Single-pitch tuning: Difference between revisions
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{{Mathematical interest}} | {{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}} | ||
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 anything. | ||
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. | ||
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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, [ ]. | ||
Being an example of a [[trivial temperament]], single-pitch tuning [[tempering out|tempers out]] | Being an example of a [[trivial temperament]], single-pitch tuning [[tempering out|tempers out]] everything, and is [[consistent]] in all [[limit]]s. | ||
== Music == | == Music == | ||
Revision as of 02:40, 31 December 2025
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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 anything.
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, [ ].
Being an example of a trivial temperament, single-pitch tuning tempers out everything, and is consistent in all limits.
Music
- Many Birds Pecking On Wood (2021) – SoundCloud | Bandcamp
- Lost at C (2021)
- 0 EDO Experiment (2024)
- 0-EDO for Orchestra (2021)