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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 primemapping of several various EDOs of all primefactors 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 primefactor 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 primefactor 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 primeaffect.
5edo is the smallest cardinality which is generally perceived as a scale, a type of pentatonic scale. The only primefactors it represents decently are 3, 7, and 37  so it is possible to use it as an approximation to a "no 5s" 7limit JI, altho the direction of errors for 3 and 7 are opposite.
5edo best representation of primefactor prime edo ~cents % EDOstep 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 "wholetone scale" derived from 12edo. Of course, several of its best representations of primefactors in 43limit 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 primefactor prime edo ~cents % EDOstep 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 stepsizes 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 primefactors which it represents well are 3 and 13. All others in the 43limit are mediocre and worse.
7edo best representation of primefactor prime edo ~cents % EDOstep 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 43limit represents well only primefactor 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 primefactor prime edo ~cents % EDOstep 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 primefactors in the 43limit 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 primefactors in the 43limit have relative errors which fall between 10% and 30%.
9edo best representation of primefactor prime edo ~cents % EDOstep 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 primefactor 13 (only ~ 0.5 cent error), and fairtomediocre representations of 7 (~ 9 cents error), 37, 17, and 3. It does quite poorly with 5, 11, and 19.
10edo best representation of primefactor prime edo ~cents % EDOstep 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 primefactors 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 gives excellent representations of the Pythagorean
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% EDOstep 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 stepsizes necessary for the diatonic scales. Thus, largely because of its practicality, over the course of four centuries (c. 16002000) it became strongly entrenched in Western music.



15edo gives fair representations of primefactors 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 "no3s" 11limit JI, and possibly to some degree even of full 11limit JI. Note that 15edo does not represent any of the primes in the 43limit with a relative error of less than 11%.


Because of the fact that the 5limit "3rds" (major3rd and minor3rd) are both represented by 5 degrees of 17edo, which is really a "neutral 3rd", it cannot emulate 5limit 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] (= 3^{4} = ratio 81:64), respectively  the Pythagorean minor3rd or "hemiditone", and pythagorean major3rd> or ditone.
Besides giving a very good representation of the Pythagorean primefactor 3 (~ +4 cents error), in the 43primelimit 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 nextsimplest EDOmeantone after 12edo, does a fairly good job of representing primefactors 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 commonpractice repertoire recognizably, it also provides a simple tuning which is fruitful for exploring approximateJI all the way up to 43limit.



22edo does a fairly good job of representing primefactors 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 43limit are the four lowest primes 3, 5, 7, and 11  in theory, exactly the ones generally most desired.

The famous "quartertone" system. Being simply a halving of the stepsize of 12edo, this tuning gives exactly the same representations to JI as some of the primemappings of 12EDO: 3, 5, 17, 19, and 43. It gives a significantly better approximation of 7, but in the opposite direction (that is, 24edo's best representation of 7 is smaller than the primefactor, whereas its secondbest 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 primefactors 5 and 7 do not sound close to those of 7limit JI. However, 24edo's biggest improvement over 12edo is its excellent mapping of 11, and it also gives significantly better representations of primefactors 13, 31, and 37.





29edo gives an excellent mapping of primefactor 3, and is thus a good representation of pythagorean tuning.

30edo is very strange in the 43limit. It gives a superb representation of only primefactor 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 1/4comma meantone. Among the lowest primes, it gives excellent representations of primefactors 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 43limit JI are exactly the same as those of 12edo: primefactors 3, 5, 17, 19, and 43  thus, its approximation of 5limit JI gives no advantage over 12edo. However, 36edo offers a superb representation of primefactor 7, making it a much better approximation of 7limit JI. And it provides significantly better representations than 12edo of primefactors 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 secondbest 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 7limit 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 primefactor 5, and the single occurence of the letter indicating the secondbest mapping of that prime.

37edo is the first cardinality which gives essentially no error for a primemapping other than 2: the relative error for 11 is +0.1%. It also gives excellent representations of primefactors 5, 7, and 13. It is thus a great approximation of "no 3s" 13limit JI. Its representation of primefactor 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 primefactor 3, and decent representations of 5, 7, and 11. It is thus a very good approximation of 11limit JI, and also gives good representations of primefactors 19, 29, and 31.

