User:Cmloegcmluin/EPD: Difference between revisions
Cmloegcmluin (talk | contribs) No edit summary |
mNo edit summary |
||
| (17 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
An '''EPD''', or '''equal pitch division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[ | {{Editable user page}} | ||
An '''EPD''', or '''equal pitch division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | |||
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). | 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" | {| class="wikitable" | ||
|+example: | |+example: 4-EPDO = 4-EDO | ||
|- | |- | ||
! quantity | ! quantity | ||
! (0) | |||
! 1 | ! 1 | ||
! 2 | ! 2 | ||
! 3 | ! 3 | ||
! 4 | ! 4 | ||
|- | |- | ||
! frequency | ! frequency (''f'', ratio) | ||
| | |(1) | ||
| | |1.19 | ||
| | |1.41 | ||
| | |1.68 | ||
| | |2 | ||
|- | |- | ||
! pitch | ! pitch (log₂''f'', octaves) | ||
| | |(2⁰⸍⁴) | ||
| | |2¹⸍⁴ | ||
| | |2²⸍⁴ | ||
| | |2³⸍⁴ | ||
| | |2⁴⸍⁴ | ||
|- | |- | ||
! length | ! length (1/''f'', ratio) | ||
| | |(1) | ||
| | |0.84 | ||
| | |0.71 | ||
| | |0.59 | ||
| | |0.5 | ||
|} | |} | ||
[[Category:Equal-step tuning]] | |||
[[Category:Equal divisions of the octave ]] | |||