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EDO prime error

[Joe Monzo]

On this page I present a detailed examination of the amount of absolute error ("absolute" in the sense of ignoring whether the error is positive or negative) of the prime-mapping of several various EDOs of all prime-factors from 3 to 43 (an arbitrarily selected stopping point), using the best approximation of each prime. Note that a composer may deliberately choose to use an EDO as a temperament of JI but employing approximations which are not the closest, but this will not be analyzed here. Also note that prime-factor 2 (the octave) always has zero error, as opposed to TOP tunings where 2 may also be tempered.

Different EDOs approximate JI intervals in different ways. In these listings, errors are given both in cents (as an absolute error measurement) and as a percentage of one degree or "step" of that EDO (as a relative error measurement, that is, relative to that EDO). These latter values are shown on the graphs. Thus, the maximum error on the graph is 50%, which would indicate that the prime-factor lies midway between two neighboring EDO degrees.

Error values tend to cluster into groups -- this clustering is plainly visible on the graphs, and I have also divided the lists accordingly, putting a blank row between clusters.

One of the primary objectives for creating this analysis was to attempt to quantify Ivor Darreg's theory of EDO "moods", wherein Darreg stated that there are no "bad" equal temperaments, each one just has its own particular "mood". Also see Darreg's It Is Time To Release Composers From Hidden Restraints. The analysis here correlates Darreg's theory with the theory of prime-affect.

5edo

5edo is the smallest cardinality which is generally perceived as a scale, a type of pentatonic scale. The only prime-factors it represents decently are 3, 7, and 37 -- so it is possible to use it as an approximation to a "no 5s" 7-limit JI, altho the direction of errors for 3 and 7 are opposite.

5edo best representation of prime-factor

prime    edo    ~cents  % EDO-step
       degrees   error      error

   2      5        0          0       

   7     14       -8.8       -4
  37     26      -11.3       -5
   3      8      +18.0       +8
  43     27      -31.5      +13

  41     27      +50.9      +21
  31     25      +55.0      +23
  19     21      -57.5      -24
  29     24      -69.6      -29
  11     17      -71.3      -30
  23     23      +91.7      +38
   5     12      +93.7      +39
  17     20     -105.0      -44
  13     19     +119.5      +50
							
6edo

6edo is the familiar "whole-tone scale" derived from 12edo. Of course, several of its best representations of prime-factors in 43-limit JI are exactly the same as those in 12edo: 5, 7, 11, 13, 23, 29, and 41, which is to say, they are all just as bad as in 12edo, and all of them have less than 25% relative error in 6edo.

6edo best representation of prime-factor

prime    edo    ~cents   % EDO-step
       degrees   error      error

   2      6       0         0

   5     14     +13.7      +7

  23     27     -28.3     -14
  41     32     -29.1     -15
  29     29     -29.6     -15
   7     17     +31.2     +16

  13     22     -40.5     -20

  11     21     +48.7     +24
  37     31     -51.3     -26
  31     30     +55.0     +27

  43     33     +88.5     +44
  17     25     +95.0     +48
  19     25     -97.5     -49
   3     10     +98.0     +49

							
7edo

7edo is the smallest cardinality that can give the impression of a diatonic scale, altho it is lacking the important distinction of the two different step-sizes of all diatonic scales. It is also the smallest cardinality that can be described as a form of meantone. It represents 29 and 43 very well, with relative errors of only 1% and 2% respectively. The only two of the lowest prime-factors which it represents well are 3 and 13. All others in the 43-limit are mediocre and worse.

7edo best representation of prime-factor

prime    edo    ~cents  % EDO-step
       degrees   error      error

   2      7       0          0

  29     34      -1.0       -1
  43     38      +2.8       +2

   3     11     -16.2       -9
  13     26     +16.6      +10

  11     24     -37.0      -22
   5     16     -43.5      -25
  19     30     +45.3      +26

  31     35     +55.0      +32
  23     32     +57.4      +34
   7     20     +59.7      +35
  17     29     +66.5      +39

  37     36     -79.9      -47
  41     38     +85.2      +50

							
8edo

8edo in the 43-limit represents well only prime-factor 9. It has fair representations of 23, 29, and 41, and mediocre and worse for all other primes in this limit. None of the low primes (3, 5, 7, 11, 13, 17) are represented well.

8edo best representation of prime-factor

prime    edo    ~cents  % EDO-step
       degrees   error      error

   2      8       0          0

  19     34      +2.5       +2

  29     39     +20.4      +14
  41     43     +20.9      +14
  23     36     -28.3      -19

  17     33     +45.0      +30
   3     13     +48.0      +32
  11     28     +48.68     +32
  37     42     +48.66     +32

  31     40     +55.0      +37
  13     30     +59.5      +40
  43     43     -61.5      -41
   5     19     +63.7      +42
   7     22     -68.8      -46
  
							
9edo

9edo represents no prime-factors in the 43-limit with less than 10% relative error. Its best representations are of 5, 11, 37, and 43, all under 20%, with another group of 17, 19, and 41 all near 20%, and another group of 3, 7, 13, 23, and 29 all just under 30%, and only 31 being quite poor at 41%. So interestingly, except for 31, its representations of all prime-factors in the 43-limit have relative errors which fall between 10% and 30%.

9edo best representation of prime-factor

prime    edo    ~cents  % EDO-step
       degrees   error      error

   2      9       0         0

   5     21     +13.7     +10
  37     47     +15.3     +11
  11     31     -18.0     -13
  43     49     +21.8     +16

  17     37     +28.4     +21
  41     48     -29.1     -22
  19     38     -30.8     -23

   3     14     -35.3     -26
   7     25     -35.5     -27
  29     44     +37.1     +28
  23     41     +38.4     +29
  13     33     -40.5     -30

  31     45     +55.0     +41
  
							
10edo

10edo gives a suberb representation of prime-factor 13 (only ~ -0.5 cent error), and fair-to-mediocre representations of 7 (~ -9 cents error), 37, 17, and 3. It does quite poorly with 5, 11, and 19.

