Private lessons available via Discord/WhatsApp/Skype: composition, music-theory, tuning-theory, piano, and all woodwinds (sax, clarinet, flute, bassoon, recorder). Current rates US$ 80 per hour (negotiable). Send an email to: monzojoe (AT) gmail.
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 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 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 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 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 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 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 gives fairly good representations of prime-factors 7, 11, and 17 (~ +13, -6, and +4 cents error, respectively) but does a terrible job with 3 and 5 (~ -47 and +50 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.
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%.
Because of the fact that the 5-limit "3rds"
(major-3rd 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), in the 43-prime-limit
17edo also gives decent representations of 13, 23,
and 41.
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. It is thus a good tuning choice
to not emulate JI.
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.
22edo does a fairly good job of representing prime-factors
3, 5, and 11, and is mediocre with 7:
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.
The famous "quarter-tone" system.
Being simply a halving of the step-size of 12edo, this tuning gives
exactly the same representations to JI as some of the prime-mappings of 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, intervals incorporating 24edo's best mappings of
prime-factors 5 and 7 do not sound close to those of 7-limit JI.
However, 24edo's biggest improvement over 12edo is its excellent
mapping of 11, and it also gives significantly better representations
of prime-factors 13, 31, and 37.
29-edo gives an excellent mapping of prime-factor 3, and is thus a
good representation of pythagorean tuning.
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.
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 mappings
except that of 5 are smaller in size than the actual primes.
Note that it also gives mediocre representations of 13, 17, and 19,
but those are in the opposite direction, larger than the primes.
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 best 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 12\36 (12 degrees of 36)
is the best mapping of 5 (exactly 400 cents, same as 12edo), the second-best mapping
of 5, to 11\36, gives an absolute error in cents
larger than anything in the table below, and a relative error of 59%,
but 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. This tuning uses a "wart"
in its name 36c, the wart being the letter "c" as the third letter of the alphabet
designating the third prime-factor 5, and the single occurence of
the letter indicating the second-best mapping of that prime.
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.
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.
43-EDO is historically important as an approximation of 1/5-comma meantone.
see also: meride
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. All 17-limit factors have less
than 6 cents absolute error, and the 13-limit factors cluster
between 8% and 22% relative error.
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".
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.
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 3, 5, and 7 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
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.
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.
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.
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.
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%.
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.
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 for
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.
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.
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.
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.
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.
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.
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.
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.
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 between 10% and 20%,
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.
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.
730edo was introduced in 1835 by
Wesley Woolhouse
as an very accurate integer measurement of the basic intervals in 5-limit JI.
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
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.
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.
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.
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.
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.
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 a very good mapping of 11.
Its approximations of 19, 23, and 43 are fairly good, of 17 and 29 mediocre,
and of 13, 37, 41, and 43 poor.
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.
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.
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.
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.
1049064edo is an excellent integer logarithmic interval measurement for 43-limit JI,
to replace cents. Its representations of all 43-limit prime-factors have less than 18% error.
Comparison of different EDO prime-error graphs
©2003 by Joe Monzo
MouseOver the cardinality names of the various
The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by selecting the highest level of financial support that you can afford. Thank you.
