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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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72edo
12edo best representation of prime-factor
prime ~cents error % EDO-step error
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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.
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
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Comparison of different EDO prime-error graphs
©2003 by Joe Monzo
