10edo: Difference between revisions

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~~'''~~10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==

~~{{Harmonics in equal|10}}~~

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===

== Intervals ==

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 {{Infobox ET}}
-'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.
+{{EDO intro|10}}
 == Theory ==
-{{Harmonics in equal|10}}
 edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
+=== Prime harmonics ===
+{{Harmonics in equal|10}}
 == Intervals ==