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EDO prime error

[Joe Monzo]

On this page I present a detailed examination of the amount of error of the representations of all prime-factors from 3 to 43 (an arbitrarily selected stopping point), for several various EDOs.

Different EDOs approximate JI intervals in different ways. In these listings, errors are given both in cents and as a percentage of one degree or "step" of that EDO, and 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.

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  ~cents error  % EDO-step error

  13  -0.527661769    0

   7  -8.825906469   -7
  37  -11.34403875   -9
  17  +15.0445905   +13
   3  +18.04499913  +15

   5  -26.31371386  -22
  23  -28.27434727  -24
  43  -31.51770564  -26

  11  +48.68205764  +41
  29  +50.42280585  +42
  41  +50.93759446  +42
  31  +54.96442754  +46
  19  -57.51301613  -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 ~cents error % EDO-step error 17 +4.135499591 +4 11 -5.86339691 -5 7 +12.99227535 +12 23 +26.27110728 +24 19 +29.75971114 +27 13 +32.19961096 +30 37 -33.16222057 -30 43 +33.93683981 +31 3 -47.40954632 -43 29 -47.75901233 -44 5 +50.0499225 +46 31 -54.12648156 -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 best representation of prime-factor prime ~cents error % EDO-step error 3 -1.955000865 -2 19 +2.486983868 +2 17 -4.9554095 -5 43 -11.51770564 -12 5 +13.68628614 +14 23 -28.27434727 -28 29 -29.57719415 -30 7 +31.17409353 +31 13 -40.52766177 -41 31 -45.03557246 -45 11 +48.68205764 +49 37 +48.65596125 +49


13edo

    13edo best representation of prime-factor prime ~cents error % EDO-step error 11 +2.528211481 +3 13 -9.758431 -11 17 -12.64771719 -14 29 -14.19257877 -15 23 +17.87949889 +19 5 -17.08294463 -19 19 -20.58993921 -22 37 +25.57903817 +28 41 +32.476056 +35 3 +36.5065376 +40 31 -37.34326477 -40 43 +42.3284482 +46 7 +46.55870892 -50


14edo

    14edo best representation of prime-factor prime ~cents error % EDO-step error 41 -0.49097697 -1 29 -1.005765582 -1 43 +2.768008643 +3 37 +5.798818388 +7 3 -16.24071515 -19 13 +16.61519537 +19 17 -19.24112379 -22 7 -25.96876361 -30 23 -28.27434727 -33 31 -30.74985818 -36 11 -37.03222808 -43 19 -40.37015899 -47 5 +42.25771471 +49


15edo

    15edo best representation of prime-factor prime ~cents error % EDO-step error 11 +8.682057635 +11 7 -8.825906469 -11 29 +10.42280585 +13 37 -11.34403875 -14 23 +11.72565273 +15 5 +13.68628614 +17 3 +18.04499913 +23 19 +22.48698387 +28 17 -24.9554095 -31 31 -25.03557246 -31 41 -29.06240554 -36 43 -31.51770564 -39 13 +39.47233823 +49


16edo

    16edo best representation of prime-factor prime ~cents error % EDO-step error 19 +2.486983868 +3 7 +6.174093531 +8 5 -11.31371386 -15 43 +13.48229436 +18 13 -15.52766177 -21 31 -20.03557246 -27 29 +20.42280585 +27 41 +20.93759446 +28 11 -26.31794236 -35 37 -26.34403875 -35 3 -26.95500087 -36 23 -28.27434727 -38 17 -29.9554095 -40


17edo

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

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 ~cents error % EDO-step error 3 +3.927352076 +6 41 -5.532993777 -8 13 +6.53116176 +9 23 +7.019770379 +10 11 +13.38793999 +19 19 -15.16007496 -21 31 -15.62380776 -22 43 -17.40005858 -25 7 +19.40938765 +27 29 +29.24633526 +41 37 +31.00890242 +44 5 -33.37253739 -47 17 -34.36717421 -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 ~cents error % EDO-step error 31 -11.70223913 -18 5 +13.68628614 +21 37 +15.32262791 +23 11 -17.98460903 -27 43 +21.81562769 +33 13 +26.1390049 +39 23 -28.27434727 -42 17 +28.37792383 +43 41 -29.06240554 -44 29 -29.57719415 -44 19 -30.84634947 -46 7 +31.17409353 +47 3 +31.37833247 +47


19edo

19edo does a fairly good job of representing prime-factors 3 and 5, and is mediocre with 7 and 11, and none of the lower primes really has a terrible representation:

    19edo best representation of prime-factor prime ~cents error 37 +1.287540193 +2 23 +3.3046001 +5 43 -6.254547748 -10 3 -7.21815876 -11 5 -7.366345444 -12 31 -8.193467201 -13 41 +13.04285762 +21 11 +17.10311027 +27 19 +18.27645755 +29 29 -19.05087836 -30 13 -19.47503019 -31 17 +21.36037997 +34 7 -21.45748542 -34


