Xenharmonic Wiki

EDO: Difference between revisions

← Older editNewer edit →
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,281 edits
m →10000 and up: 3159811
Sintel (talk | contribs)
autopatrolled, emailconfirmed
1,636 edits
Convert FAQ into regular headings. Misc. clarity and style nits
Line 23: Line 23:
Several alternate notations have been devised, including "edd" ("EDD"; equal division of the [[octave|ditave]]), "DIV," and "EQ".{{Citation needed|date=July 2021|reason=Who used this term?}}</sup>
Several alternate notations have been devised, including "edd" ("EDD"; equal division of the [[octave|ditave]]), "DIV," and "EQ".{{Citation needed|date=July 2021|reason=Who used this term?}}</sup>


== Formula ==
== 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


Line 34: Line 34:
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.
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 ==
== Properties ==
=== What are EDO scales like? ===
EDO scales are straightforward to work with due to their uniform step size.
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.
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.


=== Why would I want to use an EDO? ===
== Practical advantages ==
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.
=== 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.
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.


More generally, 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.
=== 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.


=== How do I explore so many? ===
== Approaches to exploring EDOs ==
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 the perfect fifth ([[3/2]]) can be rewarding.
 
These include {{EDOs| 12, 17, 19, 22, 26, 27, 29, 31, 39, 41, 43, 45, 46, 49, 50, and 53 }}.
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.
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.
Line 65: Line 73:
* '''Trivial''' EDOs ({{EDOs| 1, 2, 3, 4, and 6 }}) have a fifth about 100{{c}} from just, and are contained in 12edo
* '''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 ===
== 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 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, {{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.


Line 77: Line 85:
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, {{nowrap|(''a'' + ''b'')\(''n'' + ''m'')}}, will be a generator for a MOS for the same temperament, this time in {{nowrap|(''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.
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, {{nowrap|(''a'' + ''b'')\(''n'' + ''m'')}}, will be a generator for a MOS for the same temperament, this time in {{nowrap|(''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.


=== Size of an EDO ===
=== Scale size considerations ===
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.
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.


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.  
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.


To practically tune large edos through software tuning, one may take advantage of [[MIDI]] channels. See [[Tuning per channel]].  
To practically tune large edos through software tuning, one may take advantage of [[MIDI]] channels. See [[Tuning per channel]].  
Line 86: Line 97:
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.


=== What's the difference between EDOs and Equal Temperaments? ===
== EDOs versus Equal Temperaments ==
See [[EDO vs ET]].
See [[EDO vs ET]].


Retrieved from "https://en.xen.wiki/w/EDO"