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An '''EPD''', or '''equal pitch division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | |||
Because pitch is the overwhelmingly most common musical resource to divide equally, this may be abbreviated to '''ED''', or '''equal division'''. | |||
== 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: | |||
<math> | |||
c(k) = p^{\frac{k}{n}} | |||
</math> | |||
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-EPDp 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 === | |||
One period of 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). | |||
{| class="wikitable" | |||
|+example: 4-EPDO = 4-EDO | |||
|- | |||
! quantity | |||
! (0) | |||
! 1 | |||
! 2 | |||
! 3 | |||
! 4 | |||
|- | |||
! frequency (''f'', ratio) | |||
|(1) | |||
|1.19 | |||
|1.41 | |||
|1.68 | |||
|2 | |||
|- | |||
! pitch (log₂''f'', octaves) | |||
|(2⁰⸍⁴) | |||
|2¹⸍⁴ | |||
|2²⸍⁴ | |||
|2³⸍⁴ | |||
|2⁴⸍⁴ | |||
|- | |||
! length (1/''f'', ratio) | |||
|(1) | |||
|0.84 | |||
|0.71 | |||
|0.59 | |||
|0.5 | |||
|} | |||
[[Category:Equal-step tuning]] | |||
[[Category:Equal divisions of the octave ]] | |||