20edo

    20edo best representation of prime-factor prime ~cents error % EDO-step error 13 -0.527661769 -1 19 +2.486983868 +4 31 -5.035572464 -8 7 -8.825906469 -15 41 -9.062405542 -15 29 -9.577194153 -16 11 -11.31794236 -19 37 -11.34403875 -19 17 +15.0445905 +25 3 +18.04499913 +30 5 -26.31371386 -44 23 -28.27434727 -47 43 +28.48229436 +47


21edo

    21edo best representation of prime-factor prime ~cents error % EDO-step error 23 +0.297081303 +1 29 -1.005765582 -2 31 -2.178429607 -4 7 +2.602664959 +5 43 +2.768008643 +5 17 +9.330304785 +16 19 -11.79873042 -21 5 +13.68628614 +24 3 -16.24071515 -28 13 +16.61519537 +29 11 +20.11062906 +35 37 -22.77261018 -40 41 +28.0804516 +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 ~cents error % EDO-step error 31 +0.41897299 +1 17 +4.135499591 +8 5 -4.495532047 -8 11 -5.86339691 -11 29 +6.786442211 +12 3 +7.135908226 +13 41 +7.301230822 +13 7 +12.99227535 +24 43 -20.60861473 -38 37 +21.38323397 +39 13 -22.34584359 -41 19 -24.78574341 -45 23 +26.27110728 +48


24edo

    24edo best representation of prime-factor prime ~cents error % EDO-step error 11 -1.317942365 -3 37 -1.344038755 -3 3 -1.955000865 -4 19 +2.486983868 +5 17 -4.9554095 -10 31 +4.964427536 +10 13 +9.472338231 +19 43 -11.51770564 -23 5 +13.68628614 +27 7 -18.82590647 -38 29 +20.42280585 +41 41 +20.93759446 +42 23 +21.72565273 +43


26edo

    26edo best representation of prime-factor prime ~cents error % EDO-step error 7 +0.404862762 +1 11 +2.528211481 +5 43 -3.82539795 -8 31 +8.810581382 +19 3 -9.647308558 -21 13 -9.758431 -21 17 -12.64771719 -27 41 -13.67779016 -30 29 -14.19257877 -31 5 -17.08294463 -37 23 +17.87949889 +39 37 -20.57480799 -45 19 -20.58993921 -45


27edo

    27edo best representation of prime-factor prime ~cents error % EDO-step error 13 +3.916782675 +9 23 -6.052125046 -14 29 -7.354971931 -17 7 +8.951871309 +20 3 +9.156110246 +21 31 +10.51998309 +24 19 +13.59809498 +31 5 +13.68628614 +31 37 +15.32262791 +34 41 +15.3820389 +35 17 -16.06652061 -36 11 -17.98460903 -40 43 +21.81562769 +49


28edo

    28edo best representation of prime-factor prime ~cents error % EDO-step error 41 -0.49097697 -1 5 -0.599428151 -1 29 -1.005765582 -2 19 +2.486983868 +6 43 +2.768008643 +6 37 +5.798818388 +14 11 +5.824914778 +14 31 +12.10728468 +28 23 +14.58279559 +34 3 -16.24071515 -38 13 +16.61519537 +39 7 +16.88837925 +39 17 -19.24112379 -45


29edo

    29edo best representation of prime-factor prime ~cents error % EDO-step error 3 +1.493274997 +4 37 -3.068176686 -7 29 +4.905564468 +12 23 -7.584692096 -18 19 -7.857843719 -19 13 -12.94145487 -31 11 -13.38690788 -32 31 +13.58511719 +33 5 -13.89992076 -34 43 -14.9659815 -36 41 -15.26930209 -37 7 -17.10176854 -41 17 +19.18252153 +46


31edo

31edo gives excellent representations of prime-factors 5 (~ 3/4 cent error) and 7 (~ 1 cent error), and mediocre representations of 3 and 11:

    31edo best representation of prime-factor prime ~cents error % EDO-step error 5 +0.783060329 +2 7 -1.083970985 -3 3 -5.180807317 -8 47 -7.442105884 -13 43 -8.291899191 -21 23 -8.919508559 -23 11 -9.382458494 -24 13 +11.08524146 +29 17 +11.17362276 +29 19 +12.16440322 +31 31 +16.25475012 +40 29 +15.58409617 +42 37 -19.08597424 -49


36edo

    36edo best representation of prime-factor prime ~cents error % EDO-step error 3 -1.955000865 -6 7 -2.159239802 -6 19 +2.486983868 +7 29 +3.75613918 +11 41 +4.270927792 +13 17 -4.9554095 -15 23 +5.058986065 +15 13 -7.194328436 -22 43 -11.51770564 -35 31 -11.70223913 -35 5 +13.68628614 +41 37 +15.32262791 +46 11 +15.3487243 +46


37edo

    12edo best representation of prime-factor prime ~cents % EDO-step error error 11 +0.033409 +0 13 +2.715581 +8 5 +2.875475 +9 7 +4.147067 +13 19 -5.621124 -17 41 -7.440784 -23 43 +7.401213 +23 17 -7.658112 -24 29 +8.260644 +25 37 +8.115421 +25 31 -9.900437 -31 3 +11.558513 +36 23 -12.058131 -37


40edo

    40edo best representation of prime-factor prime ~cents error % EDO-step error 13 -0.527661769 -2 43 -1.517705643 -5 23 +1.725652732 +6 19 +2.486983868 +8 5 +3.686286135 +12 31 -5.035572464 -17 7 -8.825906469 -29 41 -9.062405542 -30 29 -9.577194153 -32 11 -11.31794236 -38 37 -11.34403875 -38 3 -11.95500087 -40 17 -14.9554095 -50


41edo

41edo gives a superb representation of the Pythagorean prime-factor 3, and decent representations of 5, 7, and 11.

