it's not consistent.

I really don't want to argue this point, but there are
cases (my _A Noiseless Patient Spider_ was one of them)
where 72-tET just doesn't provide enough accuracy to portray
what's happening, and 144-tET does. Once you get used to
working with it, you learn how to accomodate the inconsistencies.
But as a *general* rule, I'll agree that 72-tET works much
better.

The most interesting thing I found with this pitch-bend lattice
(as it appears on my webpage) unfortunately did not appear in
the clipped version that I posted here.

I was surprised, given the accuracy to which the pitch-bend
unit is calculated (i.e., 2^(12*4096) = 49152-EDO), that
exactly the same amount of bend recurs at certain points on
the lattice.

For example, to produce a 16/9 Bb (= 3^-2), you use the 12-EDO
Bb and put a bend of -160 on it. To produce a 1215/1024 D#-
(= 3^5 * 5^1), you use the 12-EDO D# and also put a bend of
-160 on it. I thought of that as a kind of cousin to the
periodicity-block concept, and used it to delimit the boundaries
of the lattice. Hmmm... I'd love to know what those who are
more mathematically-inclined have to say about this...

-monz

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

Monz wrote,

>For example, to produce a 16/9 Bb (= 3^-2), you use the 12-EDO
>Bb and put a bend of -160 on it. To produce a 1215/1024 D#-
>(= 3^5 * 5^1), you use the 12-EDO D# and also put a bend of
>-160 on it. I thought of that as a kind of cousin to the
>periodicity-block concept, and used it to delimit the boundaries
>of the lattice. Hmmm... I'd love to know what those who are
>more mathematically-inclined have to say about this...

You have discovered the fact that the schisma is almost exactly the same
size as the deviation of the equal-tempered fourth from the just fourth.
This was the basis of some rational approximations of equal temperament
discussed by Margo Schulter and a recent MTO article.