# 144-tone equal-temperament / 144-edo

The tuning determined by 144 logarithmically equal divisions of the "octave". (see also EDO)

144-edo has been advocated by Dan Stearns and Joe Monzo for notational purposes, as an easily-grasped representation of the virtual pitch continuum. Their adaptation of "accidental" symbols is based on the 72-edo notation created by Ezra Sims, and on Monzo's ASCII adaptation of that, called HEWM.

Below is a table of the range of the virtual pitch continuum which can be represented by each 144-edo note. Only one semitone is shown, as the data can be reproduced similarly for the other 11 semitones of 12-edo. The table gives the absolute Semitone ranges (i.e., cents as a fraction of semitones), the nearest 12-edo degree and its cents deviation (i.e., to relate 144-edo to Johnny Reinhard's 1200-edo notation), and MIDI pitch-bend values (in 12mus) for the deviation from 12-edo.

This emphasizes the use of 144-edo as a notation, rather than as an actual tuning. (The latter would give the precise cent or pitch-bend value of each 144-edo degree, which I give further below.) This is how Stearns and Monzo use 144-edo to represent pitches which they expect to actually be tuned slightly differently, usually just-intonation in Monzo's case, and poly-edos in Stearn's case. (An example of Monzo's use of 144-edo to represent just-intonation pitches which fall within the ranges given below, can be found in his score to A Noiseless Patient Spider.)

```
nearest  Semitones   MIDI
12-edo   deviation   pitch-bend
Semitones  degree  from 12-edo  (cawapus)

1.04     1      + 0.04    +  171
12  C#          <				        >
0.96     1      - 0.04    -  171
11  C#~-        <				        >
0.88     1      - 0.13    -  512
10  C#-         <				        >
0.79     1      - 0.21    -  853
9  C#~<        <				        >
0.71     1      - 0.29    - 1195
8  C#<         <				        >
0.63     1      - 0.38    - 1536
7  C#~v        <				        >
0.54     1      - 0.46    - 1877
6  C^ or C#v   <				        >
0.46     0      + 0.46    + 1877
5  C~^         <				        >
0.38     0      + 0.38    + 1536
4  C>          <				        >
0.29     0      + 0.29    + 1195
3  C~>         <				        >
0.21     0      + 0.21    +  853
2  C+          <				        >
0.13     0      + 0.13    +  512
1  C~+         <				        >
0.04     0      + 0.04    +  171
0  C           <				        >
-0.04     0      - 0.04    -  171

```

144-edo is calculated by taking the 144th root of each successive power of 2, from 0 to 143, with higher or lower "octaves" of these 144 notes assumed to be, and tuned as, equivalents. The basic step-size is 8 & 1/3 cents.