10edo best representation of prime-factor

prime     edo     ~cents  % EDO-step
        degrees    error      error

   2      10        0         0

  13      37       -0.5      -0

   7      28       -8.8      -7
  37      52      -11.3      -9
  17      41      +15.0     +13
   3      16      +18.0     +15

   5      23      -26.3     -22
  23      45      -28.3     -24
  43      54      -31.5     -26

  11      35      +48.7     +41
  29      49      +50.4     +42
  41      54      +50.9     +42
  31      50      +55.0     +46
  19      42      -57.5     -48

							
11edo

11edo gives fairly good representations of prime-factors 17 and 11 (~ +4 and -6 cents error, respectively) but does a terrible job with 3 and 5 (~ -47 and +50 cents error).

    11edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 11 0 0 17 45 +4.1 +4 11 38 -5.9 -5 41 59 +7.3 +7 7 31 +13.0 +12 23 50 +26.3 +24 19 47 +29.8 +27 13 41 +32.2 +30 37 57 -33.2 -30 43 60 +33.9 +31 3 17 -47.4 -43 29 53 -47.8 -44 5 26 +50.0 +46 31 54 -54.1 -50
12edo


12edo gives excellent representations of the Pythagorean prime-factor 3 (~ -2 cents error), and of 19 (~ +2.5 cents error) and 17 (~ -5 cents error).

12edo's best representations of 5 and 43 are mediocre, those of 7, 13, 23 and 29 quite poor, and those of 11, 31, and 37 terrible, 11 and 37 lying basically midway between two 12edo degrees (50% EDO-step error is the exact midpoint).

12edo is the smallest cardinality which gives representations of 3 and 5 with low enough error to be perceivable as an approximation to JI, and at the same time gives the distinction of the two step-sizes necessary for the diatonic scales. Thus, largely because of its practicality, over the course of four centuries (c. 1600-2000) it became strongly entrenched in Western music.

    12edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 12 0 0 3 19 -2.0 -2 19 51 +2.5 +2 17 49 -5.0 -5 43 65 -11.5 -12 5 28 +13.7 +14 23 54 -28.3 -28 41 64 -29.1 -29 29 58 -29.6 -30 7 34 +31.2 +31 13 44 -40.5 -41 31 59 -45.0 -45 37 63 +48.66 +49 11 42 +48.68 +49


13edo

    13edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 13 0 0 11 45 +2.5 +3 13 48 -9.8 -11 17 53 -12.6 -14 29 63 -14.2 -15 5 30 -17.1 -19 23 59 +17.9 +19 19 55 -20.6 -22 37 68 +25.6 +28 41 70 +32.5 +35 3 21 +36.5 +40 31 64 -37.3 -40 43 71 +42.3 +46 7 36 -45.7 -50


14edo

    14edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 14 0 0 41 75 -0.5 -1 29 68 -1.0 -1 43 76 +2.8 +3 37 73 +5.8 +7 3 22 -16.2 -19 13 52 +16.6 +19 17 57 -19.2 -22 7 39 -26.0 -30 23 63 -28.3 -33 31 69 -30.7 -36 11 48 -37.0 -43 19 59 -40.4 -47 5 33 +42.3 +49


15edo

15edo gives fair representations of prime-factors 5, 7, and 11, (as well as higher primes 23, 29, and 37) and a mediocre representation of 3, that is fairly wide at exactly 720 cents. The approximation of 5 is exactly the same one familiar from 12edo. So 15edo could be used as a fair approximation of "no-3s" 11-limit JI, and possibly to some degree even of full 11-limit JI. Note that 15edo does not represent any of the primes in the 43-limit with a relative error of less than 11%.

    15edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 15 0 0 11 52 +8.7 +11 7 42 -8.8 -11 29 73 +10.4 +13 37 78 -11.3 -14 23 68 +11.7 +15 5 35 +13.7 +17 3 24 +18.0 +23 19 64 +22.5 +28 17 61 -24.96 -31 31 74 -25.04 -31 41 80 -29.1 -36 43 81 -31.5 -39 13 56 +39.5 +49


16edo

    16edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 16 0 0 19 68 +2.5 +3 7 45 +6.2 +8 5 37 -11.3 -15 43 87 +13.5 +18 13 59 -15.5 -21 31 79 -20.0 -27 29 78 +20.4 +27 41 86 +20.9 +28 11 55 -26.32 -35 37 83 -26.34 -35 3 25 -27.0 -36 23 72 -28.3 -38 17 65 -30.0 -40


17edo

Because of the fact that the 5-limit "3rds" (major-3rd>major and minor-3rd) are both represented by 5 degrees of 17edo, which is really a "neutral 3rd", it cannot emulate 5-limit tuning well, but rather it functions primarily as a pythagorean tuning, with 4 and 6 degrees being good representations of [-3 0] (i.e., 3-3 = ratio 32:27) and [4 0] (= 34 = ratio 81:64), respectively -- the Pythagorean minor-3rd or "hemiditone", and pythagorean major-3rd> or ditone.

Besides giving a very good representation of the Pythagorean prime-factor 3 (~ +4 cents error), 17edo also gives decent representations of 13 and 23.

    17edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 17 0 0 3 27 +3.9 +6 41 91 -5.5 -8 13 63 +6.5 +9 23 77 +7.0 +10 11 59 +13.4 +19 19 72 -15.2 -21 31 84 -15.6 -22 43 92 -17.4 -25 7 48 +19.4 +27 29 83 +29.2 +41 37 89 +31.0 +44 5 39 -33.4 -47 17 69 -34.4 -49


18edo

None of 18edo's representations of lower primes is very good, the best error being about 1/5 of an 18edo degree, and most of the lower primes falling nearly midway between two neighboring 18edo degrees.

    18edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 18 0 0 31 89 -11.7 -18 5 42 +13.7 +21 37 94 +15.3 +23 11 62 -18.0 -27 43 98 +21.8 +33 13 67 +26.1 +39 23 81 -28.3 -42 17 74 +28.4 +43 41 96 -29.1 -44 29 87 -29.6 -44 19 76 -30.8 -46 7 51 +31.2 +47 3 29 +31.4 +47


19edo

19edo, the next-simplest EDO-meantone after 12edo, does a fairly good job of representing prime-factors 3 and 5, and is mediocre with 7 and 11, but none of the lower primes really has a terrible representation, all primes up to 43 having less than 35% error. Thus, in addition to rendering the entire common-practice repertoire recognizably, it also provides a simple tuning which is fruitful for exploring approximate-JI all the way up to 43-limit.