43EDO is historically important as an approximation of 1/5comma meantone.

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

53edo is the next cardinality (after 37edo) which presents a negligible error in the 43limit: it gives an outstanding representation of the Pythagorean primefactor 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 (3limit) tuning, and a very good approximation to 5limit 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 lowerprime 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 11limit JI, with errors of primefactors 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 primefactors, unfortunately giving a very poor approximation to 13.
The fact that 72edo's approximations do not temper out so many important 5limit intervals means that, even tho it is not as accurate as the lowercardinality 53edo in representing the entire 5limit lattice, its structure does emulate so many of the important lowerlimit JI commas that in a systemic sense it is easily perceived as a good approximation of 11limit JI.

81EDO is of interest as a very close approximation to goldenmeantone. It gives a very good approximation of primefactor 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 13limit JI, giving excellent representations of primefactors 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 builtin microtonal tunings in the Csound music programming language, making it very easy to compose pieces in 100edo in Csound. The only primefactors within the 43limit 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 5limit JI, with relative errors below 4% for primefactors 3 and 5. Errors for all the rest of the primes within the 43limit 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 primefactors within the 43limit, with all of its relative errors at 14% and above. Its main interest is in its extreme divisiblity: it can function as 18 bikechains of 8edo, 16 bikechains of 9edo, 12 bikechains of 12edo, 8 bikechains of 18edo, 6 bikechains of 24edo, 4 bikechains of 36edo, 3 bikechains of 48edo, or 2 bikechains 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 72edo HEWM, to indicate subtleties smaller than those available in 72edo.

152edo gives a decent approximation of 5limit JI, with relative errors below 10% for primefactors 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 43limit are errors are above 28%. Its main interest is that it can work as an adaptiveJI in the form of 8 bikechains 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 43limit.

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

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

217edo is another cardinality which gives a negligible error for a 43limit primefactor: 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 adaptiveJI in the form of 7 bikechains of 31edo, two of which are nearly identical to Vicentino's adaptiveJI of 1555. Use of this tuning for the commonpractice repertoire would allow much of it to be performed in adaptiveJI.

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

301edo is based on the base10 logarithm of 2, obviating the need for calculating logarithms. It has extremely low error for its mapping of primefactor 7, and also good representations of 3 and 5  thus is works well as an integer unit of measurement for 7limit JI, and was named heptaméride by Sauveur. It also functions as 7 bikechains of 43edo, which is very close to 1/5comma 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 41primelimit. Since 311 is a fairly high cardinality, the absolute error amounts are all fairly small. What is important is that all primefactors up to the 41limit 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 41limit 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 23primelimit. Since 581 is a fairly high cardinality, the absolute error amounts in 43limit are all ~1 cent or less. What is important is that all primefactors up to the 23limit are approximated with very small relative error (i.e., percentage of a 581edo 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 23limit 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 primefactor 3, at +0.29%. It is useful as an accurate integer measurement of 5limit JI.

730edo was introduced in 1835 by Wesley Woolhouse as an very accurate integer measurement of the basic intervals in 5limit 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 12edo 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 primefactor 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 12edo semitone into exactly 100 equal parts.
On all the equaltemperaments 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 equaltemperament. Because 1200edo 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 floatingpoint 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 primefactors 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 7limit JI, and fairly good for 23limit 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 bikechains of all of these tunings.

6691edo is useful as a unit of integer logarithmic interval measurement for 11limit JI, and is especially good for 7limit JI. It has outstanding representations of primefactors 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 primefactor 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 13limit and also for primefactor 19, fairly good representations for 17, 31, 41, and 43, mediocre for 23 and 29, and in the 43limit only 37 having a poor representation.

54624edo is a superb integer logarithmic interval measurement for 11limit 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 13limit 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 23limit 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 43limit JI, to replace cents. Its representations of all 43limit primefactors have less than 18% error.

Comparison of different EDO primeerror graphs
©2003 by Joe Monzo
MouseOver the cardinality names of the various

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