13edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|13}} | {{EDO intro|13}} | ||
== Theory == | == Theory == | ||
In 13edo, the steps less than 600¢ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place. | In 13edo, the steps less than 600¢ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place. | ||
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== JI | == Approximation to JI == | ||
=== Selected 13-odd-limit intervals === | |||
[[File:13ed2-001.svg|alt=alt : Your browser has no SVG support.]] | [[File:13ed2-001.svg|alt=alt : Your browser has no SVG support.]] | ||
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13edo can be approximated by a circle of [[64/49]] subminor fourths (which can be tuned by tuning two [[7/4]] subminor sevenths). A stack of 13 of these subfourths closes with an error of +10.526432¢, or +11% of 13-edo's step size. | 13edo can be approximated by a circle of [[64/49]] subminor fourths (which can be tuned by tuning two [[7/4]] subminor sevenths). A stack of 13 of these subfourths closes with an error of +10.526432¢, or +11% of 13-edo's step size. | ||
== | == 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 EDOs, 13-EDO 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 13-EDO is the only EDO that tempers the ratios of the Fibonacci sequence into a single interval. | 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, 13-EDO 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 13-EDO is the only EDO that tempers the ratios of the Fibonacci sequence into a single interval. | ||