    19edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 19 0 0 37 99 +1.3 +2 23 86 +3.3 +5 43 103 -6.3 -10 3 30 -7.2 -11 5 44 -7.4 -12 31 94 -8.2 -13 41 102 +13.0 +21 11 66 +17.1 +27 19 81 +18.3 +29 29 92 -19.1 -30 13 70 -19.5 -31 17 78 +21.4 +34 7 53 -21.5 -34


20edo

    20edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 20 0 0 13 74 -0.5 -1 19 85 +2.5 +4 31 99 -5.0 -8 7 56 -8.8 -15 41 107 -9.1 -15 29 97 -9.6 -16 11 69 -11.32 -19 37 104 -11.34 -19 17 82 +15.0 +25 3 32 +18.0 +30 5 46 -26.3 -44 23 90 -28.3 -47 43 109 +28.5 +47


21edo

    21edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 21 0 0 23 95 +0.3 +1 29 102 -1.0 -2 31 104 -2.2 -4 7 59 +2.6 +5 43 114 +2.8 +5 17 86 +9.3 +16 19 89 -11.8 -21 5 49 +13.7 +24 3 33 -16.2 -28 13 78 +16.6 +29 11 73 +20.1 +35 37 109 -22.8 -40 41 113 +28.1 +49


22edo

22edo does a fairly good job of representing prime-factors 3, 5, and 11, and is mediocre with 7:

    22edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 22 0 0 31 109 +0.4 +1 17 90 +4.1 +8 5 51 -4.5 -8 11 76 -5.9 -11 29 107 +6.8 +12 3 35 +7.1 +13 41 118 +7.3 +13 7 62 +13.0 +24 43 119 -20.6 -38 37 115 +21.4 +39 13 81 -22.3 -41 19 93 -24.8 -45 23 100 +26.3 +48


23edo

23edo is generally considered to be a very strange EDO. As can easily be seen on the graph, its worst representations in the 43-limit are the four lowest primes 3, 5, 7, and 11 -- in theory, exactly the ones generally most desired.

    23edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 23 0 0 17 94 -0.6 -1 23 104 -2.2 -4 31 114 +2.8 +5 13 85 -5.7 -11 37 120 +9.5 +18 43 125 +10.2 +20 41 123 -11.7 -22 29 112 +13.9 +27 19 98 +15.5 +30 5 53 -21.1 -40 7 65 +22.5 +43 11 80 +22.6 +43 3 36 -23.7 -45


24edo

The famous "quarter-tone" system. Being simply a halving of the step-size of 12edo, this tuning gives some of the exact same representations to JI as 12-EDO: 3, 5, 17, 19, and 43. It gives a significantly better approximation of 7, but in the opposite direction (that is, 24-edo's best representation of 7 is smaller than the prime-factor, whereas its second-best is exactly the same as 12edo and is larger), therefore, because its representation of 5 is larger than the prime, the intervals do not sound close to 7-limit JI. However, 24edo gives significantly better representations of prime-factors 11, 13, 31, and 37.

    24edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 24 0 0 11 83 -1.32 -3 37 125 -1.34 -3 3 38 -2.0 -4 19 102 +2.5 +5 17 98 -4.955 -10 31 119 +4.964 +10 13 89 +9.5 +19 43 130 -11.5 -23 5 56 +13.7 +27 7 67 -18.8 -38 29 117 +20.4 +41 41 129 +20.9 +42 23 109 +21.7 +43


25edo

    25edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 25 0 0 5 58 -2.3 -5 41 134 +2.9 +6 23 113 -4.3 -9 31 124 +7.0 +15 7 70 -8.8 -18 17 102 -9.0 -19 19 106 -9.5 -20 37 130 -11.3 -24 43 136 +16.5 +34 3 40 +18.0 +38 29 121 -21.6 -45 11 86 -23.3 -49 13 93 +23.5 +49


26edo

    26edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 26 0 0 7 73 +0.4 +1 11 90 +2.5 +5 43 141 -3.8 -8 31 129 +8.8 +19 3 41 -9.6 -21 13 96 -9.8 -21 17 106 -12.6 -27 41 139 -13.7 -30 29 126 -14.2 -31 5 60 -17.1 -37 23 118 +17.9 +39 37 135 -20.57 -45 19 110 -20.59 -45


27edo

    27edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 27 0 0 13 100 +3.9 +9 23 122 -6.1 -14 29 131 -7.4 -17 7 76 +9.0 +20 3 43 +9.2 +21 31 134 +10.5 +24 19 115 +13.6 +31 5 63 +13.7 +31 37 141 +15.3 +34 41 145 +15.4 +35 17 110 -16.1 -36 11 93 -18.0 -40 43 147 +21.8 +49


28edo

    28edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 28 0 0 41 150 -0.5 -1 5 65 -0.6 -1 29 136 -1.0 -2 19 119 +2.5 +6 43 152 +2.8 +6 37 146 +5.799 +14 11 97 +5.825 +14 31 139 +12.1 +28 23 127 +14.6 +34 3 44 -16.2 -38 13 104 +16.6 +39 7 79 +16.9 +39 17 114 -19.2 -45


29edo

    29edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 29 0 0 3 46 +1.5 +4 37 151 -3.1 -7 29 141 +4.9 +12 23 131 -7.6 -18 19 123 -7.9 -19 13 107 -12.9 -31 11 100 -13.4 -32 31 144 +13.6 +33 5 67 -13.9 -34 43 157 -15.0 -36 41 155 -15.3 -37 7 81 -17.1 -41 17 119 +19.2 +46


30edo

30edo is very strange in the 43-limit. It gives a superb representation of only prime-factor 13; all other primes have more than 20% error, and their errors are fairly evenly distributed.

