144-EDO

An abbreviation coined by Dan Stearns for the tuning determined by 144 equal divisions of the "octave". (see also EDO)

144-EDO has been advocated by Stearns and I for notational purposes, as an easily-grasped representation of the virtual pitch continuum. Our adaptation of "accidental" symbols is based on the 72edo notation created by Ezra Sims, and my 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. I cover only one semitone, as it can be reproduced similarly for the other 11.

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 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 Dan Stearns and I use 144-EDO to represent pitches which we expect to actually be tuned slightly differently, usually JI in my case, and poly-EDOs in Dan's case.

(An example of my use of 144-EDO to represent JI pitches which fall within the ranges given below, can be found in my 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 12 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 '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

 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).

[from Joe Monzo, JustMusic: A New Harmony]

---

updated: 2002.2.11

(to download a zip file of the entire Dictionary, click here)

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

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