EDO

An equal division of the octave (EDO, EE-dee-oh; edo, EE-doh) is a tuning system obtained by dividing the octave into a whole number of equal steps. A tuning with n equal divisions of the octave is usually called "n-edo" (or "n-EDO"). In terms of frequency, the octave with frequency ratio 2/1 is logarithmically divided into n steps, each with frequency ratio 2^1/n. For instance, the predominant tuning system in the world today is 12edo (12-EDO), with consecutive steps having a frequency ratio of 2^1/12. This implies that the interval between any two consecutive pitches is identical. Equal divisions of the octave are the most common equal-step tunings, with other nonoctave tunings existing as well.

History

Tuning theorists first used the term "equal temperament" for edos designed to approximate low-complexity just intervals. The same term is still used today for all rank-1 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.).

The acronym "EDO" was coined by Daniel Anthony Stearns in 1999, originally standing for "equidistant divisions of the octave". More recently, the anacronym "edo", spelled in lowercase and pronounced as a regular word, has also become common.

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

Calculating the step size

To find the step size of n-edo in terms of cents, divide 1200 by n. The size s of k steps of n-edo (k\n) is

$$ s = 1200 \cdot k/n $$

To find the step size of n-edo in terms of frequency ratio, take the n-th root of 2. For example, the step of 12edo is 2^1/12 (≈ 1.059). So the ratio r of the k steps of n-edo is

$$ r = 2^{k/n} $$

In particular, when k is 0, r is simply 1, because any number to the 0th power is 1. And when k = n, r is simply 2, because any number to the 1st power is itself.

Properties

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.

Practical advantages

Free modulation

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 JI, an unequal regular temperament, or a well temperament, especially with smaller numbers of tones). For those accustomed to the "equality" of 12-TET, the equality of the alternative EDOs can be reassuringly familiar.

Fretted instruments

If you are a guitarist (or a player of some other fretted string instrument, like a bass guitar, Appalachian dulcimer, ukulele, banjo, mandolin, sitar, saz, pipa, or zhong ruan), an EDO will provide you with the simplest possible fretboard layout, as all of the frets will go straight across the fretboard, regardless of how you want to tune the open strings. Fret crowding can become an issue with smaller divisions, especially high up the neck. For these cases, 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.

Approaches to exploring EDOs

If you are interested in exploring the unique merits and challenges of each EDO, irrespective of any desire to approximate just intonation (or any other a priori musical goal), starting at the bottom and working your way up can be a most illuminating exercise. Macrotonal EDOs have a step size larger than that of 12edo, and thus have fewer than 12 steps per octave, so they may be preferable to those who want simplicity.

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 12, 19, 24, 26, 31, 36, 38, 43, 45, 48, and 50 temper out 81/80, and are thus meantone systems. 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 12, 17, 19, 22, 26, 27, 29, 31, 39, 41, 43, 45, 46, 49, and 53, can be notated with some variant on the A–G "circle of fifths" notation, while other EDOs, including 24, 34, 36, 38, 44, 48, or 51, involve multiple such circles.

Some EDOs, such as 26, 27, 32, 33, or 37 have fifths which are reasonably good but quite audibly not just. Other EDOs, such as 11, 13, 14, 15, 16, 18, 20, 21, 23, or 25, are of interest to the avid seeker of totally unusual sounds that have next-to-no connection with the common practice.

If your interest lies in the nuanced approximation of just intonation through EDOs, then delving into EDOs characterized by a strong zeta peak could be especially captivating. Such EDOs, including 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 temperaments.

EDOs with a less pronounced, yet still noteworthy zeta peak—specifically 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 R. H. M. Bosanquet. Kite Giedraitis has proposed these six categories, based on the size of the fifth. From narrowest to widest:

• Superflat EDOs (9, 11, 13b, 16, 18b, and 23) have a fifth narrower than four-sevenths of an octave (4\7 = 685.714 ¢)
• Perfect EDOs (7, 14, 21, 28, and 35) have a fifth equal to 4\7 = 685.714 ¢
• Diatonic EDOs (12, 17, 19, 22, 24, etc.) have a fifth between 685.714 ¢ and 720 ¢
• Pentatonic EDOs (5, 10, 15, 20, 25, and 30) have a fifth of three-fifths of an octave (3\5 = 720 ¢)
• Supersharp EDOs (8, 13, and 18) have a fifth wider than 720 ¢
• Trivial EDOs (1, 2, 3, 4, and 6) have a fifth about 100 ¢ from just, and are contained in 12edo

Structural properties

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 × 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.