    30edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 30 0 0 13 111 -0.5 -1 43 163 +8.5 +21 11 104 +8.7 +22 7 84 -8.8 -22 29 146 +10.4 +26 41 161 +10.9 +27 37 156 -11.3 -28 23 136 +11.7 +29 5 70 +13.7 +34 31 149 +15.0 +37 17 123 +15.0 +38 19 127 -17.5 -44 3 48 +18.0 +45


31edo

31edo is historically important as a very good approximation to 1/4-comma meantone. Among the lowest primes, it gives excellent representations of prime-factors 5 (~ 3/4 cent error) and 7 (~ 1 cent error), a passable representation of 3, and mediocre representation of 11 and 23; all of these are smaller than the actual primes. Note that it also gives mediocre representations of 13, 17, and 19, but they are in the opposite direction, larger than the primes.

    31edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 31 0 0 5 72 +0.8 +2 7 87 -1.1 -3 41 166 -3.3 -8 3 49 -5.2 -13 43 168 -8.3 -21 23 140 -8.9 -23 11 107 -9.4 -24 13 115 +11.1 +29 17 127 +11.2 +29 19 132 +12.2 +31 29 151 +15.6 +40 31 154 +16.3 +42 37 161 -19.1 -49


36edo

Similarly to 24edo, 36edo has been advocated as a fairly easy means of exploring microtonality by simply dividing each degree of 12edo into 3 equal parts. Again, as with 24edo, some of its representations of 43-limit JI are exactly the same as those of 12edo: prime-factors 3, 5, 17, 19, and 43 -- thus, its approximation of 5-limit JI gives no advantage over 12edo. However, 36edo offers a superb representation of prime-factor 7, making it a much better approximation of 7-limit JI. And it provides significantly better representations than 12edo of prime-factors 13, 23, 29, and 41.

As a side note, not shown in this analysis, while 12deg36 is the best mapping of 5 (400 cents, same as 12edo), the second-best mapping of 5, to 11deg36, known as 36c, while giving an absolute error in cents larger than anything in the table below, and a relative error of 59%, puts the error of 5 in the same negative direction as with those of 3 and 7, and provides an alternate flavor for approximations of 5- and especially 7-limit JI.

    36edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 36 0 0 3 57 -2.0 -6 7 101 -2.2 -6 19 153 +2.5 +7 29 175 +3.8 +11 41 193 +4.3 +13 17 147 -5.0 -15 23 163 +5.1 +15 13 133 -7.2 -22 43 195 -11.5 -35 31 178 -11.7 -35 5 84 +13.7 +41 37 188 +15.32 +46 11 125 +15.35 +46


37edo

37edo is the first cardinality which gives essentially no error for a prime-mapping other than 2: the relative error for 11 is +0.1%. It also gives excellent representations of prime-factors 5, 7, and 13. It is thus a great approximation of "no 3s" 13-limit JI. Its representation of prime-factor 3 is quite mediocre, with a relative error of more than 1/3 of a step of 37edo.

    37edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 37 0 0 11 128 +0.03 +0.1 13 137 +2.7 +8 5 86 +2.9 +9 7 104 +4.15 +13 19 157 -5.6 -17 41 198 -7.44 -23 43 201 +7.40 +23 17 151 -7.7 -24 29 180 +8.3 +25 37 193 +8.1 +25 31 183 -9.9 -31 3 59 +11.6 +36 23 167 -12.1 -37


40edo

    40edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 40 0 0 13 148 -0.5 -2 43 217 -1.5 -5 23 181 +1.7 +6 19 170 +2.5 +8 5 93 +3.7 +12 31 198 -5.0 -17 7 112 -8.8 -29 41 214 -9.1 -30 29 194 -9.6 -32 11 138 -11.32 -38 37 208 -11.34 -38 3 63 -12.0 -40 17 163 -15.0 -50


41edo

41edo gives a superb representation of the Pythagorean prime-factor 3, and decent representations of 5, 7, and 11. It is thus a very good approximation of 11-limit JI, and also gives good representations of prime-factors 19, 29, and 31.

    41edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 41 0 0 3 65 +0.5 +2 7 115 -3.0 -10 31 203 -3.6 -12 11 142 +4.8 +16 19 174 -4.8 -17 29 199 -5.2 -18 5 95 -5.8 -20 13 152 +8.3 +28 41 220 +10.0 +34 37 214 +12.07 +41 17 168 +12.12 +41 23 185 -13.6 -47 43 222 -14.0 -48


43edo

43-EDO is historically important as an approximation of 1/5-comma meantone.
    43edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 43 0 0 37 224 -0.2 -1 31 213 -0.8 -3 29 209 +3.0 +11 13 159 -3.3 -12 3 68 -4.3 -15 5 100 +4.4 +16 17 176 +6.7 +24 11 149 +6.8 +24 7 121 +7.9 +28 43 233 -9.2 -33 19 183 +9.5 +34 41 230 -10.5 -37 23 195 +13.6 +49

see also: meride


46edo

46edo provides excellent approximations to basic intervals in the 11-limit, as can be seen on the equal-temperament gallery page, and also an excellent representation of 17 and good representations of 13, 23, and 31, as shown in the following table. All 17-limit factors have less than 6 cents absolute error, and the 13-limit factors cluster between 8% and 22% relative error.

    46edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 46 0 0 17 188 -0.6 -2 23 208 -2.2 -8 3 73 +2.4 +9 31 228 +2.8 +11 11 159 -3.5 -13 7 129 -3.6 -14 5 107 +5.0 +19 13 170 -5.7 -22 37 240 +9.5 +37 43 250 +10.2 +39 19 195 -10.6 -40 41 246 -11.7 -45 29 223 -12.2 -47


53edo

53edo is the next cardinality (after 37edo) which presents a negligible error in the 43-limit: it gives an outstanding representation of the Pythagorean prime-factor 3 (~ 1/15 cent error), with a relative error of only -0.3%. It also provides a very good representation of 5 (~ 1.4 cents error), and mediocre ones of 7 and 11. It is thus an extremely good approximation to pythagorean (3-limit) tuning, and a very good approximation to 5-limit JI, and has been advocated often (particularly in the late 1800s and early 1900s) as a means for expansion of the tonal palette beyond 12edo, and is still advocated in the early 21st century by many microtonalists as an "ultimate tuning".

