EDO: Difference between revisions

The acronym "EDO" (''EE-dee-oh'') was coined by [[Daniel Anthony Stearns]]^{[''year needed'']}. More recently, the [[Wikipedia: Anacronym|anacronym]] "edo" (''EE-doh''), spelled in lowercase, has become increasingly widespread.

To find the step size ([[Interval size measure #Ratio|interval ratio]]) for an <span><math>n</math></span>edo, take the <span><math>n</math></span>th root of 2. For example, the step of 12edo is <span><math>2^{\frac{1}{12}}</math></span> (≈ 1.059). So the formula for the <span><math>k</math></span>th step of an <span><math>n</math></span>edo is:

If you're a classically-trained musician and you'd like to start with some EDOs that have some relationship to common-practice tonal music, starting with reasonably-low EDOs that give a good approximation to [[3/2]] (the perfect 5th) can be rewarding. These include {{EDOs| 12, 17, 19, 22, 29, 31, 39, 41, 43, 45, 46, 49, 50, and 53 }}. All of these can be notated with some variant on the [[Circle-of-fifths notation|A-G "circle of fifths" notation]], while other EDOs, including {{EDOs| 24, 34, 36, 38, 44, 48, or 51 }} involve more than one such circle.

You will quickly find that the ''factorization'' of the total number of notes in each EDO has consequences for its structure and the way it relates to other EDOs. For example, 6 = 2 x 3, so 6edo contains all of the intervals in both 2edo and 3edo. On the other hand, 7 is a prime number, so no 7edo intervals are redundant with those of smaller EDOs. See [[Prime EDO]] for more details.

The [[Moments of Symmetry]] paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.

Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). This really amounts to the consideration of adding the associated [[vals]], which are the mappings to primes larger than 2. An EDO is defined by a certain number of steps equating to 2; if we have more steps equating to 3, we get a 3-limit val, and so forth. So, for example, 12edo can be written {{val| 12 }}, saying that twelve steps maps to 2, but the 3-limit val for 12 is {{val| 12 19 }}, telling us that 19 steps maps to 3, and the 5-limit val is {{val| 12 19 28 }}, telling us that 28 steps maps to 5.

If we add 12 and 19 we get another good division, 12 + 19 = 31. We can understand why this works if we look at it as adding vals; {{val| 12 19 28 }} + {{val| 19 30 44 }} = {{val| 31 49 72 }}. The relative error in terms of [[relative cent]]s is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is [-1.955 13.686] (the same as absolute cents) and the error of 19edo is [-11.429 -11.663], and this sums to [-13.384 2.023]. In relative cents the error of the fifth for 31edo is not much increased from 19edo, and on converting to absolute cents we find it is even better, and the error of the major third is much smaller due to the cancellation. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 31edo) can have a much smaller error due to the cancellation. 24edo's flat fifth and 29edo's sharp fifth can be added to form 53edo.

We may also look at addition of EDOs in terms of MOS; if ''a''\''n'' is a generator for an ''n''-edo MOS, and ''b''\''m'' for an ''m''-edo MOS, where both of these are generators for the same linear temperament, then the mediant, (''a'' + ''b'')\(''n'' + ''m''), will be a generator for a MOS for the same temperament, this time in (''n'' + ''m'')-edo. A visual way of putting this is that through this addition of ''n'' and ''m'', one becomes the accidentals or black keys, and the other the naturals or white keys. The choice of accidental/natural or black keys/white keys is a question of emphasis on the part of the composer or designer. Furthermore, one may add more than two numbers, hierarchically expanding the possibilities to double flats and sharps and beyond. This can be useful in designing keyboards and systems of notation.

When an edo divides the octave into fewer than 12 divisions (so that each step exceeds 100 cents), you might call it a [[macrotonal EDO]]. Of these, 1, 2, 3, 4 and 6 divide 12 and so are already available to anyone wishing to explore them. [[5edo|5]], [[7edo|7]] and [[9edo|9]] have arguably been used in various kinds of musical traditions in different parts of the world. [https://soundcloud.com/scottthompson-3/the-13-edos-of-xmas ''The 13 EDOs of Xmas'' by Scott Thompson] is a humorous demonstration of EDOs 1–13.

On the other hand, if you use the edo to tune a scale or [[regular temperament]], the size of the edo does not matter so much (at least conceptually), as you don't need to use all of it. Some of the EDOs which can be used to tune various temperaments are listed on the [[optimal patent val]] page. Tuning a scale in just intonation by one of these EDOs can be regarded as automatically tempering it to the corresponding regular temperament.

=== 0…99 ===

=== 100…199 ===

=== 200…299 ===

=== 300…499 ===

=== 500…999 ===

=== 1000…1999 ===

The regular EDOs, up to 72edo:

== Links and articles ==

[[Category:Equal divisions of the octave| ]] <!-- main article -->