10edo: Difference between revisions

{{EDO intro|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 {{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.  

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.

! Approximate ratios<ref>based on treating 10edo as a 2.15.7.13-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>

! colspan="3" | [[Ups and downs notation]]

([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)

See below: 3L 4s Mosh notation

=== 3L 4s (mosh) notation ===

! Degree

! Associated ratio

This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

 

====Evo-SZ flavor====

 

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

! rowspan="2" | Optimal<br>8ve stretch (¢)

{{Uniform map|13|9.5|10.5}}

! [[Harmonic limit|Prime<br>limit]]

! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>

! Periods <br> per 8ve

 

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1

* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1

* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]

[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

[[Category:Macrotonal]]