10edo: Difference between revisions

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== Theory ==

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.

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 {{nowrap|1 2 1 2 1 2 1}} ([[3L 4s]], or "mosh"), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. 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.

Taking the the 360 cent large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L 4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale.

~~Rather surprisingly~~, 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the harmonic series.

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|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.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===

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 == Theory ==
+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.
-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 {{nowrap|1 2 1 2 1 2 1}} ([[3L&nbsp;4s]], or "mosh"), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. 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.
+Taking the the 360 cent large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale.
-Rather surprisingly, 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the harmonic series.
+While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].
+One way to interpret it in terms of a [[Temperament|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.
+Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].
 === Prime harmonics ===