10edo: Difference between revisions

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10EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|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 happy 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, 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.

10EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|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 happy 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, 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.

== Intervals ==

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! Degree

! Cents

! Approximate Ratios<ref>based on treating ~~10-EDO~~ as a 2.7.13.15 subgroup temperament</ref>

! Approximate Ratios<ref>based on treating 10EDO as a 2.7.13.15 subgroup temperament</ref>

! Additional Ratios <br> of 3, 5 and 9<ref>adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals</ref>

! Interval Names

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{| class="wikitable"

|+

!N

! N

! L-Nedo

! s-Nedo

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* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]

* [https://youtu.be/CPmlCcJZGZQ "Vidya"] by Sevish (from his 2017 album "Harmony Hacker")

* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10~~-~~РДО) | Strunka (10~~-~~EDO) - YouTube] by Dmitriy Bazhenov

* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) | Strunka (10-EDO) - YouTube] by Dmitriy Bazhenov

[[Category:Equal divisions of the octave]]

@@ Line 17: / Line 17: @@
 {{Odd harmonics in edo|edo=10}}
-EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|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 happy 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, 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.
+EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|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 happy 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, 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.
 == Intervals ==
@@ Line 23: / Line 23: @@
 ! Degree
 ! Cents
-! Approximate Ratios<ref>based on treating 10-EDO as a 2.7.13.15 subgroup temperament</ref>
+! Approximate Ratios<ref>based on treating 10EDO as a 2.7.13.15 subgroup temperament</ref>
 ! Additional Ratios <br> of 3, 5 and 9<ref>adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals</ref>
 ! Interval Names
@@ Line 155: / Line 155: @@
 {| class="wikitable"
 |+
-!N
+! N
 ! L-Nedo
 ! s-Nedo
@@ Line 492: / Line 492: @@
 * [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
 * [https://youtu.be/CPmlCcJZGZQ "Vidya"] by Sevish (from his 2017 album "Harmony Hacker")
-* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10&#45;РДО) &#124; Strunka (10&#45;EDO) &#45; YouTube] by Dmitriy Bazhenov
+* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &#124; Strunka (10-EDO) &#45; YouTube] by Dmitriy Bazhenov
 [[Category:Equal divisions of the octave]]