6edo: Difference between revisions
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'''6 equal divisions of the octave''' (''' | '''6 equal divisions of the octave''' ('''6edo''') is the [[tuning system]] derived by dividing the [[octave]] into 6 equal steps of 200 [[cent]]s each, or the sixth root of 2. It is also known as the '''whole tone scale'''. As a subset of [[12edo|12edo]], it can be notated on a five-line staff with standard notation. It is the first [[edo]] that is not a [[The_Riemann_zeta_function_and_tuning #Zeta_edo_lists|zeta peak]], has lower [[Consistency_levels_of_small_edos|consistency]] than the one that precedes it, and the highest edo that has no single period mode of symmetry scales other than using the single step as a generator. This means it is relatively poor for it's size at creating traditional tonal music, with 5edo and 7edo both having much better representations of the third harmonic, but has still seen more use than most edos other than 12, since it can be played on any 12 tone instrument. | ||
== Theory == | == Theory == | ||
{{primes in | {{primes in equal|6}} | ||
While | While 6edo does not well approximate the 3rd harmonic, it does contain a good approximation of the 9th harmonic. Therefore, 6edo can be treated as a 2.5.7.9 subgroup temperament. | ||
Related | Related edos: | ||
* Subsets: [[2edo| | * Subsets: [[2edo|2edo]], [[3edo|3edo]] | ||
* Supersets: [[12edo| | * Supersets: [[12edo|12edo]], [[18edo|18edo]], [[24edo|24edo]]... | ||
* Neighbours: [[5edo| | * Neighbours: [[5edo|5edo]], [[7edo|7edo]] | ||
== Intervals == | == Intervals == | ||
| Line 76: | Line 76: | ||
== Commas == | == Commas == | ||
6edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 6 10 14 17 21 22 }}. | |||
{| class="commatable wikitable center-1 center-2 right-4 center-5" | {| class="commatable wikitable center-1 center-2 right-4 center-5" | ||
Revision as of 00:51, 15 January 2022
6 equal divisions of the octave (6edo) is the tuning system derived by dividing the octave into 6 equal steps of 200 cents each, or the sixth root of 2. It is also known as the whole tone scale. As a subset of 12edo, it can be notated on a five-line staff with standard notation. It is the first edo that is not a zeta peak, has lower consistency than the one that precedes it, and the highest edo that has no single period mode of symmetry scales other than using the single step as a generator. This means it is relatively poor for it's size at creating traditional tonal music, with 5edo and 7edo both having much better representations of the third harmonic, but has still seen more use than most edos other than 12, since it can be played on any 12 tone instrument.
Theory
While 6edo does not well approximate the 3rd harmonic, it does contain a good approximation of the 9th harmonic. Therefore, 6edo can be treated as a 2.5.7.9 subgroup temperament.
Related edos:
Intervals
| Steps | Cents | Interval | Approximate JI Ratios* | ||
|---|---|---|---|---|---|
| 0 | 0 | unison | P1 | D | 1/1 |
| 1 | 200 | major 2nd | M2 | E | 8/7, 9/8, 10/9 |
| 2 | 400 | major 3rd | M3 | F# | 5/4, 9/7 |
| 3 | 600 | aug 4th, dim 5th | A4, d5 | G#, Ab | 7/5, 10/7 |
| 4 | 800 | minor 6th | m6 | Bb | 8/5, 14/9 |
| 5 | 1000 | minor 7th | m7 | C | 7/4, 9/5, 16/9 |
| 6 | 1200 | perfect 8ve | P8 | D | 2/1 |
* based on treating 6edo as a 2.5.7.9 subgroup temperament; other approaches are possible.
Commas
6edo tempers out the following commas. This assumes val ⟨6 10 14 17 21 22].
| Prime Limit |
Ratio[1] | Monzo | Cents | Color name | Name(s) |
|---|---|---|---|---|---|
| 3 | 32/27 | [5 -3⟩ | 294.13 | Wa | Pythagorean minor third |
| 5 | 25/24 | [-3 -1 2⟩ | 70.67 | Yoyo | Classic chromatic semitone |
| 5 | 128/125 | [7 0 -3⟩ | 41.06 | Trigu | Diesis, augmented comma |
| 5 | 3125/3072 | [-10 -1 5⟩ | 29.61 | Laquinyo | Small diesis, magic comma |
| 5 | (12 digits) | [17 1 -8⟩ | 11.45 | Saquadbigu | Würschmidt comma |
| 5 | (30 digits) | [-44 -3 21⟩ | 6.72 | Trila-septriyo | Mutt comma |
| 7 | 49/48 | [-4 -1 0 2⟩ | 35.70 | Zozo | Slendro diesis |
| 7 | 50/49 | [1 0 2 -2⟩ | 34.98 | Biruyo | Tritonic diesis, jubilisma |
| 7 | 3136/3125 | [6 0 -5 2⟩ | 6.08 | Zozoquingu | Hemimean |
| 7 | 6144/6125 | [11 1 -3 -2⟩ | 5.36 | Sarurutrigu | Porwell |
| 7 | 2401/2400 | [-5 -1 -2 4⟩ | 0.72 | Bizozogu | Breedsma |
| 11 | 121/120 | [-3 -1 -1 0 2⟩ | 14.37 | Lologu | Biyatisma |
| 11 | 176/175 | [4 0 -2 -1 1⟩ | 9.86 | Lorugugu | Valinorsma |
| 11 | 385/384 | [-7 -1 1 1 1⟩ | 4.50 | Lozoyo | Keenanisma |
| 13 | 13/12 | [-2 -1 0 0 0 1⟩ | 138.57 | tho 2nd | Tridecimal neutral second |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
Music
| Title | Composer | Year | Genre | Additional links |
|---|---|---|---|---|
| "Dvandva" | Milan Guštar | 1987/2007 | Folk | |
| The Good Boundless | Chris Vaisvil | 2011 (?) | Jazz | Lyrics (personal website) |
| Prelude in 6ET | Aaron Andrew Hunt | 2015 | Neobaroque | |
| Invention in 6ET | Aaron Andrew Hunt | 2015 | Neobaroque | |
| "Exiting" (from Edolian) | NullPointerException Music | 2020 | Classical | |
| "Bowser breaks into Arnold Schoenberg's house and steals six of the twelve Tone Crystals (every other one), activating The 666666-Year-Curse Mechanism" (from STAFFcirc vol. 7) | Chimeratio | 2021 | Electronic | Album (Bandcamp) |