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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]] |
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| == Specification ==
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| Its full specification is n-EPDp: n equal (pitch) divisions of interval p.
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| == Formula ==
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| 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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| <math>
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| c(k) = p^{\frac{k}{n}}
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| </math>
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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.
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| == Relationship to other tunings ==
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| === Vs. rank-1 temperaments & equal multiplications ===
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| 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.
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| === Vs. APS ===
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| 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¢).
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| == Examples ==
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| 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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| {| class="wikitable"
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| |+example: 4-EPDO = 4-EDO
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| |-
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| ! quantity
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| ! (0)
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| ! 1
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| ! 2
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| ! 3
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| ! 4
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| |-
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| ! frequency (f)
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| |(1)
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| |1.19
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| |1.41
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| |1.68
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| |2
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| |-
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| ! pitch (log₂f)
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| |(2⁰⸍⁴)
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| |2¹⸍⁴
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| |2²⸍⁴
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| |2³⸍⁴
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| |2⁴⸍⁴
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| |-
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| ! length (1/f)
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| |(1)
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| |0.84
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| |0.71
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| |0.59
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| |0.5
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| |}
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| [[Category:Equal-step tuning]]
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| [[Category:Equal divisions of the octave ]]
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