10edo: Difference between revisions

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10edo can be thought of as two circles of [[5edo]] separated by 120 cents. 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{{c}} 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, and can be seen as a neutralized diatonic scale.

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scale|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, and can be seen as a neutralized diatonic scale.

It shares [[5edo]]'s approximation quality in the 2.3.7 subgroup (though the detuned fifth could be seen as a bigger problem with the more fine division of steps), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all.

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This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.

By treating ~~360c~~ as 11/9, we arrive at 11/8 = ~~600c~~ (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.

By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.

10edo is a [[~~The Riemann zeta function and tuning #Zeta edo lists|~~zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 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]].

10edo is a [[zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 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]].

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

@@ Line 11: / Line 11: @@
 edo can be thought of as two circles of [[5edo]] separated by 120 cents. 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{{c}} 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, and can be seen as a neutralized diatonic scale.
+Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scale|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, and can be seen as a neutralized diatonic scale.
 It shares [[5edo]]'s approximation quality in the 2.3.7 subgroup (though the detuned fifth could be seen as a bigger problem with the more fine division of steps), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all.
@@ Line 17: / Line 17: @@
 This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.
-By treating 360c as 11/9, we arrive at 11/8 = 600c (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.
+By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.
-edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 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]].
+edo is a [[zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 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]].
 Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].