21edo: Difference between revisions

{{EDO intro|21}}

21edo provides both [[7edo]] as a subset and the familiar 400-[[cent]] major third, while also giving some higher-[[limit]] [[JI]] possibilities. The system can be treated as three intertwining 7edo or "equiheptatonic" scales, or as seven [[3edo]] ''augmented'' triads. The [[7/4]] at 971.43{{cent}} is only off in 21edo by 2.60{{cent}} from just (968.83{{cent}}), which is better than any other [[edo]] below 26.

In diatonically-related terms, 21edo possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.

Of harmonics 3, 5, 7, 11, and 13, the only harmonic 21edo approximates with anything approaching a near-just flavor is the 7th harmonic. On the other hand, 21edo provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3{{cent}} or less), as well as a very reasonable approximation of the 27th harmonic (around 8{{cent}} sharp). As such, treating 21edo as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate JI interpretation of the tuning, since almost every interval in 21edo can be described as a ratio within the 29-odd-limit. 21edo also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.  

! [[5L 3s]] octotonic <br> notation

 

===Sagittal notation===

This notation uses the same sagittal sequence as [[16edo#Sagittal notation|16-EDO]], is a subset of the notation for [[42edo#Sagittal notation|42b]], and is a superset of the notation for [[7edo#Sagittal notation|7-EDO]].

The rank-3 scale [[diasem]] (323132313 or 313231323 in 21edo) is the 21edo tempering of [[Zarlino]] diatonic with 1\21 comma steps added, resulting in two "major seconds" (171{{cent}} and 228{{cent}}), two "minor thirds" (286{{cent}} and 343{{cent}}) and two "fourths" (457{{cent}} and 514{{cent}}). 21edo is the smallest edo to support a non-degenerate diasem (with L:M:S ratio 3:2:1).

The [[patent val]] for 21 EDO tempers out [[128/125]] and [[2187/2000]] in the [[5-limit]], and supplies the [[optimal patent val]] for the 5-limit [[laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-, 11- and 13-limit [[gorgo]], and 11- and 13-limit spartan. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.

{{Uniform map|13|20.5|21.5}}

21edo [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 21 33 49 59 73 78 }}.)

| Apotome

| Diesis, Augmented Comma

| [[Ampersand]], Ampersand's Comma

| [[Escapade]] Comma

| Septimal Quarter Tone

| Septimal Kleisma, Marvel Comma

| Mirkwai

| [[Akjaysma]], 5\7 Octave Comma

| Wizardharry

[[List of 21edo rank two temperaments by badness]]

! Periods per octave

| [[Escapade family#Escapade|Escapade]]

| [[Gamelismic clan#Miracle|Miracle]]

| [[Slendric]]/[[Gamelismic clan#Gorgo|Gorgo]]/[[Gamelismic clan#Gidorah|Gidorah]]

| [[Mint temperaments#Subklei|Subklei]]

| [[Subgroup temperaments#Tridec|Tridec]]

| [[Marvel temperaments#Triton|Triton]]

|  

| [[Augmented family|Augmented]]/[[August]]

| [[Apotome family|Whitewood]]

== Books / Literature ==

* [https://www.youtube.com/watch?v=pOJhks06uLI ''Gordon Guide''] (2024) – gorgo[11] in 21edo tuning