EDO: Difference between revisions

An equal (pitch) division of the octave is the most common type of [[EPD|equal (pitch) division]], which is a kind of [[equal-step tuning]]. Therefore, it is also a kind of [[arithmetic tuning|arithmetic]] and [[harmonotonic tuning|harmonotonic]] tuning.

Tuning theorists first used the term "equal temperament" for edos designed to approximate [[low-complexity JI|low-complexity just intervals]]. The same term is still used today for all rank-1 [[regular temperament|temperaments]]. For example, [[15edo]] can be referred to as 15-tone equal temperament (15-TET, 15-tET, 15tet, etc.), or more simply 15 equal temperament (15-ET, 15et, etc.).

With the development of [[edonoi|equal divisions of non-octave intervals (edonoi)]], some people started writing "ed2" ("ED2"), especially when naming a specific tuning.

EDO scales are straightforward to work with due to their uniform step size. Some musicians find the consistency bland, while others appreciate the stable foundation it provides for composition. The only property shared by all of them is the equality of their step-sizes; otherwise, their individual properties are as different as can be. Lower-numbered EDOs, especially 5 to 24, possess very strong and unique "characters", which some composers find inspiring.

EDOs allow for modulation to every single key in the tuning, without any alteration in harmonic properties, thus making transposition totally seamless. This also makes them somewhat easier to learn, as you do not have to memorize the harmonic and melodic variations that appear in various keys (which you would have to learn in [[Just intonation|JI]], an unequal [[regular temperament]], or a [[well temperament]], especially with smaller numbers of tones). For those accustomed to the "equality" of [[12edo|12-TET]], the equality of the alternative EDOs can be reassuringly familiar.

Fret crowding can become an issue with smaller divisions, especially high up the neck. For these cases, [[ed4|equal divisions of the double octave]] or higher multiples offer a compromise solution, for example the [[Kite Guitar]] with frets at every other step of [[41edo]].

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 the perfect fifth ([[3/2]]) can be rewarding. Classical music also relies on the fact that [[5/4]] is the major third of the diatonic scale, which occurs only when [[81/80]], the syntonic comma, is [[tempered out]]. EDOs {{EDOs| 12, 19, 24, 26, 31, 36, 38, 43, 45, 48, and 50}} temper out 81/80, and are thus [[meantone]] systems. EDOs {{EDOs| 17, 22, 27, 29, 34, 39, 41, 44, 46, 49, 51, and 53}} have an accurate perfect fifth, but do not temper out 81/80, and thus require new ways of thinking about harmony. Many EDOs, such as {{EDOs| 12, 17, 19, 22, 26, 27, 29, 31, 39, 41, 43, 45, 46, 49, and 53 }}, can be notated with some variant on the [[chain-of-fifths notation|A–G "circle of fifths" notation]], while other EDOs, including {{EDOs| 24, 34, 36, 38, 44, 48, or 51}}, involve multiple such circles.

If your interest lies in the nuanced approximation of just intonation through EDOs, then delving into EDOs characterized by a strong [[the Riemann zeta function and tuning #Zeta EDO lists|zeta peak]] could be especially captivating. Such EDOs, including {{EDOs| 12, 19, 22, 27, 31, 34, 41, 46, 53, 58, 60, 65, 68, 72, 77, 80, 84, 87, 94, and 99 }}, offer rich avenues for exploration in the quest for harmonic purity and transparent [[temperament]]s.

EDOs with a less pronounced, yet still noteworthy [[the Riemann zeta function and tuning#Local zeta edos|zeta peak]]—specifically {{EDOs| 10, 14, 15, 16, 17, 21, 24, 26, 29, 32, 36, 37, 38, 39, 43, 45, 48, 49, 50, 56, 62, 63, 82, 89, and 96 }}—present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.

EDOs can be further subdivided and classified according to the size of the fifth, such as with [[Margo Schulter]]'s [[gentle region]] or the distinction between negative, positive, doubly negative and doubly positive of {{w|R. H. M. Bosanquet}}. [[Kite Giedraitis]] has proposed these six categories, based on the size of the fifth. From narrowest to widest:

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, {{nowrap| 6 {{=}} 2 × 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]] and [[Highly composite EDO]] for more details.

EDOs with fewer than 12 divisions have steps exceeding 100 cents. Of these, 1, 2, 3, 4, and 6 divide 12 and so are already available. {{EDOs| 5, 7, and 9 }} have arguably been used in various musical traditions worldwide.  

When using EDOs to tune scales or [[regular temperament]]s, the size becomes less conceptually important since not all notes need to be used. 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.  

== EDOs versus equal temperaments ==

{{Main| EDO vs ET }}