User:Cmloegcmluin/EPD: Difference between revisions

Line 1:

An '''EPD''', or '''equal pitch division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.

Its full specification is n-EDp: n equal (pitch) divisions of interval p. It is equivalent to a rank-1 temperament of p/n, or an [[Equal-step_tuning#Equal_multiplications|equal multiplication]] of p/n.

== Specification ==

~~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¢~~).

Its full specification is n-EPDp: n equal (pitch) divisions of interval p.

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).

== 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>2^{\frac{1}{12}}</math>. So the formula for the kth step of an n-EPDp is:

Line 14:

This way, when <math>k</math> is <math>0</math>, <math>c(k)</math> is simply <math>1</math>, because any number to the 0th power is 1. And when <math>k</math> is <math>n</math>, <math>c(k)</math> is simply <math>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 [[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"

@@ Line 1: / Line 1: @@
 An '''EPD''', or '''equal pitch division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-Its full specification is n-EDp: n equal (pitch) divisions of interval p. It is equivalent to a rank-1 temperament of p/n, or an [[Equal-step_tuning#Equal_multiplications|equal multiplication]] of p/n.
+== Specification ==
-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¢).
+Its full specification is n-EPDp: n equal (pitch) divisions of interval p.
-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).
+== 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:
@@ Line 14: / Line 14: @@
 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"