User:Cmloegcmluin/EPD: Difference between revisions
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An '''EPD''', or '''equal pitch division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | An '''EPD''', or '''equal pitch division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | ||
== Specification == | |||
Its full specification is n-EPDp: n equal (pitch) divisions of interval p. | |||
== Formula == | |||
To find the step size for an n-EPDp, take the nth root of p. For example, the step of 12-EDO is <span><math>2^{\frac{1}{12}}</math></span>. So the formula for the kth step of an n-EPDp is: | To find the step size for an n-EPDp, take the nth root of p. For example, the step of 12-EDO is <span><math>2^{\frac{1}{12}}</math></span>. So the formula for the kth step of an n-EPDp is: | ||
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This way, when <span><math>k</math></span> is <span><math>0</math></span>, <span><math>c(k)</math></span> is simply <span><math>1</math></span>, because any number to the 0th power is 1. And when <span><math>k</math></span> is <span><math>n</math></span>, <span><math>c(k)</math></span> is simply <span><math>p</math></span>, because any number to the 1st power is itself. | This way, when <span><math>k</math></span> is <span><math>0</math></span>, <span><math>c(k)</math></span> is simply <span><math>1</math></span>, because any number to the 0th power is 1. And when <span><math>k</math></span> is <span><math>n</math></span>, <span><math>c(k)</math></span> is simply <span><math>p</math></span>, because any number to the 1st power is itself. | ||
== Relationship to other tunings == | |||
=== vs. rank-1 temperaments & equal multiplications === | |||
An n-EPDn is equivalent to a [[Tour_of_Regular_Temperaments#Equal_temperaments_.28Rank-1_temperaments.29|rank-1 temperament]] of p/n, or an [[Equal-step_tuning#Equal_multiplications|equal multiplication]] of p/n. | |||
=== vs. APS === | |||
An EPD will be equivalent to some [[APS|APS, or arithmetic pitch sequence]], which has had its count of pitches specified by prefixing "n-". Specifically, n-EPDx = n-APS(x/n), for example 12-EPD1200¢ = 12-APS(1200¢/12=100¢). | |||
== Examples == | |||
The most common example of this type of tuning is 12-EDO, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12-EPDO, for 12 equal '''pitch''' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12-EDO is the better name). | |||
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Revision as of 20:38, 24 March 2021
An EPD, or equal pitch division, is a kind of arithmetic and harmonotonic tuning.
Specification
Its full specification is n-EPDp: n equal (pitch) divisions of interval p.
Formula
To find the step size for an n-EPDp, take the nth root of p. For example, the step of 12-EDO is [math]\displaystyle{ 2^{\frac{1}{12}} }[/math]. So the formula for the kth step of an n-EPDp is:
[math]\displaystyle{ c(k) = p^{\frac{k}{n}} }[/math]
This way, when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ 0 }[/math], [math]\displaystyle{ c(k) }[/math] is simply [math]\displaystyle{ 1 }[/math], because any number to the 0th power is 1. And when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ n }[/math], [math]\displaystyle{ c(k) }[/math] is simply [math]\displaystyle{ p }[/math], because any number to the 1st power is itself.
Relationship to other tunings
vs. rank-1 temperaments & equal multiplications
An n-EPDn is equivalent to a rank-1 temperament of p/n, or an equal multiplication of p/n.
vs. APS
An EPD will be equivalent to some APS, or arithmetic pitch sequence, which has had its count of pitches specified by prefixing "n-". Specifically, n-EPDx = n-APS(x/n), for example 12-EPD1200¢ = 12-APS(1200¢/12=100¢).
Examples
The most common example of this type of tuning is 12-EDO, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12-EPDO, for 12 equal pitch divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12-EDO is the better name).
| quantity | (0) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| frequency (f) | (1) | 1.19 | 1.41 | 1.68 | 2 |
| pitch (log₂f) | (2⁰⸍⁴) | 2¹⸍⁴ | 2²⸍⁴ | 2³⸍⁴ | 2⁴⸍⁴ |
| length (1/f) | (1) | 0.84 | 0.71 | 0.59 | 0.5 |