EDO: Difference between revisions
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== History == | == History == | ||
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.). | 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.). | ||
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<math>\displaystyle c = 2^{k/n}</math> | <math>\displaystyle c = 2^{k/n}</math> | ||
In particular, when ''k'' is 0, ''c'' is simply 1, because any number to the 0th power is 1. And when ''k'' = ''n'', ''c'' is simply 2, because any number to the 1st power is itself. | In particular, when ''k'' is 0, ''c'' is simply 1, because any number to the 0th power is 1. And when {{nowrap|''k'' {{=}} ''n''}}, ''c'' is simply 2, because any number to the 1st power is itself. | ||
== EDO FAQ == | == EDO FAQ == | ||
=== What are EDO scales like? === | === What are EDO scales like? === | ||
Very straightforward to work with, the step size being so even and all. Some find the monotony bland, others find it a safe stable footing for musicmaking. 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. The lower-numbered EDOs, especially 5 to 24, possess very strong and unique "characters", which some composers have found to be inspiring in their own right. | Very straightforward to work with, the step size being so even and all. Some find the monotony bland, others find it a safe stable footing for musicmaking. 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. The lower-numbered EDOs, especially 5 to 24, possess very strong and unique "characters", which some composers have found to be inspiring in their own right. | ||
=== Why would I want to use an EDO? === | === Why would I want to use an EDO? === | ||
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. Speaking of string instruments fretted for EDOs, since ascending through the EDOs will crowd a fretboard relatively quickly, especially as one approaches the 30-something EDOs, [[ed4|equal divisions of the double octave]] (or higher multiple of the octave) are a relatively tidy compromise solution to the problem of laying out high-EDO fretboards. | 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. Speaking of string instruments fretted for EDOs, since ascending through the EDOs will crowd a fretboard relatively quickly, especially as one approaches the 30-something EDOs, [[ed4|equal divisions of the double octave]] (or higher multiple of the octave) are a relatively tidy compromise solution to the problem of laying out high-EDO fretboards. | ||
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=== How do I explore so many? === | === How do I explore so many? === | ||
It depends entirely on your desires as a musician! | It depends entirely on your desires as a musician! | ||
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. | ||
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 | 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 fifth) can be rewarding. These include {{EDOs| 12, 17, 19, 22, 26, 27, 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 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 [[The Riemann zeta function and tuning|local zeta peak]] could be especially captivating. Such EDOs, including | 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|local 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|temperaments]]. | ||
EDOs with a less pronounced, yet still noteworthy [[The Riemann zeta function and tuning|local zeta peak]]—specifically | EDOs with a less pronounced, yet still noteworthy [[The Riemann zeta function and tuning|local 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 [[R. H. M. Bosanquet]]. [[Kite Giedraitis]] has proposed these six categories, based on the size of the fifth. From narrowest to widest: | 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 ({{EDOs| 9, 11, 13b, 16, 18b & 23 }}) have a fifth narrower than four-sevenths of an octave ({{nowrap|4\7 {{=}} 685.714{{c}}}}) | ||
* ''' | * '''Perfect''' EDOs ({{EDOs| 7, 14, 21, 28 & 35 }}) have a fifth equal to {{nowrap|4\7 {{=}} 685.714{{c}}}} | ||
* ''' | * '''Diatonic''' EDOs ({{EDOs| 12, 17, 19, 22, 24, etc. }}) have a fifth between 685.714{{c}} and 720{{c}} | ||
* ''' | * '''Pentatonic''' EDOs ({{EDOs| 5, 10, 15, 20, 25 & 30 }}) have a fifth of three-fifths of an octave ({{nowrap|3\5 {{=}} 720{{c}}}} | ||
* ''' | * '''Supersharp''' EDOs ({{EDOs| 8, 13 & 18 }}) have a fifth wider than 720{{c}} | ||
* ''' | * '''Trivial''' EDOs ({{EDOs| 1, 2, 3, 4 and 6 }}) have a fifth about 100{{c}} from just, and are contained in 12edo | ||
=== Non-tuning properties === | === Non-tuning 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 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]] and [[Highly composite EDO]] for more details. | |||
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]] 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. | ||
=== Adding 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 {{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. | 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. | ||
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=== Size of an EDO === | === Size of an EDO === | ||
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. {{EDOs| 5, 7, and 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. | |||
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. | |||
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. | 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. | ||
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| 10009, 10600, 11664, 12276, 12348, 12500, 13382, 14124, 14348, 14618, 14842, 15601, 15900, 16384, 16625, 16808, 17461, 18355, 20203, 20567, 21489, 25281, 25282, 28000, 28742, 30103, 30631, 31867, 31920, 32768, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 99693, 99694, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 571611, 762148, 1714833, 1905370, 2547047, 2667518, 2901533, 4191814, 6000000, 11358058, 402653184, 5407372813 }} | | 10009, 10600, 11664, 12276, 12348, 12500, 13382, 14124, 14348, 14618, 14842, 15601, 15900, 16384, 16625, 16808, 17461, 18355, 20203, 20567, 21489, 25281, 25282, 28000, 28742, 30103, 30631, 31867, 31920, 32768, 34691, 46032, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 99693, 99694, 102557, 103169, 111202, 148418, 190537, 196608, 241200, 253389, 258008, 324296, 571611, 762148, 1714833, 1905370, 2547047, 2667518, 2901533, 4191814, 6000000, 11358058, 402653184, 5407372813 }} | ||
The largest physically possible EDO in a frequency range can be found through molecular physics such as the [[mean free path]] combined with the | The largest physically possible EDO in a frequency range can be found through molecular physics such as the [[mean free path]] combined with the {{w| Speed of sound}} in a given substance. | ||
== Non-integer EDO == | == Non-integer EDO == | ||