## Joe Monzo's 7-limit pitch-bend lattice Text and diagrams © 2000 by Joseph L. Monzo

Below is my lattice diagram showing a 3-dimensional vector-space whose coordinates represent a 7-limit just-intonation pitch-set.

```

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

(c) 2000 by Joe Monzo

```

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

```

72-EDO meaning

sharper  flatter   - than 12-tET note

+        -      ~1/12-tone  ~ 17 cents
>        <      ~1/6-tone   ~ 33 cents
^        v      ~1/4-tone   ~ 50 cents
#        b      ~1/2-tone   ~100 cents

```

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

```

7-limit JI meaning

sharper  flatter     - than Pythagorean note

+        -        80:81    syntonic comma   ~ 21.5 cents
>        <        63:64    septimal comma   ~ 27.3 cents
^        v        35:36    septimal diesis  ~ 48.8 cents
#        b      2048:2187  apotome          ~113.7 cents

```

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; I have coined the term cawapu to designate these units.

```

B<           F#<             C#<             G#<             D#<             A#<             E#<             B#<             Fx<
50/27 ------- 25/18 --------- 25/24 --------- 25/16 --------- 75/64 -------- 225/128 ------- 675/512 ------ 2025/1024 ----- 6075/4096
-1361         -1281 `        '-1201 `        '-1121 `        '-1041 `       ' -961 `        ' -881 `        ' -801 `         '-721
/|\    ` D# '   /|\    ` A# '   /|\    ` E# '   /|\    `B# '    /|\   ` Fx '    /|\   ` Cx '    /|\   `  Gx '   /
/ | \   25/21   / | \   25/14   / | \   75/56   / | \  225/224  / | \  675/448  / | \ 2025/1792 / | \ 6075/3584 /
/  |  \   +76   /  |  \  +156   /  |  \  +236   /  |  \  +316   /  |  \  +396   /  |  \  +476   /  |  \  +556   /
/  Cv   \  /|\  /  Gv   \  /|\  /  Dv   \  /|\  /  Av   \  /|\  /  Ev   \  /|\  /  Bv   \  /|\  / F#v   \  /|   /
/  35/18 -------- 35/24 --------- 35/32 --------- 105/64 -------- 315/256 ------  945/512 ------ 2835/2048\/ |  /
/  '-1998` /\ | /\'-1918 ` /\ | /\'-1837 ` /\ | /\'-1757 ` /\ | /\'-1677 ` /\ | /\'-1597 ` /\ | /\'-1517 ` /\ | /
/ '   / \  /` \|/' \  / \  /` \|/' \  / \  /` \|/' \  / \  /` \|/' \  / \  /` \|/' \  / \  /` \|/' \  /    /` \|/
D-    /   \/    A-   \/   \/    E-   \/   \/    B-   \/   \/   F#-   \/   \/   C#-   \/   \/   G#-   \/    /   D#-
10/9 --/-----\-- 5/3 --/-----\-- 5/4 --/-----\- 15/8 --/-----\- 45/32 -/-----\ 135/128 /-----\ 405/256 /----- 1215/1024
-721 `/    /  \'-641 `/  \ /  \'-561 `/  \ /  \'-481 `/  \ /  \'-400 `/  \ /  \'-320 `/  \ /  \'-240 `/  \ /   '-160
/|\ / `  B+ ' \ /|\ / ` F#+ ' \ /|\ / ` C#+ ' \ /|\ / ` G#+ ' \ /|\ / ` D#+ ' \ /|\ / ` A#+ ' \ /|\ / ` E#+ '   /
/ | /   40/21 --\-|-/-- 10/7 ---\-|-/-- 15/14 --\-|-/-- 45/28 --\-|-/- 135/112 -\-|-/- 405/224 -\-|-/ 1215/896  /
/  |/ \  +636   / \|/ \  +716   / \|/ \  +796   / \|/ \  +876   / \|/ \  +957   / \|/ \  +1037  / \|/ \ +1117   /
/  Ab<  \  /|\  /  Eb<  \  /|\  /   Bb< \  /|\  /   F<  \  /|\  /   C<  \  /|\  /   G<  \  /|\  /   D<  \  /|   /
/   14/9 ---------- 7/6 ----------- 7/4 ---------- 21/16 --------- 63/32 -------- 189/128 -------- 567/512\/ |  /
/   -1437  /\ | /\ -1357   /\ | /\ -1277   /\ | /\  -1197  /\ | /\  -1117  /\ | /\  -1037  /\ | /\  -957   /\ | /
/  '  / \ `/  \|/  '  / \` /  \|/  '  / \` /  \|/  '  / \ `/  \|/  '  / \ `/  \|/  '  / \ `/  \|/  '  /  ` /  \|/
Bb '   /   \/ `  F '  \/   \/ `  C '  \/   \/ `  G '  \/   \/ `  D '  \/   \/ `  A '  \/   \/  ` E '  \/    / `  B
16/9 --/-----\-- 4/3 --/-----\-- 1/1 --/-----\-- 3/2 --/-----\-- 9/8 --/-----\- 27/16 -/-----\- 81/64 -/------ 243/128
-160.  /    /  \'-80 ` /  \ /  \'  0 ` /  \ /  \' +80 `/  \ /  \'+160 `/  \ /  \'+240 `/  \ /  \'+320 `/  \ /   '+400
/|\ /    G> ' \ /|\ / `  D> ' \ /|\ / ` A> '  \ /|\ / `  E> ' \ /|\ /  ` B> ' \ /|\ / ` F#> ' \ /|\ / `  C#>'   /
/ | / ' 32/21 --\-|-/--- 8/7 ---\-|-/-- 12/7 ---\-|-/--- 9/7 ---\-|-/-- 27/14 --\-|-/-- 81/56 --\-|-/- 243/224  /
/  |/ \  -1197  / \|/ \ +1277   / \|/ \ +1357   / \|/ \  +1437  / \|/ \  +1517  / \|/ \ +1597   / \|/ \   +1677 /
/  Fb-  \   |   /  Cb-  \   |   /  Gb-  \   |   /  Db-  \   |   /  Ab-  \   |   /  Eb-  \   |   /  Bb-  \   |   /
/  56/45 --------- 28/15 ---------- 7/5 ---------- 21/20 --------- 63/40 -------- 189/160 ------- 567/320 \  |  /
/   -876    \ | /   -796    \ | /   -716    \ | /   -636    \ | /   -556    \ | /   -476    \ | /   -396    \ | /
/   '     `   \|/   '     `   \|/   '     `   \|/   '     `   \|/   '     `   \|/   '     `   \|/   '     `   \|/
Gb+ '         ` Db+ '         ` Ab+ '         ` Eb+ '         ` Bb+ '          ` F+ '         `  C+ '         `  G+
64/45 --------- 16/15 ---------- 8/5 ----------- 6/5 ----------- 9/5 ---------- 27/20 --------- 81/80 -------- 243/160
+400            +481            +561            +641            +721            +801            +881            +961

```

Updated: 2002.1.5

 For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here or try some definitions.