Chain-of-fifths notation: Difference between revisions

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To notate edos, one of the intervals must be selected as the fifth. Edos that are best supported by this system are those whose fifth does not deviate too much from the pure fifth [[3/2]] (702{{cent}}) and that can be represented by only one ring of fifths. [[24edo]], as a counter-example to this, contains two rings. If we as well demand that whole tones (2 × P5 - P8), diatonic semitones (3 × P8 - 5 × P5), and chromatic semitones (shifts caused by one accidental, 7 × P5 - 4 × P8), use a positive number of steps, we exclude all edos below 12 and also {{EDOs| 13, 16, 18, and 23 }}. They make more sense notated as subsets. For example, 13edo can be notated as a subset of [[26edo]].

The '''neutral circle-of-fifths notation''' (aka '''quartertone notation''') uses an extended accidental set including '''demisharps''' and '''demiflats'''. It works for any ~~tuning system~~ generated by a neutral third. ~~The~~ [[mohaha]] temperament and its typical edo tunings ([[17edo]], 24edo, [[31edo]], [[38edo]], [[45edo]]) ~~are well represented by this system~~.

Any regular rank-2 temperament generated by the 8ve and the 5th (i.e. one with the unsplit [[pergen]]) can be notated this way. Because it's rank-2, the circle of fifths is actually a (theoretically infinite) chain of fifths.

The '''neutral circle-of-fifths notation''' (aka '''quartertone notation''') uses an extended accidental set including '''demisharps''' and '''demiflats'''. It works for any rank-2 temperament generated by an octave and a neutral third, i.e. those with a pergen of (P8, P5/2). It also works for any edo of [[sharpness]] 2. Examples are the [[mohaha]] temperament and its typical edo tunings ([[17edo]], 24edo, [[31edo]], [[38edo]], [[45edo]]).

== Edos up to 100 ==

Edos up to 100 are listed in the following tables. The unit (if not stated otherwise) is ''steps'' of the corresponding edo which is given in the first column of each row. The list contains only those edos whose all degrees can be reached by stacking the [[direct approximation]] of the fifth in the respective edo.

Edos up to 100 are listed in the following tables. The unit (if not stated otherwise) is ''steps'' of the corresponding edo which is given in the first column of each row. The list contains only those edos whose all degrees can be reached by stacking the [[direct approximation]] of the fifth in the respective edo. The last two columns are the edo's [[Sharpness|pentasharpness and sharpness]] respectively.

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 To notate edos, one of the intervals must be selected as the fifth. Edos that are best supported by this system are those whose fifth does not deviate too much from the pure fifth [[3/2]] (702{{cent}}) and that can be represented by only one ring of fifths. [[24edo]], as a counter-example to this, contains two rings. If we as well demand that whole tones (2 × P5 - P8), diatonic semitones (3 × P8 - 5 × P5), and chromatic semitones (shifts caused by one accidental, 7 × P5 - 4 × P8), use a positive number of steps, we exclude all edos below 12 and also {{EDOs| 13, 16, 18, and 23 }}. They make more sense notated as subsets. For example, 13edo can be notated as a subset of [[26edo]].
-The '''neutral circle-of-fifths notation''' (aka '''quartertone notation''') uses an extended accidental set including '''demisharps''' and '''demiflats'''. It works for any tuning system generated by a neutral third. The [[mohaha]] temperament and its typical edo tunings ([[17edo]], 24edo, [[31edo]], [[38edo]], [[45edo]]) are well represented by this system.
+Any regular rank-2 temperament generated by the 8ve and the 5th (i.e. one with the unsplit [[pergen]]) can be notated this way. Because it's rank-2, the circle of fifths is actually a (theoretically infinite) chain of fifths.
+The '''neutral circle-of-fifths notation''' (aka '''quartertone notation''') uses an extended accidental set including '''demisharps''' and '''demiflats'''. It works for any rank-2 temperament generated by an octave and a neutral third, i.e. those with a pergen of (P8, P5/2). It also works for any edo of [[sharpness]] 2. Examples are the [[mohaha]] temperament and its typical edo tunings ([[17edo]], 24edo, [[31edo]], [[38edo]], [[45edo]]).
 == Edos up to 100 ==
-Edos up to 100 are listed in the following tables. The unit (if not stated otherwise) is ''steps'' of the corresponding edo which is given in the first column of each row. The list contains only those edos whose all degrees can be reached by stacking the [[direct approximation]] of the fifth in the respective edo.
+Edos up to 100 are listed in the following tables. The unit (if not stated otherwise) is ''steps'' of the corresponding edo which is given in the first column of each row. The list contains only those edos whose all degrees can be reached by stacking the [[direct approximation]] of the fifth in the respective edo. The last two columns are the edo's [[Sharpness|pentasharpness and sharpness]] respectively.
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