User:Cmloegcmluin/EPD: Difference between revisions

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

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

n-~~EDp~~: n equal (pitch) divisions of interval p (e.g. 12-EDO) (equivalent to rank-1 temperament of p/n)

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

<math>

c(k) = p^{\frac{k}{n}}

</math>

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-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).

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! 4

|-

! frequency (f)

! frequency (''f'', ratio)

|(1)

|1.19

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|2

|-

! pitch (~~log₂f~~)

! pitch (log₂''f'', octaves)

|(2⁰⸍⁴)

|2¹⸍⁴

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|2⁴⸍⁴

|-

! length (1/f)

! length (1/''f'', ratio)

|(1)

|0.84

@@ Line 1: / Line 1: @@
-An '''EPD''', or '''equal pitch division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.
+{{Editable user page}}
+An '''EPD''', or '''equal pitch division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-n-EDp: n equal (pitch) divisions of interval p (e.g. 12-EDO) (equivalent to rank-1 temperament of p/n)
+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).
@@ Line 15: / Line 42: @@
 ! 4
 |-
-! frequency (f)
+! frequency (''f'', ratio)
 |(1)
 |1.19
@@ Line 22: / Line 49: @@
 |2
 |-
-! pitch (log₂f)
+! pitch (log₂''f'', octaves)
 |(2⁰⸍⁴)
 |2¹⸍⁴
@@ Line 29: / Line 56: @@
 |2⁴⸍⁴
 |-
-! length (1/f)
+! length (1/''f'', ratio)
 |(1)
 |0.84