Below is an overview of the complete 144-edo scale as a tuning. Notes which are presented horizontally adjacent are enharmonically equivalent.

```
144-edo Monzo ASCII adaptation of Stearns 'crosshatch-Sims' notation

9 different symbols in addition to the letter-names:

+        >       ^     #
-        <       v     b
16&2/3Â¢  33&1/3Â¢   50Â¢   100Â¢
1/12      1/6     1/4    1/2   of a 12-edo "tone"

and ~	used in connection with all 8 as a 'mitigator'
of their actions, reduces the 'adjustment'
of the main symbol by 8&1/3Â¢ (= 1/24-tone)

DE-  SEMI-  /-------------- NOTATION ------------------\
GREE TONES

0   0.00  C
-----------------------------------------------------------
143  11.92  C-~
-----------------------------------------------------------
142  11.83  C-
-----------------------------------------------------------
141  11.75  C<~
-----------------------------------------------------------
140  11.67  C<
-----------------------------------------------------------
139  11.58  Cv~
-----------------------------------------------------------
138  11.50  Cv  B^
-----------------------------------------------------------
137  11.42      B^~
-----------------------------------------------------------
136  11.33      B>
-----------------------------------------------------------
135  11.25      B>~
-----------------------------------------------------------
134  11.17      B+
-----------------------------------------------------------
133  11.08      B+~
-----------------------------------------------------------
132  11.00      B
===========================================================
131  10.92      B-~
-----------------------------------------------------------
130  10.83      B-
-----------------------------------------------------------
129  10.75      B<~
-----------------------------------------------------------
128  10.67      B<
-----------------------------------------------------------
127  10.58      Bv~
-----------------------------------------------------------
126  10.50      Bv (Bb^  A#^)
-----------------------------------------------------------
125  10.42          Bb^~ A#^~
-----------------------------------------------------------
124  10.33          Bb>  A#>
-----------------------------------------------------------
123  10.25          Bb>~ A#>~
-----------------------------------------------------------
122  10.17          Bb+  A#+
-----------------------------------------------------------
121  10.08          Bb+~ A#+~
-----------------------------------------------------------
120  10.00          Bb   A#
===========================================================
119   9.92          Bb-~ A#-~
-----------------------------------------------------------
118   9.83          Bb-  A#-
-----------------------------------------------------------
117   9.75          Bb<~ A#<~
-----------------------------------------------------------
116   9.67          Bb<  A#<
-----------------------------------------------------------
115   9.58          Bbv~ A#V~
-----------------------------------------------------------
114   9.50         (Bbv  A#v)  A^
-----------------------------------------------------------
113   9.42                     A^~
-----------------------------------------------------------
112   9.33                     A>
-----------------------------------------------------------
111   9.25                     A>~
-----------------------------------------------------------
110   9.17                     A+
-----------------------------------------------------------
109   9.08                     A+~
-----------------------------------------------------------
108   9.00                     A
===========================================================
107   8.92                     A-~
-----------------------------------------------------------
106   8.83                     A-
-----------------------------------------------------------
105   8.75                     A<~
-----------------------------------------------------------
104   8.67                     A<
-----------------------------------------------------------
103   8.58                     Av~
-----------------------------------------------------------
102   8.50                     Av (Ab^  G#^)
-----------------------------------------------------------
101   8.42                         Ab^~ G#^~
-----------------------------------------------------------
100   8.33                         Ab>  G#>
-----------------------------------------------------------
99   8.25                         Ab>~ G#>~
-----------------------------------------------------------
98   8.17                         Ab+  G#+
-----------------------------------------------------------
97   8.08                         Ab+~ G#+~
-----------------------------------------------------------
96   8.00                         Ab   G#
===========================================================
95   7.92                         Ab-~ G#-~
-----------------------------------------------------------
94   7.83                         Ab-  G#-
-----------------------------------------------------------
93   7.75                         Ab<~ G#<~
-----------------------------------------------------------
92   7.67                         Ab<  G#<
-----------------------------------------------------------
91   7.58                         Abv~ G#v~
-----------------------------------------------------------
90   7.50                        (Abv  G#v) G^
-----------------------------------------------------------
89   7.42                                   G^~
-----------------------------------------------------------
88   7.33                                   G>
-----------------------------------------------------------
87   7.25                                   G>~
-----------------------------------------------------------
86   7.17                                   G+
-----------------------------------------------------------
85   7.08                                   G+~
-----------------------------------------------------------
84   7.00                                   G
===========================================================
83   6.92                                   G-~
-----------------------------------------------------------
82   6.83                                   G-
-----------------------------------------------------------
81   6.75                                   G<~
-----------------------------------------------------------
80   6.67                                   G<
-----------------------------------------------------------
79   6.58                                   Gv~
-----------------------------------------------------------
78   6.50                             (F#^) Gv  (Gb^)
-----------------------------------------------------------
77   6.42                              F#^~      Gb^~
-----------------------------------------------------------
76   6.33                              F#>       Gb>
-----------------------------------------------------------
75   6.25                              F#>~      Gb>~
-----------------------------------------------------------
74   6.17                              F#+       Gb+
-----------------------------------------------------------
73   6.08                              F#+~      Gb+~
-----------------------------------------------------------
72   6.00                              F#        Gb
===========================================================
71   5.92                              F#-~      Gb-~
-----------------------------------------------------------
70   5.83                              F#-       Gb-
-----------------------------------------------------------
69   5.75                              F#<~      Gb<~
-----------------------------------------------------------
68   5.67                              F#<       Gb<
-----------------------------------------------------------
67   5.58                              F#v~      Gbv~
-----------------------------------------------------------
66   5.50                             (F#v) F   (Gbv)
-----------------------------------------------------------
65   5.42                                   F^~