The MOS 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.

Adding 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 ⟨12], saying that twelve steps maps to 2, but the 3-limit val for 12 is ⟨12 19], telling us that 19 steps maps to 3, and the 5-limit val is ⟨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; ⟨12 19 28] + ⟨19 30 44] = ⟨31 49 72]. The relative error in terms of relative cents 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.

Scale size considerations

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. 5, 7, and 9 have arguably been used in various musical traditions worldwide.

When using EDOs to tune scales or regular temperaments, 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.

To practically tune large edos through software tuning, one may take advantage of MIDI channels; see Tuning per channel.

All of these tools are also applicable to equal divisions of other (nonoctave) intervals as well.

EDOs versus equal temperaments

Main article: EDO vs ET

Individual pages for EDOs

📝Note:

Before creating an EDO page, please make sure that it satisfies the notability guidelines. Also, if the EDO is greater than or equal to 1000, please add it to the list below.

1000…1999

1000, 1001, 1012, 1015, 1019, 1051, 1053, 1059, 1063, 1065, 1080, 1092, 1106, 1125, 1131, 1147, 1152, 1166, 1171, 1178, 1193, 1200, 1205, 1210, 1224, 1230, 1236, 1240, 1244, 1260, 1272, 1277, 1289, 1308, 1312, 1323, 1330, 1337, 1342, 1361, 1376, 1381, 1395, 1407, 1419, 1429, 1440, 1448, 1489, 1506, 1517, 1520, 1525, 1536, 1547, 1553, 1554, 1559, 1578, 1583, 1590, 1600, 1609, 1612, 1619, 1637, 1641, 1643, 1650, 1665, 1672, 1700, 1730, 1759, 1776, 1778, 1783, 1789, 1793, 1794, 1802, 1803, 1817, 1848, 1861, 1879, 1880, 1889, 1911, 1920, 1944, 1955, 1957, 1983, 1984

2000…9999

2000, 2016, 2019, 2022, 2023, 2024, 2029, 2048, 2053, 2072, 2081, 2100, 2101, 2113, 2118, 2129, 2153, 2190, 2200, 2207, 2243, 2320, 2444, 2460, 2477, 2513, 2520, 2544, 2549, 2554, 2619, 2684, 2711, 2730, 2777, 2809, 2814, 2819, 2897, 2901, 2912, 2960, 2964, 3041, 3072, 3079, 3125, 3178, 3361, 3395, 3422, 3476, 3498, 3558, 3566, 3578, 3600, 3643, 3684, 3696, 3776, 3889, 3920, 4004, 4007, 4079, 4096, 4172, 4190, 4231, 4296, 4320, 4327, 4349, 4380, 4501, 4650, 4973, 5040, 5280, 5544, 5585, 5809, 5902, 5941, 6079, 6349, 6380, 6650, 6664, 6691, 7033, 7315, 7980, 8103, 8192, 8269, 8404, 8539, 8736, 8745, 9539

10000 and up

10009, 10600, 10729, 11664, 12276, 12348, 12500, 14124, 14348, 14618, 14842, 15601, 15900, 16218, 16625, 16808, 17100, 17461, 18355, 20203, 20567, 28000, 28472, 28742, 30103, 30631, 31867, 31920, 32436, 33616, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 2547047, 2901533, 3159811, 6000000, 11358058, 402653184, 5407372813

Non-integer EDO

A non-integer EDO can be defined as using a non-integer divisor to divide the octave. Typically, non-integer EDOs are understood as not containing the exact octave, so that they remain equal tunings. If the exact octave is retained and if the generator resets itself at each period, then this results in a MOS scale with only 1 small step.

All fractional EDOs are integer equal divisions of another integer interval. For example, (25/2)edo is equivalent to 25ed4. In general:

[math]\displaystyle{ \displaystyle (p/q) \text{edo} = p \text{-ed} 2^q }[/math]

for integers p and q. Many irrational EDOs cannot be converted to integer equal divisions of another integer interval, so they are things of their own.