    53edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 53 0 0 3 84 -0.1 -0.3 41 284 +1.1 +5 5 123 -1.4 -6 37 276 -2.3 -10 13 196 -2.8 -12 19 225 -3.2 -14 7 149 +4.8 +21 23 240 +5.7 +25 11 183 -7.9 -35 17 217 +8.3 +36 43 288 +9.2 +41 31 263 +9.7 +43 29 257 -10.7 -47


55edo

The striking thing to observe about 55edo is that, despite its status as a "standard" tuning during the meantone era (see my webpage on Mozart's Tuning), none of its lower-prime representations is especially good in terms of relative error: the best approximations are about 1/5 of a 55edo degree off.

    55edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 55 0 0 3 87 -3.8 -17 29 267 -4.12 -19 17 225 +4.14 +19 23 249 +4.5 +20 11 190 -5.9 -27 5 128 +6.4 +29 41 295 +7.3 +33 19 234 +7.9 +36 7 154 -8.8 -40 43 298 -9.7 -44 13 204 +10.38 +48 37 287 +10.47 +48 31 272 -10.49 -48


72edo

72edo gives a very good representation of 11-limit JI, with errors of prime-factors 3, 5, 7, and 11 (as well as 37 and 19) all under 20%. Also, because the errors of the four lowest prime-factors are all in the same direction (negative), intervals among these primes tend to sound very much like just intervals. 72edo also gives fairly good representations of many of the higher prime-factors, unfortunately giving a very poor approximation to 13.

The fact that 72edo's approximations do not temper out so many important 5-limit intervals means that, even tho it is not as accurate as the lower-cardinality <#53">53edo in representing the entire 5-limit lattice, its structure does emulate so many of the important lower-limit JI commas that in a systemic sense it is easily perceived as a good approximation of 11-limit JI.

    72edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 72 0 0 11 249 -1.32 -8 37 375 -1.34 -8 3 114 -1.96 -12 7 202 -2.16 -13 19 306 +2.49 +15 5 167 -2.98 -18 29 350 +3.76 +23 41 386 +4.27 +26 17 294 -4.96 -30 31 357 +4.96 +30 23 326 +5.06 +30 43 391 +5.15 +31 13 266 -7.19 -43


81edo

81-EDO is of interest as a very close approximation to golden-meantone. It gives a very good approximation of prime-factor 5 and some other higher primes (17, 19, 37, 41), and it represents 11 and 13 with middling accuracy, but as with many meantones its approximation of 3 is quite mediocre.

    81edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 81 0 0 37 422 +0.51 +3 41 434 +0.57 +4 5 188 -1.13 -8 19 344 -1.22 -8 17 331 -1.25 -8 11 280 -3.17 -21 13 300 +3.92 +26 31 401 -4.29 -29 3 128 -5.66 -38 7 227 -5.86 -40 23 366 -6.05 -41 43 440 +7.00 +47 29 393 -7.35 -50


87edo

87edo is a very good approximation to 13-limit JI, giving excellent representations of prime-factors 3, 5, 11, and 13, and a mediocre representation of 7 that is still not too bad. It also gives excellent representations of primes 31, 41, and 43, and a mediocre 37.

    87edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 87 0 0 5 202 -0.11 -1 31 431 -0.21 -2 11 301 +0.41 +3 13 322 +0.85 +6 43 472 -1.17 -9 41 466 -1.48 -11 3 138 +1.49 +11 37 453 -3.07 -22 7 244 -3.31 -24 29 423 +4.91 +36 17 356 +5.39 +39 19 370 +5.94 +43 23 394 +6.21 +45


100edo

100edo is significant as one of the built-in microtonal tunings in the Csound music programming language, making it very easy to compose pieces in 100edo in Csound. The only prime-factors within the 43-limit at which it excels are 11, 13, and 37; all the rest are near or above 20% relative error.

    100edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 100 0 0 13 370 -0.53 -4 37 521 +0.66 +5 11 346 +0.68 +6 5 232 -2.31 -19 29 486 +2.42 +20 19 425 +2.49 +21 41 536 +2.94 +24 17 409 +3.04 +25 7 281 +3.17 +26 23 452 -4.27 -36 43 543 +4.48 +37 31 495 -5.04 -42 3 158 -5.96 -50


118edo

118edo give an outstanding approximation of 5-limit JI, with relative errors below 4% for prime-factors 3 and 5. Errors for all the rest of the primes within the 43-limit are fairly evenly distributed between 19% and 35%, with prime 31 the only outlier at 40%.

    118edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 118 0 0 5 274 +0.13 +1 3 187 -0.26 -3 41 632 -1.94 -19 11 408 -2.17 -21 23 534 +2.23 +22 29 573 -2.46 -24 19 501 -2.60 -26 7 331 -2.72 -27 37 615 +2.89 +28 43 640 -3.04 -30 17 482 -3.26 -32 13 437 +3.54 +35 31 585 +4.12 +40


144edo

144edo does not give any especially good mappings to prime-factors within the 43-limit, with all of its relative errors at 14% and above. Its main interest is in its extreme divisiblity: it can function as 18 bike-chains of 8edo, 16 bike-chains of 9edo, 12 bike-chains of 12edo, 8 bike-chains of 18edo, 6 bike-chains of 24edo, 4 bike-chains of 36edo, 3 bike-chains of 48edo, or 2 bike-chains of 72edo. It was used by Dan Stearns and Joe Monzo as a notation, not as an actual tuning, with one extra accidental as a supplement to 72-edo HEWM, to indicate subtleties smaller than those available in 72edo.

    144edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 144 0 0 13 533 +1.14 +14 11 498 -1.32 -16 37 750 -1.34 -16 3 228 -1.96 -23 7 404 -2.16 -26 19 612 +2.49 +30 5 334 -2.98 -36 43 781 -3.18 -38 23 651 -3.27 -39 31 713 -3.37 -40 17 589 +3.38 +41 29 700 +3.76 +45 41 771 -4.06 -49


152edo

152edo gives a decent approximation of 5-limit JI, with relative errors below 10% for prime-factors 3 and 5, and also for 31. Errors 11, 37, and 41 are all mediocre at around 20%, and for all the rest of the primes within the 43-limit are errors are above 28%. Its main interest is that it can work as an adaptive-JI in the form of 8 bike-chains of 19edo.