-----------------------------------------------------------
64   5.33                                   F>
-----------------------------------------------------------
63   5.25                                   F>~
-----------------------------------------------------------
62   5.17                                   F+
-----------------------------------------------------------
61   5.08                                   F+~
-----------------------------------------------------------
60   5.00                                   F
===========================================================
59   4.92                                   F-~
-----------------------------------------------------------
58   4.83                                   F-
-----------------------------------------------------------
57   4.75                                   F<~
-----------------------------------------------------------
56   4.67                                   F<
-----------------------------------------------------------
55   4.58                                   Fv~
-----------------------------------------------------------
54   4.50                              E^   Fv
-----------------------------------------------------------
53   4.42                              E^~
-----------------------------------------------------------
52   4.33                              E>
-----------------------------------------------------------
51   4.25                              E>~
-----------------------------------------------------------
50   4.17                              E+
-----------------------------------------------------------
49   4.08                              E+~
-----------------------------------------------------------
48   4.00                              E
===========================================================
47   3.92                              E-~
-----------------------------------------------------------
46   3.83                              E-
-----------------------------------------------------------
45   3.75                              E<~
-----------------------------------------------------------
44   3.67                              E<
-----------------------------------------------------------
43   3.58                              Ev~
-----------------------------------------------------------
42   3.50                   (D#^  Eb^) Ev
-----------------------------------------------------------
41   3.42                    D#^~ Eb^~
-----------------------------------------------------------
40   3.33                    D#>  Eb>
-----------------------------------------------------------
39   3.25                    D#>~ Eb>~
-----------------------------------------------------------
38   3.17                    D#+  Eb+
-----------------------------------------------------------
37   3.08                    D#+~ Eb+~
-----------------------------------------------------------
36   3.00                    D#   Eb
===========================================================
35   2.92                    D#-~ Eb-~
-----------------------------------------------------------
34   2.83                    D#-  Eb-
-----------------------------------------------------------
33   2.75                    D#<~ Eb<~
-----------------------------------------------------------
32   2.67                    D#<  Eb<
-----------------------------------------------------------
31   2.58                    D#v~ Ebv~
-----------------------------------------------------------
30   2.50                D^ (D#v  Ebv)
-----------------------------------------------------------
29   2.42                D^~
-----------------------------------------------------------
28   2.33                D>
-----------------------------------------------------------
27   2.25                D>~
-----------------------------------------------------------
26   2.17                D+
-----------------------------------------------------------
25   2.08                D+~
-----------------------------------------------------------
24   2.00                D
===========================================================
23   1.92                D-~
-----------------------------------------------------------
22   1.83                D-
-----------------------------------------------------------
21   1.75                D<~
-----------------------------------------------------------
20   1.67                D<
-----------------------------------------------------------
19   1.58                Dv~
-----------------------------------------------------------
18   1.50     (C#^  Db^) Dv
-----------------------------------------------------------
17   1.42      C#^~ Db^~
-----------------------------------------------------------
16   1.33      C#>  Db>
-----------------------------------------------------------
15   1.25      C#>~ Db>~
-----------------------------------------------------------
14   1.17      C#+  Db+
-----------------------------------------------------------
13   1.08      C#+~ Dv+~
-----------------------------------------------------------
12   1.00      C#   Db
===========================================================
11   0.92      C#-~ Db-~
-----------------------------------------------------------
10   0.83      C#-  Db-
-----------------------------------------------------------
9   0.75      C#<~ Db<~
-----------------------------------------------------------
8   0.67      C#<  Db<
-----------------------------------------------------------
7   0.58      C#v~ Dbv~
-----------------------------------------------------------
6   0.50  C^ (C#v  Dbv)
-----------------------------------------------------------
5   0.42  C^~
-----------------------------------------------------------
4   0.33  C>
-----------------------------------------------------------
3   0.25  C>~
-----------------------------------------------------------
2   0.17  C+
-----------------------------------------------------------
1   0.08  C+~
-----------------------------------------------------------
0   0.00  C
===========================================================

index of enharmonicity =  1.395833...

= 201 different symbols
for 144 discrete pitches per 'octave'

using simplest possible notation for any degree
enharmonics only where necessary
- thus excluding those in ()

144-edo = (2^4)*(3^2) degrees

all possibilities of the matrix:

2^ 3^   # of degrees:

|1 0| = 2
|2 0| = 4
|3 0| = 8
|4 0| = 16

|0 1| = 3
|1 1| = 6
|2 1| = 12
|3 1| = 24
|4 1| = 48

|0 2| = 9
|1 2| = 18
|2 2| = 36
|3 2| = 72
|4 2| = 144

arranged in order of number of degrees:

|1 0| = 2     tritone
|0 1| = 3     augmented triad
|2 0| = 4     diminished 7th tetrad
|1 1| = 6     whole tones
|3 0| = 8
|0 2| = 9
|2 1| = 12    semitones
|4 0| = 16
|1 2| = 18    third-tones
|3 1| = 24    quarter-tones
|2 2| = 36    sixth-tones
|4 1| = 48    eighth-tones
|3 2| = 72    twelfth-tones
|4 2| = 144

144-edo contains all these divisions within it.

in addition, because 11 * 13 = 143,
144-eq approximates 11-edo and 13-edo extremely well:

11-edo:

2^( 1/11)  = ~109.09Â¢
2^(13/144) = ~108.33Â¢

2^(1/11) / 2^(13/144) = ~0.76Â¢ = ~3/4Â¢  difference

13-edo:

2^( 1/13)  = ~92.31Â¢  = ~92&1/3Â¢
2^(11/144) = ~91.67Â¢  = ~91&2/3Â¢

2^(1/13) / 2^(11/144) = ~0.64Â¢ = ~2/3Â¢  difference

summary of resources:

13 smaller systems into which 144-edo divides exactly
2 into which it divides almost exactly

```

Note also that in addition to the regular 12edo "5th" of 700 cents, 144edo also provides a quasi-meantone-type "5th" of ~6912/3 cents. This is very close to the "5th" sizes of 33edo (= ~69010/11 cents) and 1/2-comma meantone (= ~6911/5 cents), which are the bottom limit for meantone-like systems (because the note functioning as a "tone", the ii = 2nd degree of the diatonic scale, is tempered a full comma flat so that it is already exactly or nearly exactly the ratio 10/9 instead of 9/8, and thus leaves no room for a "mean" tone).

Note also that 144-edo provides a very likely explanation for the tetrachord divisions of the ancient Greek music-theorist Aristoxenus.

[Joe Monzo]
. . . . . . . . .

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