13edo: Difference between revisions
m Section titles |
m →Approximation to irrational intervals: direct mapping -> direct approximation; style |
||
| Line 1,094: | Line 1,094: | ||
== Approximation to irrational intervals == | == Approximation to irrational intervals == | ||
=== Acoustic phi === | === Acoustic phi === | ||
13edo has a very close approximation of [[acoustic phi]] (9\13), with only -2.3 cents of error. [[23edo]] and [[36edo]] are even closer, but unlike all closer | 13edo has a very close approximation of [[acoustic phi]] (9\13), with only -2.3 cents of error. [[23edo]] and [[36edo]] are even closer, but unlike all closer edos, 13edo has no other interval that represents any ratio from the Fibonacci sequence (3/2, 5/3, 8/5, 13/8, 21/13, etc.) except of course for 1/1 and 2/1. In a way, one could say that 13edo is the only edo that tempers the ratios of the Fibonacci sequence into a single interval. | ||
See also: [[9edϕ]] | See also: [[9edϕ]] | ||
| Line 1,101: | Line 1,101: | ||
As a coïncidence, 13edo also has a very close appoximation of [[logarithmic phi]] (21\13), with only -3.2 cents of error. | As a coïncidence, 13edo also has a very close appoximation of [[logarithmic phi]] (21\13), with only -3.2 cents of error. | ||
Not until [[144edo|144]] do we find a better | Not until [[144edo|144]] do we find a better edo in terms of relative error on these two intervals. | ||
However, it should be noted that when we are hearing logarithmic phi, we are in fact hearing 2<sup>ϕ</sup> ≃ 3.070. While this interval can still be used in a way or another as a useful tone in a piece of music, it doesn't correspond to anything. When it comes to acoustic phi, we are truly hearing the mathematical constant ϕ ≃ 1.6180. | However, it should be noted that when we are hearing logarithmic phi, we are in fact hearing 2<sup>ϕ</sup> ≃ 3.070. While this interval can still be used in a way or another as a useful tone in a piece of music, it doesn't correspond to anything. When it comes to acoustic phi, we are truly hearing the mathematical constant ϕ ≃ 1.6180. | ||
| Line 1,108: | Line 1,108: | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|+Direct | |+Direct approximation | ||
|- | |- | ||
! Interval | ! Interval | ||