Single-pitch tuning: Difference between revisions
Degeneracy |
→Music: Chris Vaisvil's ''0-EDO for Orchestra'' (2021): Really in 1edo, so moved there |
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== In equal tunings == | == In equal tunings == | ||
{{Infobox ET|0edo}} | {{Infobox ET|0edo}}{{todo|fix template|description=The step size is wrong; it shouldn't be zero cents, but rather infinite or undefined.}} | ||
Single-pitch tuning can be specified as '''0 equal divisions of the octave''' ('''0edo'''), or 0 equal divisions of | Single-pitch tuning can be specified as '''0 equal divisions of the octave''' ('''0edo'''), or 0 equal divisions of any finite interval. | ||
The way to approach the idea of 0edo that leads to single-pitch tuning is to see what happens as ''n'' gets smaller in ''n''-edo. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller, steps become sparser, and in the limit, the steps go to infinity and only one pitch is left. It is thus a degenerate case. | The way to approach the idea of 0edo that leads to single-pitch tuning is to see what happens as ''n'' gets smaller in ''n''-edo. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller, steps become sparser, and in the limit, the steps go to infinity and only one pitch is left. It is thus a degenerate case. | ||
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== In regular temperament theory == | == In regular temperament theory == | ||
Single-pitch tuning corresponds to the [[regular temperament]] in any given [[subgroup]] where all [[prime]]s in that subgroup are [[tempering out|tempered out]], resulting in a rank-0 temperament with no [[generator]]. The mapping for this is the 0-val, {{val| 0 0 … 0 }}, or more precisely, the rank-0 matrix, [ ]. | Single-pitch tuning corresponds to the [[regular temperament]] in any given [[subgroup]] where all [[prime]]s in that subgroup are [[tempering out|tempered out]], resulting in a rank-0 temperament with no [[generator]]. The mapping for this is the 0-val, {{val| 0 0 … 0 }}, or more precisely, the rank-0 matrix, [ ]. Since it maps all intervals to the same pitch, it [[tempering out|tempers out]] all commas and is [[consistent]] in all [[limit]]s. | ||
Single-pitch tuning can also be considered a rank-0 temperament in the empty subgroup, which contains no primes. It tempers no commas and the pitch represents only the [[1/1|unison]], so it is also empty-subgroup JI. (Tempering everything and tempering nothing are the same in this case, because there is nothing to temper.) This is closer to representing how single-pitch tuning is actually used, when it is used at all. | |||
Both are examples of [[trivial temperament]]s. | |||
== Music == | == Music == | ||
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; [[No Clue Music]] | ; [[No Clue Music]] | ||
* [https://www.youtube.com/watch?v=g-WXe2mvAPk ''0 EDO Experiment''] (2024) | * [https://www.youtube.com/watch?v=g-WXe2mvAPk ''0 EDO Experiment''] (2024) | ||
[[Category:Limiting cases]] | [[Category:Limiting cases]] | ||
[[Category:Trivial temperaments]] | [[Category:Trivial temperaments]] | ||