Single-pitch tuning: Difference between revisions
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Single-pitch tuning | '''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. | ||
==In equal | == In equal tunings == | ||
{{Infobox ET|0edo}} | {{Infobox ET|0edo}} | ||
'''0 equal divisions of the octave''' ('''0edo''') | |||
Single-pitch tuning can be specified as '''0 equal divisions of the octave''' ('''0edo'''), or 0ed-''p'' for any positive, finite number ''p''. | |||
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 you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one pitch without any octaves, which is, arguably, pure rhythm. | 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 you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one pitch without any octaves, which is, arguably, pure rhythm. | ||
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 could not use it as a tuning system. | 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 could not use it as 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 | == 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]]. This is sometimes called the '''Om temperment'''. The mapping for this is the 0-val, {{val| 0 0 … 0 }}. | |||
The name | The name ''Om'' is a reference to {{w|Om|that syllable's use in Hindu meditation practices}}; [[Keenan Pepper]] gave it this name because there is only one temperament-distinct pitch in the whole system, in the same way that ''Om'' in the meditation sense is the only word you need to create the whole universe. | ||
Being an example of a [[trivial temperament]], single-pitch tuning [[tempering out|tempers out]] all [[comma]]s and is [[consistent]] in all [[limit]]s. | |||
== Music == | |||
; [[Cryptovolans]], [[Reuben Gingrich]] | |||
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp]) | |||
; [[Elliott Carter]] | |||
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950) | |||
; [[Herman Miller]] | |||
* ''[https://soundcloud.com/morphosyntax-1/lost-at-c Lost at C]'' (2021) | |||
; [[No Clue Music]] | |||
* [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 22:29, 10 August 2025
|
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. 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 0ed-p for any positive, finite number p.
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 you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one pitch without any octaves, which is, arguably, pure rhythm.
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 could not use it as 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. This is sometimes called the Om temperment. The mapping for this is the 0-val, ⟨0 0 … 0].
The name Om is a reference to that syllable's use in Hindu meditation practices; Keenan Pepper gave it this name because there is only one temperament-distinct pitch in the whole system, in the same way that Om in the meditation sense is the only word you need to create the whole universe.
Being an example of a trivial temperament, single-pitch tuning tempers out all commas and is consistent in all limits.
Music
- Many Birds Pecking On Wood (2021) (Bandcamp)
- Lost at C (2021)
- 0 EDO Experiment (2024)