Single-pitch tuning: Difference between revisions
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→Music: Chris Vaisvil's ''0-EDO for Orchestra'' (2021): Really in 1edo, so moved there |
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{{ | {{Mathematical interest}} | ||
'''Single-pitch tuning''' is a [[tuning system]] that contains only a single pitch. It contrasts [[1edo]] because it does not even have [[2/1|octaves]]. It also contrasts unpitched tuning, where no recognizable pitches exist. Single-pitch tuning can be described in terms of various models of tuning. | '''Single-pitch tuning''' is a [[tuning system]] that contains only a single pitch, and no single interval above or below it. It contrasts [[1edo]] because it does not even have [[2/1|octaves]]. It also contrasts unpitched tuning, where no recognizable pitches exist. Single-pitch tuning can be described in terms of various models of tuning. | ||
== 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 | 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 | 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. | ||
An alternative interpretation is that given that ''n''-edo means that you are dividing the octave into equal divisions with a logarithmic size of 1/''n'' octaves, and that [[1/0]] is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus | One musical application of 0edo is to use it with pure rhythm, or with change of timbre, though the latter can be disputed if harmonics are counted as distinct pitches. In the most purist sense, 0edo would be sonically similar to Morse code, only using one sinewave at a fixed frequency at different amplitudes with time. | ||
An alternative interpretation is that given that ''n''-edo means that you are dividing the octave into equal divisions with a logarithmic size of 1/''n'' octaves, and that [[1/0]] is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus not a tuning system. | |||
As a result of the step size of 0edo being infinite, the [[relative interval error|relative error]] of all intervals is zero. | As a result of the step size of 0edo being infinite, the [[relative interval error|relative error]] of all intervals is zero. | ||
== 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]] | 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 == | ||
; [[Cryptovolans]], [[Reuben Gingrich]] | ; [[Cryptovolans]], [[Reuben Gingrich]] | ||
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben | * ''Many Birds Pecking On Wood'' (2021) – [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben SoundCloud] | [https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp] | ||
; [[Elliott Carter]] | ; [[Elliott Carter]] | ||
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* [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:Trivial temperaments]] | |||
[[Category:Limiting | |||
[[Category:Trivial | |||
Latest revision as of 03:39, 17 May 2026
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
Single-pitch tuning is a tuning system that contains only a single pitch, and no single interval above or below it. It contrasts 1edo because it does not even have octaves. It also contrasts unpitched tuning, where no recognizable pitches exist. Single-pitch tuning can be described in terms of various models of tuning.
In equal tunings
| 0edo | 1edo → |
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.
One musical application of 0edo is to use it with pure rhythm, or with change of timbre, though the latter can be disputed if harmonics are counted as distinct pitches. In the most purist sense, 0edo would be sonically similar to Morse code, only using one sinewave at a fixed frequency at different amplitudes with time.
An alternative interpretation is that given that n-edo means that you are dividing the octave into equal divisions with a logarithmic size of 1/n octaves, and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus not a tuning system.
As a result of the step size of 0edo being infinite, the relative error of all intervals is zero.
In regular temperament theory
Single-pitch tuning corresponds to the regular temperament in any given subgroup where all primes in that subgroup are tempered out, resulting in a rank-0 temperament with no generator. The mapping for this is the 0-val, ⟨0 0 … 0], or more precisely, the rank-0 matrix, [ ]. Since it maps all intervals to the same pitch, it tempers out all commas and is consistent in all limits.
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 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 temperaments.
Music
- Many Birds Pecking On Wood (2021) – SoundCloud | Bandcamp
- Lost at C (2021)
- 0 EDO Experiment (2024)