    152edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 152 0 0 31 753 -0.30 -4 5 353 +0.53 +7 3 241 +0.68 +9 37 792 +1.29 +16 11 526 +1.31 +17 43 825 +1.64 +21 7 427 +2.23 +28 17 621 -2.32 -29 19 646 +2.49 +32 41 814 -2.75 -35 29 738 -3.26 -41 23 688 +3.30 +42 13 562 -3.69 -47


161edo

161edo is included here only because the graph exhibits an interesting pattern in the errors, where about half of the primes have fairly low error and the other half higher, in an almost alternating pattern, and the lowest primes having the lowest error while the amount of error increases with the prime series thru the 43-limit.

    161edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 171 0 0 7 452 +0.12 +2 11 557 +0.23 +3 17 658 -0.61 -8 19 684 +0.62 +8 29 782 -1.01 -13 3 255 -1.33 -18 5 374 +1.26 +17 13 596 +1.71 +23 37 839 +2.07 +28 23 728 -2.19 -29 43 874 +2.77 +37 31 798 +2.79 +37 41 863 +3.24 +43
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171edo

171edo was advocated by German theorist Martin Vogel in many of his numerous books as an excellent approximation to 7-limit JI. Its prime-mappings to 3, 5, 7, and 17 have very low error, and it was used by Vogel as an integer unit of measurement for 7-limit JI. It can also function as 9 bike-chains of 19-edo, allowing its use as an adaptive-JI.

    171edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 171 0 0 3 271 -0.20 -3 17 699 +0.31 +4 5 397 -0.35 -5 7 480 -0.40 -6 43 928 +0.76 +11 41 916 -0.99 -14 31 847 -1.18 -17 37 891 +1.29 +18 13 633 +1.58 +22 29 831 +2.00 +29 19 726 -2.78 -40 11 592 +3.07 +44 23 774 +3.30 +47
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205edo

205edo is a tuning used by H-Pi instruments for its flagship microtonal keyboard the Tonal Plexus, and named by them a "mem". It gives a negligible error for its mapping of prime-factor 5 at + 0.47%, and also low error for prime-factor 3 as well as several higher primes, thus it is a good approximation of 5-limit JI. It can function as 5 bike-chains of 41-edo.

    205edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 205 0 0 5 476 +0.03 +0.47 37 1068 +0.36 +6 17 838 +0.41 +7 3 325 +0.48 +8 29 996 +0.67 +11 19 871 +1.02 +17 11 709 -1.07 -18 41 1098 -1.75 -30 23 927 -1.93 -33 43 1112 -2.25 -38 31 1016 +2.28 +39 13 759 +2.40 +41 7 576 +2.88 +49


217edo

217edo is another cardinality which gives a negligible error for a 43-limit prime-factor: the error of its mapping of 13 is +0.5%. Other excellent representations are those for 3, 17, and 31, with fairly good representations of 5 and 7. Its main interest is that it can work as an adaptive-JI in the form of 7 bike-chains of 31edo, two of which are nearly identical to Vicentino's adaptive-JI of 1555. Use of this tuning for the common-practice repertoire would allow much of it to be performed in adaptive-JI.

    217edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 217 0 0 13 803 +0.03 +0.5 17 887 +0.11 +2 31 1075 -0.34 -6 3 344 +0.35 +6 5 504 +0.78 +14 29 1054 -1.01 -18 7 609 -1.08 -20 19 922 +1.10 +20 11 751 +1.68 +30 23 982 +2.14 +39 41 1163 +2.27 +41 37 1130 -2.50 -45 43 1177 -2.76 -50


270edo

270edo is very useful as an integer unit of interval measurement up to the 13-prime-limit. Since 270 is a fairly high cardinality, the absolute error amounts are all fairly small. What is important is that all prime-factors up to the 13-limit are approximated with extremely small relative error (i.e., percentage of a 270-edo degree), all less than 13%, which thus obviates the need for decimal places when mapping 13-limit JI to 270edo. Joe Monzo advocates its use for this, with the name "tredek", as a replacement for cents.

    270edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 270 0 0 7 758 +0.06 +1 11 934 -0.21 +5 19 1147 +0.26 +6 3 428 +0.27 +6 5 617 +0.35 +8 43 1465 -0.41 -9 13 999 -0.53 -12 29 1312 +1.53 +35 23 1221 -1.61 -36 31 1338 +1.63 +37 17 1104 +1.71 +39 37 1407 +1.99 +45 41 1447 +2.05 +46


301edo

301edo is based on the base-10 logarithm of 2, obviating the need for calculating logarithms. It has extremely low error for its mapping of prime-factor 7, and also good representations of 3 and 5 -- thus is works well as an integer unit of measurement for 7-limit JI, and was named heptaméride by Sauveur. It also functions as 7 bike-chains of 43edo, which is very close to 1/5-comma meantone. In fact, several of the temperings of 43edo from the related just intervals are measureable in nearly exact heptamérides.

    301edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 301 0 0 7 845 -0.06 -1 37 1568 -0.18 -5 3 477 -0.29 -7 5 699 +0.40 +10 13 1114 +0.67 +17 31 1491 -0.85 -21 29 1462 -1.01 -25 11 1041 -1.15 -29 43 1633 -1.22 -31 17 1230 -1.30 -33 19 1279 +1.49 +37 41 1613 +1.50 +38 23 1362 +1.63 +41


311edo

311edo is very useful as an integer unit of interval measurement all the way up to the 41-prime-limit. Since 311 is a fairly high cardinality, the absolute error amounts are all fairly small. What is important is that all prime-factors up to the 41-limit are approximated with very little relative error (i.e., percentage of a 311 degree), with all errors evenly distributed between 8% and 24%, which thus obviates the need for decimal places when mapping 41-limit JI to 311edo. Joe Monzo advocates its use for this, with the name "gene", as a replacement for cents.