Non-integer EDOs can be written in decimal form, such as 12.1edo. This is often meant to be approximate, used in the context of octave stretch of an integer EDO, rather than as a fractional EDO.

Scale tree

The scale tree, or Stern-Brocot tree, provides a visual map of the world of EDOs, based on fifth size.

The diatonic EDOs, up to 72edo:

Pergens

See also: Pergen § Pergens and EDOs

Pergens provide a JI-agnostic way to name the rank-2 scales of an EDO. This table lists every possible period/generator combination for EDOs 5–24, and for each coprime combination, the simplest pergen that it can represent. Non-coprime combinations such as 6\12, 4\12 are marked as "-". Note that the "b" after a number refers to using the second-best approximation of the perfect fifth; for more info see Wart notation.

Pergens in EDOs from 5 to 24

EDO	Period	Generator in EDO steps
	in EDO steps	1	2	3	4	5	6	7	8	9	10	11
5	5 = P8	P4/2	P5
6	6 = P8	P4/2	-
	3 = P8/2	P5
7	7 = P8	P4/3	P5/2	P5
8	8 = P8	P4/3	-	P5
	4 = P8/2	P5	-
9	9 = P8	P4/4	P4/2	-	P5
	3 = P8/3	P5
10	10 = P8	P4/4	-	P5/2	-
	5 = P8/2	P5	P4/2
11	11 = P8	P4/5	P5/3	P5/2	P11/4	P5
12	12 = P8	P4/5	-	-	-	P5
	6 = P8/2	P5	-	-
	4 = P8/3	P5	-
	3 = P8/4	P5
13b	13 = P8	P4/6	P4/3	P4/2	P12/5	P12/4	P5
14	14 = P8	P4/6	-	P4/2	-	P11/4	-
	7 = P8/2	P5	P4/3	P4/2
15	15 = P8	P4/6	P4/3	-	P12/6	-	-	P11/3
	5 = P8/3	P5	P4/2
	3 = P8/5	P4/3
16	16 = P8	P4/7	-	P5/3	-	P12/5	-	P5
	8 = P8/2	P5	-	P5/3	-
	4 = P8/4	P5	-
17	17 = P8	P4/7	P5/5	P11/8	P11/6	P5/2	P11/4	P5	P11/3
18b	18 = P8	P4/8	-	-	-	P5/2	-	P12/4	-
	9 = P8/2	P5	P4/4	-	P4/2
	6 = P8/3	P5/2	-	-
	3 = P8/6	P5
19	19 = P8	P4/8	P4/4	P11/9	P4/2	P12/6	P12/5	ccP5/7	P5	P11/3
20	20 = P8	P4/8	-	P5/4	-	-	-	P11/4	-	c3P5/8
	10 = P8/2	M2/4	-	P5/4	-	-
	5 = P8/4	P4/2	P5
	4 = P8/5	P5/4	-
21	21 = P8	P4/9	P5/6	-	P5/3	P11/6	-	-	c3P4/9	-	P11/3
	7 = P8/3	P5/2	P5	P4/3
	3 = P8/7	P5/3
22	22 = P8	P4/9	-	P4/3	-	P12/7	-	P12/5	-	P5	-
	11 = P8/2	M2/4	P5	P4/3	P12/5	P12/7
23	23 = P8	P4/10	P4/5	P11/11	P12/9	P4/2	P12/6	ccP4/8	ccP4/7	P12/4	P5	P11/3
24	24 = P8	P4/10	-	-	-	P4/2	-	P5/2	-	-	-	c4P5/10
	12 = P8/2	M2/4	-	-	-	P4/2
	8 = P8/3	P5/2	-	P4/2
	6 = P8/4	P4/2	-	-
	4 = P8/6	P4/2	-
	3 = P8/8	P5
		1	2	3	4	5	6	7	8	9	10	11

See also

Related topics

• Equal-step tuning
• Prime equal division
• Highly composite equal division
• List of rank one temperaments by step size

Technical data

• Absolute errors of small EDOs
• Relative errors of small EDOs
• Minimal consistent EDOs
• Consistency limits of small EDOs
• Monotonicity limits of small EDOs
• List of distinct EDO scales

Opinions

• Collection of EDO impressions

Other

• Category: Equal divisions of the octave

External links

• Ivor Darreg, Teen Tunes