    41edo best representation of prime-factor prime ~cents error % EDO-step error 3 +0.484023525 +2 7 -2.972247933 -10 31 -3.57215783 -12 11 +4.779618611 +16 19 -4.830089303 -17 29 -5.186950251 -18 5 -5.825908987 -20 13 +8.252826036 +28 41 +9.961984702 +34 37 +12.07059539 +41 17 +12.11776123 +41 23 -13.64020093 -47 43 -13.95673003 -48


43edo

    43edo best representation of prime-factor prime ~cents error % EDO-step error 37 -0.181248057 -1 31 -0.849525953 -3 29 +2.980945382 +11 13 -3.318359444 -12 3 -4.280582261 -15 5 +4.383960554 +16 17 +6.672497476 +24 11 +6.821592519 +24 7 +7.918279577 +28 43 -9.192124247 -33 19 +9.463728054 +34 41 -10.45775438 -37 23 +13.58611785 +49

see also: meride


46edo

46edo provides excellent approximations to basic intervals in not only in the 5-limit (as in this lattice), but also in 7- and 11-limit, as can be seen on the equal-temperament gallery page and in the following table; all 11-limit factors have less than 5 cents error:

    46edo best representation of prime-factor prime ~cents error % EDO-step error 17 -0.607583413 -2 23 -2.187390747 -8 3 +2.392825222 +9 31 +2.790514492 +11 11 -3.491855408 -13 7 -3.608515165 -14 5 +4.990633961 +19 13 -5.745053074 -22 37 +9.525526463 +37 43 +10.22142479 +39 19 -10.55649439 -40 41 -11.67110119 -45 29 -12.18588981 -47


53edo

53edo gives an extremely good representation of the Pythagorean prime-factor 3 (~ 1/15 cent error), a very good representation of 5 (~ 1.4 cents error), and mediocre ones of 7 and 11:

    53edo best representation of prime-factor prime ~cents error 3 -0.068208413 -0.3 41 +1.126273704 +5 5 -1.408053487 -6 37 -2.287434981 -10 13 -2.791812713 -12 19 -3.173393491 -14 7 +4.758999191 +21 23 +5.687916883 +25 11 -7.92171595 -35 17 +8.252137669 +36 43 +9.237011339 +41 31 +9.681408668 +43 29 -10.70926962 -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 <../monzo/55edo/55edo>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 ~cents error % EDO-step error 3 -3.773182684 -17 29 -4.122648699 -19 17 +4.135499591 +19 23 +4.452925459 +20 11 -5.86339691 -27 5 +6.413558862 +29 41 +7.301230822 +33 19 +7.941529322 +36 7 -8.825906469 -40 43 -9.699523824 -44 13 +10.38142914 +48 37 +10.47414306 +48 31 -10.49011792 -48


72edo

    12edo best representation of prime-factor prime ~cents error % EDO-step error

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 JI commas that in a systemic sense it is easily perceived as a good approximation of 11-limit JI.


311edo

311edo is very useful as an integer unit of interval measurement. 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), which thus obviates the need for decimal places when mapping to 311edo. Joe Monzo advocates its use, with the name "gene", as a replacement for cents.

    311edo best representation of prime-factor prime ~cents % EDO-step error error 3 +0.295803 +8 7 -0.337160 -9 19 -0.406907 -11 5 -0.461624 -12 11 +0.450546 +12 37 -0.540180 -14 13 +0.629895 +16 23 +0.664559 +17 29 +0.647886 +17 17 -0.775345 -20 41 -0.766586 -20 31 +0.945135 +24 43 +1.665574 +43


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 ~cents error % EDO-step error 7 -0.075906469 -5 13 +0.097338231 +6 29 +0.110305847 +7 23 -0.149347268 -10 37 +0.218461245 +14 11 +0.244557635 +16 17 -0.2679095 -17 31 +0.276927536 +18 5 -0.376213865 -24 3 -0.392500865 -25 43 -0.580205643 -37 41 +0.625094458 +40 19 -0.638016132 -41


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.

    1200edo best representation of prime-factor prime ~cents % EDO-step error error 3 +0.044999 +4 17 +0.044590 +4 31 -0.035572 -4 41 -0.062406 -6 7 +0.174094 +17 23 -0.274347 -27 5 -0.313714 -31 11 -0.317942 -32 37 -0.344039 -34 29 +0.422806 +42 13 +0.472338 +47 43 +0.482294 +48 19 +0.486984 +49




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.

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
311 768 1200
. . . . . . . . .

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