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An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') is a [[ | An '''equal division of the octave''' ('''EDO''', ''EE-dee-oh''; '''edo''', ''EE-doh'') is a [[tuning system]] obtained by dividing the [[2/1|octave]] into a whole number of [[equal-step tuning|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<sup>1/n</sup>. For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO), with consecutive steps having a frequency ratio of 2<sup>1/12</sup>. This implies that the [[interval]] between any two consecutive pitches is identical. Equal divisions of the octave are the most common [[equal-step tuning]]s, with other [[nonoctave]] tunings existing as well. | ||
A tuning with ''n'' equal divisions of the octave is usually called "''n''-edo" ("''n''-EDO"). For instance, the predominant tuning system in the world today is [[12edo]] (12-EDO). | |||
== History == | == History == | ||
Tuning theorists first used the term "equal temperament" for edos designed to approximate [[ | 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.). | ||
The acronym "EDO" was coined by [[Daniel Anthony Stearns]] in 1999, originally standing for "equidistant divisions of the octave"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_65#65 Yahoo! Tuning Group | ''Where F + f = O'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_117#117 Yahoo! Tuning Group | ''f + F and WFS/MOS'']</ref>. More recently, the {{w|anacronym}} "edo", spelled in lowercase and pronounced as a regular word, has also become common. | The acronym "EDO" was coined by [[Daniel Anthony Stearns]] in 1999, originally standing for "equidistant divisions of the octave"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_65#65 Yahoo! Tuning Group | ''Where F + f = O'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_117#117 Yahoo! Tuning Group | ''f + F and WFS/MOS'']</ref>. More recently, the {{w|anacronym}} "edo", spelled in lowercase and pronounced as a regular word, has also become common. | ||
With the development of [[ | With the development of [[edonoi|equal divisions of non-octave intervals (edonoi)]], some people started writing "ed2" ("ED2"), especially when naming a specific tuning. | ||
== Calculating the step size == | == Calculating the step size == | ||
To find the step size of ''n''-edo in terms of [[cent]]s, divide 1200 by ''n''. The size ''s'' of ''k'' steps of ''n''-edo (''k''\''n'') is | To find the step size of ''n''-edo in terms of [[cent]]s, 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<sup>1/12</sup> (≈ 1.059). So the ratio '' | 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<sup>1/12</sup> (≈ 1.059). So the ratio ''r'' of the ''k'' steps of ''n''-edo is | ||
$$ r = 2^{k/n} $$ | |||
In particular, when ''k'' is 0, '' | In particular, when ''k'' is 0, ''r'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''r'' is simply 2, because any number to the 1st power is itself. | ||
== Properties == | == Properties == | ||
EDO scales are straightforward to work with due to their uniform step size. | 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. | ||
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 == | == 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 [[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. | |||
=== Fretted instruments === | === Fretted instruments === | ||
If you are a [[guitar]]ist (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. | If you are a [[guitar]]ist (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. | 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]]. | ||
For these cases, [[ed4|equal divisions of the double octave]] or higher multiples offer a compromise solution | |||
== Approaches to exploring EDOs == | == 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. | 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 EDO]]s 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. | 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. | ||
Some EDOs, such as {{EDOs| 26, 27, 32, 33, or 37 }} have fifths which are reasonably good but quite audibly not just. Other EDOs, such as {{EDOs| 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. | Some EDOs, such as {{EDOs| 26, 27, 32, 33, or 37 }} have fifths which are reasonably good but quite audibly not just. Other EDOs, such as {{EDOs| 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 [[ | 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 [[ | 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 | 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: | ||
* '''Superflat''' EDOs ({{EDOs| 9, 11, 13b, 16, 18b, and 23 }}) have a fifth narrower than four-sevenths of an octave ({{nowrap|4\7 {{=}} 685.714{{c}}}}) | * '''Superflat''' EDOs ({{EDOs| 9, 11, 13b, 16, 18b, and 23 }}) have a fifth narrower than four-sevenths of an octave ({{nowrap|4\7 {{=}} 685.714{{c}}}}) | ||