12edo gives excellent representations of the Pythagorean
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 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
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
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 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 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 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
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 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 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
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 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 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
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
46edo
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 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
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 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
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 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 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 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 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 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 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
171edo
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
205edo
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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
1049064edo
1049064edo best representation of prime-factor
prime edo ~cents % EDO-step
degrees error error
2 1049064 0 0
23 4745506 +0.0000001 0
31 5197269 +0.000002 0
43 5692499 +0.00001 1
37 5465050 +0.000017 1
7 2945095 +0.000018 2
29 5096333 +0.00003 3
41 5620415 +0.00007 6
13 3881998 -0.000106 9
3 1662727 -0.000115 10
17 4288010 -0.000135 12
5 2435851 -0.0002 17
11 3629165 -0.0001963 17
19 4456348 +0.0001956 17
a&b temperament [a&b are numbers]
55-edo (comma) (Mozart's tuning)
1/1 or 1:1 (ratio)
2/1 or 2:1 (ratio)
a&b temperament [a&b are numbers]
apotome (Greek interval)
aristoxenean (temperament family)
atomic (temperament family)
augmented / diesic (temperament family)
augmented-2nd / aug-2 / #2 (interval)
augmented-4th / aug-4 / #4 (interval)
augmented-5th / aug-5 / #5 (interval)
augmented-6th / aug-6 / #6 (interval)
augmented-9th / aug-9 / #9 (interval)
blackjack (tuning)
cent / ¢ (unit of interval measurement)
centitone / iring (unit of interval measurement)
chromatic-semitone / augmented-prime (interval)
daseian (musical notation)
dekamu / 10mu (MIDI-unit)
diapason (Greek interval)
diapente (Greek interval)
diatessaron (Greek interval)
diatonic semitone (minor-2nd) (interval)
diesic (temperament family)
diezeugmenon (Greek tetrachord)
diminished-5th / dim5 / -5 / b5 (interval)
diminished-7th / dim7 / o7 (interval)
doamu / 2mu (MIDI-unit)
dodekamu / 12mu (MIDI-unit)
dominant-7th (dom-7, x7) (chord)
dorian (mode)
eleventh / 11th (interval)
enamu / 1mu (MIDI-unit)
endekamu / 11mu (MIDI-unit)
enharmonic semitone (interval)
ennealimmal (temperament family)
enneamu / 9mu (MIDI-unit)
farab (unit of interval measurement)
fifth / 5th (interval)
flu (unit of interval measurement)
Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)
fourth / 4th (interval)
Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)
grad (unit of interval measurement)
hexamu / 6mu (MIDI-unit)
Hurrian Hymn (Monzo reconstruction)
hypate (Greek note)
hypaton (Greek tetrachord)
hyperbolaion / hyperboleon (Greek tetrachord)
hypophrygian (Greek mode)
imperfect (interval quality)
iring / centitone (unit of interval measurement)
1/1 or 1:1 (ratio)
2/1 or 2:1 (ratio)
jot (unit of interval measurement)
JustMusic: A New Harmony [Monzo's book]
JustMusic prime-factor notation [Monzo essay]
kwazy (temperament family)
leimma / limma (Greek interval)
lichanos (Greek note)
limma / leimma (Greek interval)
locrian (mode)
lydian (mode)
magic (temperament family)
Mahler 7th/1 [Monzo score and analysis]
marvel (temperament family)
meantone (temperament family)
mem (unit of interval measurement)
méride (unit of interval measurement)
mese (Greek note)
meson (Greek tetrachord)
millioctave / m8ve (unit of interval measurement)
mina (unit of interval measurement)
minerva (temperament family)
miracle (temperament family)
mixolydian (mode)
monzo (prime-exponent vector)
Monzo, Joe (music-theorist)
morion / moria (unit of interval measurement)
mutt (temperament family)
mystery (temperament family)
octamu / oktamu / 8mu (MIDI-unit)
octave (interval)
oktamu / octamu / 8mu (MIDI-unit)
orwell (temperament family)
p4, perfect 4th, perfect fourth (interval)
p5, perfect 5th, perfect fifth (interval)
pantonality of Schoenberg [Monzo essay]
paramese (Greek note)
paranete (Greek note)
parhypate (Greek note)
pentamu / 5mu (MIDI-unit)
prime-factor notation (JustMusic) [Monzo essay]
proslambanomenos (Greek note)
savart (unit of interval measurement)
schismic / skhismic (temperament family)
Schoenberg's pantonality [Monzo essay]
second / 2nd (interval)
semisixths (temperament family)
semitone (unit of interval measurement)
seventh / 7th (interval)
sixth / 6th (interval)
sk (unit of interval measurement)
skhismic / schismic (temperament family)
sruti tuning [Monzo essay]
studloco (tuning)
subminor 3rd (interval)
Sumerian tuning [speculations by Monzo]
synemmenon (Greek tetrachord)
temperament-unit / tu (unit of interval measurement)
tenth / 10th (interval)
tetrachord-theory tutorial [by Monzo]
tetradekamu / 14mu (MIDI-unit)
tetramu / 4mu (MIDI-unit)
third / 3rd (interval)
thirteenth / 13th (interval)
tina (unit of interval measurement)
tone (interval, and other definitions)
tredek (unit of interval measurement)
triamu / 3mu (MIDI-unit)
tridekamu / 13mu (MIDI-unit)
trihemitone (Greek interval)
trite (Greek note)
tu / temperament-unit (unit of interval measurement)
Türk sent (unit of interval measurement)
twelfth / 12th (interval)
whole-tone (interval)
woolhouse-unit (unit of interval measurement)