    311edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 311 0 0 3 493 +0.30 +8 7 873 -0.34 -9 19 1321 -0.41 -11 11 1076 +0.45 +12 5 722 -0.46 -12 37 1620 -0.54 -14 13 1151 +0.63 +16 29 1511 +0.65 +17 23 1407 +0.66 +17 41 1666 -0.77 -20 17 1271 -0.78 -20 31 1541 +0.95 +24 43 1688 +1.67 +43


581edo

581edo is very useful as an integer unit of interval measurement up to the 23-prime-limit. Since 581 is a fairly high cardinality, the absolute error amounts in 43-limit are all ~1 cent or less. What is important is that all prime-factors up to the 23-limit are approximated with very small relative error (i.e., percentage of a 581-edo degree), all less than 20%, with the relative error for primes 5, 7, 11, 13, and 19 all under 8%, and the relative error for primes 3, 17, and 23 which thus obviates the need for decimal places when mapping 23-limit JI to 581edo. Joe Monzo advocates its use for this, with the name "spook", as a replacement for cents.

    581edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 581 0 0 5 1349 -0.08 -4 13 2150 +0.09 +4 19 2468 -0.09 -5 11 2010 +0.15 +7 7 1631 -0.15 -7 3 921 +0.28 +14 17 2375 +0.38 +18 23 2628 -0.39 -19 41 3113 +0.54 +26 37 3027 +0.64 +31 43 3153 +0.70 +34 31 2878 -0.80 -39 29 2822 -1.01 -49


612edo

612edo has negligible relative error for its mapping of prime-factor 3, at +0.29%. It is useful as an accurate integer measurement of 5-limit JI.

    612edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 612 0 0 3 970 +0.01 +0.29 5 1421 -0.04 -2 31 3032 +0.06 +3 29 2973 -0.17 -8 7 1718 -0.20 -10 43 3321 +0.25 +13 11 2117 -0.34 -17 41 3279 +0.35 +18 37 3188 -0.36 -19 19 2600 +0.53 +27 13 2265 +0.65 +33 23 2768 -0.82 -42 17 2502 +0.93 +47


730edo

730edo was introduced in 1835 by Wesley Woolhouse as an very accurate integer measurement of the basic intervals in 5-limit JI.

    730edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 730 0 0 5 1695 -0.01 -0.75 19 3101 +0.0212 +1.29 41 3911 -0.0213 -1.3 3 1157 -0.04 -2.26 37 3803 +0.16 +10 17 2984 +0.25 +15 43 3961 -0.28 -17 23 3302 -0.33 -20 13 2701 -0.53 -32 29 3546 -0.54 -33 7 2049 -0.61 -37 11 2525 -0.63 -39 31 3617 +0.72 +44


768edo

768edo is important for users of MIDI, because it is a standard tuning resolution on many MIDI instruments and soundcards. One degree of 768edo is a hexamu.

At this level of resolution, even the largest error shown is still less than 2/3 cent. But it is important to know how 768edo maps the primes because a MIDI user may get different results than expected because of rounding errors or data truncation.

    768edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 768 0 0 7 2156 -0.08 -5 13 2842 +0.10 +6 29 3731 +0.11 +7 23 3474 -0.15 -10 37 4001 +0.22 +14 11 2657 +0.24 +16 17 3139 -0.27 -17 31 3805 +0.28 +18 5 1783 -0.38 -24 3 1217 -0.39 -25 43 4167 -0.58 -37 41 4115 +0.63 +40 19 3262 -0.64 -41


1000edo

1000edo is important because of its use by some tuning theorists as a unit of logarithmic interval measurement, known as the millioctave. It is a division of the octave into exactly 1000 equal parts. Besides its ease of decimal calculation, one main reason for its employment is to avoid referring intervals to the familiar 12-edo tuning.

    1000edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 1000 0 0 29 4858 +0.023 +2 3 1585 +0.045 +4 5 2322 +0.086 +7 19 4248 +0.087 +7 31 4954 -0.236 -20 43 5426 -0.318 -26 7 2807 -0.426 -35 11 3459 -0.518 -43 23 4524 +0.526 +44 13 3700 -0.528 -44 41 5358 +0.538 +45 37 5209 -0.544 -45 17 4087 -0.555 -46


1024edo

1024edo is important as the tuning resolution that was incorporated into a wide variety of electronic instruments manufactured by Yamaha in the 1980s and 1990s, which were some of the earliest relatively inexpensive electronic instruments capable of microtonality. It was named yu (for "Yamaha unit") by Joe Monzo. It was chosen as an easily computible division, since the octave is divided into 2^10 (= 1024) equal parts. It happens to have negligible error in its mapping of prime-factor 3, with only -0.16% relative error -- thus, it is both an outstanding logarithmic measurement for pythagorean tuning, and capable of rendering pythogorean tuning on the Yamaha instruments with extreme accuracy.

    1024edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 1024 0 0 3 1623 -0.002 -0.16 31 5073 -0.11 -10 19 4350 +0.14 +12 23 4632 -0.15 -13 41 5486 -0.16 -13 13 3789 -0.29 -25 7 2875 +0.31 +27 5 2378 +0.41 +35 29 4975 +0.50 +43 17 4186 +0.51 +44 11 3542 -0.54 -46 37 5334 -0.56 -48 43 5556 -0.58 -50


1200edo

1200edo is important because of its widespread use as a unit of logarithmic interval measurement, known as the cent. It is a division of the 12-edo semitone into exactly 100 equal parts.

On all the equal-temperaments presented on this page, the error is given as both an absolute value, in cents, and as a relative value, in the percentage of one step of that equal-temperament. Because 1200-edo defines the cent, both sets of values here are equivalent. This analysis shows it as a mapping, as with all the other EDOs on this page, but in general usage, cents are normally used as an integer measurement only informally in actual musical performance; otherwise they are treated as a floating-point value.

