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.
#redirect [[Equal-step 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:
 
<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-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).
 
{| class="wikitable"
|+example: 4-EPDO = 4-EDO
|-
! 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
|}
 
[[Category:Equal-step tuning‏‎]]
[[Category:Equal divisions of the octave‏‎ ]]

Revision as of 19:55, 14 February 2023

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