DKW theory: Difference between revisions
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== DKW theory in the 5-limit == | == DKW theory in the 5-limit == | ||
By far the most important three-prime subgroup is the [[5-limit]], i.e. the 2.3.5 subgroup. It turns out that the 2.3.5 tonality diamond's ordering is type ''N-VI'', and that its signature is '''C''' = [[9/8]] : '''B'' = [[16/15]] : '''A''' = [[25/24]]. | By far the most important three-prime subgroup is the [[5-limit]], i.e. the 2.3.5 subgroup. It turns out that the 2.3.5 tonality diamond's ordering is type ''N-VI'', and that its signature is '''C''' = [[9/8]] : '''B''' = [[16/15]] : '''A''' = [[25/24]]. | ||
Below are commas with simple expressions in terms of these intervals, with "complexity" measured as the sum of the absolute values of powers of '''A''', '''B''', and '''C'''. | Below are commas with simple expressions in terms of these intervals, with "complexity" measured as the sum of the absolute values of powers of '''A''', '''B''', and '''C'''. | ||
Note all commas considered here (aside from complexity-1) are those that contain at least one minus sign in their signature expression. | |||
{| class="wikitable center-all" | |||
|- | |||
! Comma | |||
! Defined<br>temperament | |||
! Complexity | |||
! Subgroup monzo<br>(9/8.16/15.25/24) | |||
|- | |||
| [[9/8]] | |||
| [[Very low accuracy temperaments #Antitonic|Antitonic]] | |||
| 1 | |||
| {{Monzo| 1 0 0 }} | |||
|- | |||
| [[16/15]] | |||
| [[Father]] | |||
| 1 | |||
| {{Monzo| 0 1 0 }} | |||
|- | |||
| [[25/24]] | |||
| [[Dicot]] | |||
| 1 | |||
| {{Monzo| 0 0 1 }} | |||
|- | |||
| [[27/25]] | |||
| [[Bug]] | |||
| 2 | |||
| {{Monzo| 1 0 -1 }} | |||
|- | |||
| [[135/128]] | |||
| [[Mavila]] | |||
| 2 | |||
| {{Monzo| 1 -1 0 }} | |||
|- | |||
| [[128/125]] | |||
| [[Augmented (temperament)|Augmented]] | |||
| 2 | |||
| {{Monzo| 0 1 -1 }} | |||
|- | |||
| [[648/625]] | |||
| [[Diminished (temperament)|Diminished]] | |||
| 3 | |||
| {{Monzo| 1 0 -2 }} | |||
|- | |||
| [[2048/2025]] | |||
| [[Diaschismic]] | |||
| 3 | |||
| {{Monzo| -1 2 0 }} | |||
|- | |||
| [[3125/3072]] | |||
| [[Magic]] | |||
| 3 | |||
| {{Monzo| 0 -1 2 }} | |||
|- | |||
| [[81/80]] | |||
| [[Meantone]] | |||
| 3 | |||
| {{Monzo| 1 -1 -1 }} | |||
|} | |||
[[Category:Regular temperament tuning]] | [[Category:Regular temperament tuning]] | ||
[[Category:Diamond]] | [[Category:Diamond]] | ||
{{Todo| add examples }} | {{Todo| add examples }} | ||