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&2/3¢ 33&1/3¢ 50¢ 100¢
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&1/3¢ (= 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 ()
readily divided into smaller systems:
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) = ~109.09¢
2^(13/144) = ~108.33¢
2^(1/11) / 2^(13/144) = ~0.76¢ = ~3/4¢ difference
13-edo:
2^( 1/13) = ~92.31¢ = ~92&1/3¢
2^(11/144) = ~91.67¢ = ~91&2/3¢
2^(1/13) / 2^(11/144) = ~0.64¢ = ~2/3¢ 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.
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