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== Structural properties == | == 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, {{nowrap|6 {{=}} 2 | 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. | ||
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. | 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. | ||
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=== Scale size considerations === | === Scale size considerations === | ||
EDOs with fewer than 12 divisions have steps exceeding 100 cents. | 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. | ||
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. | 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. | ||
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 | 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. | All of these tools are also applicable to equal divisions of other ([[nonoctave]]) intervals as well. | ||
== EDOs versus | == EDOs versus equal temperaments == | ||
{{Main| EDO vs ET }} | |||
== Individual pages for EDOs == | == Individual pages for EDOs == | ||
Note: Before creating an EDO page, please make sure that it satisfies the [[Xenharmonic Wiki:Notability guidelines|notability guidelines]]. Also, if the EDO is greater than or equal to 1000, please add it to the list below. | |||
=== 0…999 === | === 0…999 === | ||
{| class="wikitable center-all mw-collapsible" | {| class="wikitable center-all mw-collapsible" | ||
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=== 1000…1999 === | === 1000…1999 === | ||
{{EDOs | {{EDOs | ||
| 1000, 1001, 1012, 1015, 1019, 1051, 1053, 1059, 1063, 1065, 1080, 1092, 1106, 1125, 1131, 1147, 1152, 1166, 1171, 1178, 1193, 1200, 1210, 1224, 1230, 1236, 1240, 1244, 1260, 1272, 1277, 1289, 1308, 1323, 1330, 1337, 1342, 1361, 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 }} | | 1000, 1001, 1012, 1015, 1019, 1051, 1053, 1059, 1063, 1065, 1080, 1092, 1106, 1125, 1131, 1147, 1152, 1166, 1171, 1178, 1193, 1200, 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…9999 === | ||
{{EDOs | {{EDOs | ||
| 2000, 2016, 2019, 2022, 2023, 2024 | | 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 === | === 10000 and up === | ||
{{EDOs | {{EDOs | ||
| 10009 | | 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 | ||
}} | }} | ||
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[[File:The_Scale_Tree.png|alt=The Scale Tree.png|800x1023px|The Scale Tree.png]] | [[File:The_Scale_Tree.png|alt=The Scale Tree.png|800x1023px|The Scale Tree.png]] | ||
The | The diatonic EDOs, up to [[72edo]]: | ||
[[File:Scale_Tree_close-up.png|alt=Scale Tree close-up.png|Scale Tree close-up.png]] | [[File:Scale_Tree_close-up.png|alt=Scale Tree close-up.png|Scale Tree close-up.png]] | ||
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{{See also| Pergen #Pergens and EDOs }} | {{See also| Pergen #Pergens and EDOs }} | ||
[[Pergen]]s 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 {{nowrap|P = 6\12}}, {{nowrap|G = 4\12}} are marked as "-". | [[Pergen]]s 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 {{nowrap|P = 6\12}}, {{nowrap|G = 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]]. | ||
{| class="wikitable" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ style="font-size: 105%; white-space: nowrap;" | Pergens in EDOs from 5 to 24 | |||
|- | |- | ||
! EDO | ! EDO | ||
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|} | |} | ||
== | == See also == | ||
; Related topics | |||
* [[Equal-step tuning]] | |||
* [[Prime equal division]] | |||
* [[ | * [[Highly composite equal division]] | ||
* [[ | |||
* [[ | |||
* [[List of rank one temperaments by step size]] | * [[List of rank one temperaments by step size]] | ||
* [[ | |||
; Technical data | |||
* [[Absolute errors of small EDOs]] | |||
* [[Relative errors of small EDOs]] | |||
* [[Minimal consistent EDOs]] | * [[Minimal consistent EDOs]] | ||
* [[ | * [[Consistency limits of small EDOs]] | ||
* [[ | * [[Monotonicity limits of small EDOs]] | ||
* [[ | * [[List of distinct EDO scales]] | ||
* [[ | |||
* [https://www.webcitation.org/5xZz8RtQB Teen Tunes | ; Opinions | ||
* [[Collection of EDO impressions]] | |||
; Other | |||
* [[:Category: Equal divisions of the octave]] | |||
== External links == | |||
* [[Ivor Darreg]], [https://www.webcitation.org/5xZz8RtQB Teen Tunes] | |||
== Notes == | == Notes == | ||
<references /> | <references /> | ||
[[Category:Equal divisions of the octave| ]] <!-- main article --> | [[Category:Equal divisions of the octave| ]] <!-- main article --> | ||
[[Category:Acronyms]] | [[Category:Acronyms]] | ||
[[Category:Lists of scales]] | [[Category:Lists of scales]] | ||