    1200edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 1200 0 0 31 5945 -0.036 -3.6 17 4905 +0.0446 +4.46 3 1902 +0.045 +4.5 41 6429 -0.06 -6 7 3369 +0.17 +17 23 5428 -0.27 -27 5 2786 -0.31 -31 11 4151 -0.32 -32 37 6251 -0.34 -34 29 5830 +0.42 +42 13 4441 +0.47 +47 43 6512 +0.48 +48 19 5098 +0.49 +49


2460edo

2460edo is a unit of integer logarithmic interval measurement called the mina, named by Gene Ward Smith, and used by George Secor and Dave Keenan in their invention of sagittal notation. Its mappings to prime-factors 3, 5, 7, 13, 19, and 23 all have relative error under 10%, and those of primes 11 and 17 between 15% and 20%. Thus it is an excellent integer measurement for 7-limit JI, and fairly good for 23-limit JI. It is exactly divisible into, and thus contains, all of the following EDOs within it: 5, 6, 10, 12, 15, 20, 30, 41, 60, 82, 123, 164, 205, 246, 410, 492, 615, 820, and 1230, and therefore may be used as bike-chains of all of these tunings.

    2460edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 2460 0 0 3 3899 -0.00 -0.8 23 11128 +0.02 +4 5 5712 +0.03 +6 13 9103 -0.04 -8 7 6906 -0.05 -9 19 10450 +0.05 +9.8 17 10055 -0.08 -16 11 8510 -0.10 -20 37 12815 -0.12 -26 31 12187 -0.16 -32 29 11951 +0.18 +37 43 13349 +0.19 +39 41 13180 +0.21 +42


6691edo

6691edo is useful as a unit of integer logarithmic interval measurement for 11-limit JI, and is especially good for 7-limit JI. It has outstanding representations of prime-factors 3, 5, and 7, and is very good with 11. Its approximations of 19, 23, and 43 are fairly good, of 17 and 29 mediocre, and of 13, 37, 41, and 43 poor.

    6691edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 6691 0 0 7 18784 -0.0021 -1.2 3 10605 +0.0029 +1.59 5 15536 -0.0037 -2.09 11 23147 -0.0102 -5.7 19 28423 +0.0210 +12 43 36307 -0.0247 -14 23 30267 -0.0274 -15 17 27349 -0.0384 -21 29 32505 +0.0447 +25 13 24760 +0.0642 +36 41 35847 -0.0682 -38 37 34856 -0.0811 -45 31 33149 +0.0847 +47


8539edo

8539edo is advocated by Joe Monzo as a integer logarithmic interval measurement to replace cents, the tina. Described first by George Secor then Gene Ward Smith, it was discovered by Monzo independently using a weighted error analysis, where prime-factor 3 was given by far the greatest weight, with the weights decreasing thru 5, 7, and 11, with primes 13 to 43 left unweighted. It gives excellent representations thru the 13-limit and also for prime-factor 19, fairly good representations for 17, 31, 41, and 43, mediocre for 23 and 29, and in the 43-limit only 37 having a poor representation.

    8539edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 8539 0 0 7 23972 -0.0005 -0 3 13534 +0.0007 +1 19 36273 -0.0075 -5 13 31598 -0.0077 -5 5 19827 +0.0079 +6 11 29540 -0.012 -9 31 42304 +0.017 +12 43 46335 +0.018 +13 41 45748 -0.019 -14 17 34903 +0.022 +15 29 41482 -0.042 -30 23 38627 +0.043 +30 37 44484 +0.067 +48


54624edo

54624edo is a superb integer logarithmic interval measurement for 11-limit JI, to replace cents. Its representations of 3, 5, 7, 11, and 31 are outstanding, those of 23 and 41 extremely good, of 13, 19, and 37 fair, and of 17, 29, and 43 poor.

    54624edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 54624 0 0 5 126833 -0.00001 -0.03 11 188968 +0.00016 +0.73 3 86577 +0.00018 +0.84 31 270618 -0.0004 -2 7 153349 +0.001 +4 23 247095 -0.0011 -5 41 292651 +0.0017 +8 13 202133 +0.004 +18 37 284561 -0.004 -18 19 232039 +0.0046 +21 43 296404 -0.0063 -29 29 265362 -0.0078 -35 17 223274 +0.0094 +43


96478edo

96478edo is a superb integer logarithmic interval measurement for 13-limit JI, to replace cents. Its representations of 3, 5, 7, 11, 13, and 31 are outstanding, that of 41 quite good, of 17, 23, and 43 mediocre, and of 19, 29, and 37 poor.

    96478edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 96478 0 0 3 152914 -0.00015 -1 7 270848 +0.00015 +1 5 224015 +0.00026 +2 13 357011 -0.00029 -2 11 333759 -0.00054 -4 31 477971 +0.0006 +5 41 516886 +0.00122 +10 43 523515 -0.00213 -17 23 436424 -0.0026 -21 17 394350 -0.003 -24 29 468688 -0.00361 -29 37 502598 +0.00446 +36 19 409832 +0.0056 +45


937060edo

937060edo is an outstanding integer logarithmic interval measurement for 23-limit JI, to replace cents. Its representations of 3, 5, 7, 11, 13, 17, 19 , and 23 are all under 7% error, its representation of 31 is mediocre, and those of 29, 37, 41 and 43 quite poor.

    937060edo best representation of prime-factor prime edo ~cents % EDO-step degrees error error 2 937060 0 0 7 2630660 -0.000004 -0.33 11 3241695 +0.00001 +0.74 23 4238849 +0.00004 +3.35 3 1485205 +0.00005 +3.91 13 3467534 -0.00005 -4.23 19 3980563 +0.00006 +4.43 5 2175786 +0.00008 +6 17 3830198 +0.00009 +7 31 4642379 -0.00025 -19 29 4552220 +0.00042 +33 37 4881570 -0.00047 -37 41 5020348 +0.00041 +32 43 5084736 +0.00045 +35




Comparison of different EDO prime-error graphs

©2003 by Joe Monzo



MouseOver the cardinality names of the various EDOs to see a bingo-card lattice of them showing the error from the closest EDO approximation to JI.

5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
81 87 100 118 144 152 161 171 181 205
217 270 301 311 581 600 612 730 768 1000
1024 1200 2460 6691 8539 10600 30103 46032 54624 96478
495399 937060 1049064

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