Chain-of-fifths notation: Difference between revisions
Followed Flora |
Circle -> chain per discussion on Discord |
||
| Line 1: | Line 1: | ||
The ''' | The '''chain-of-fifths notation''' (aka '''extended Pythagorean notation''') is suitable for a variety of [[tuning system]]s which are [[octave]] repeating and generated by the fifth. A good number of [[edo]]s and [[regular temperament]]s can be notated this way, as it generalizes the traditional classical notation system for the [[Pythagorean tuning]], the [[meantone]] tunings, and [[12edo]]. It uses the seven natural notes of the [[diatonic]] scale and accidentals (<span style="font-size:larger">♯, ♭</span> and their multiples) to sharpen and flatten these seven notes by the [[chromatic semitone|augmented unison aka the chromatic semitone]]. Any regular rank-2 temperament generated by the 8ve and the 5th (i.e. one with the unsplit [[pergen]]) can be notated this way. | ||
Chain-of-fifths notation only works for [[Ring number|single-ring]] edos. A counter-example is [[24edo]], which is double-ring. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. For any multi-sharpness edos, this notation causes the notes to run out of order. For example, 17edo would run C Db C# D Eb D# E… For negative sharpness edos the accidentals will be inverse. One can avoid these by using [[ups and downs notation]], or for certain edos by using half-sharps (see below). Edos whose fifth has a high relative error makes more sense considered as [[dual-fifth]], and notated as subsets. For example, 13edo can be notated as a subset of 26edo. | |||
The '''neutral chain-of-fifths notation''' (aka '''chain-of-half-fifths notation''', '''chain-of-neutral-thirds notation''', or less accurately, '''quartertone notation''') uses an extended accidental set including '''half-sharps''' and '''half-flats'''. 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), such as the [[mohaha]] temperament. It also works for certain edos of even sharpness (except sharp-0 edos, in which sharps and flats have no effects). Not all even-sharpness edos allow this notation. For example, 34edo (sharp-4) does not, because its half-fifth is 10\34, and 10 and 34 are not coprime. The GCD is 2, thus there are two rings of half-fifths. In other words, the edo must be [[Ring number #Generalizations|single-ring]] with respect to the half-fifth. All edos with sharpness 2 or -2 qualify. If a qualifying edo's sharpness is not ±2, the notes will run out of order. For example, 41edo (sharp-4) has C Ddb Ct Db C# Dd C#t D. | |||
Chain-of-third-fifths notation, chain-of-quarter-fifths notation, etc., are theoretical possibilities. In practice, ups and downs are usually used for third-sharps or quarter-sharps. | |||
== Edos up to 100 == | == Edos up to 100 == | ||
| Line 14: | Line 12: | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|- | |- | ||
|+Diatonic edos fit for | |+Diatonic edos fit for chain-of-fifths notation | ||
|- | |- | ||
! Edo | ! Edo | ||
| Line 176: | Line 174: | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|- | |- | ||
|+Diatonic edos fit for neutral | |+Diatonic edos fit for neutral chain-of-fifths notation | ||
|- | |- | ||
! Edo | ! Edo | ||
| Line 259: | Line 257: | ||
== Expansions == | == Expansions == | ||
* [[Syntonic-rastmic subchroma notation]] – built on neutral | * [[Syntonic-rastmic subchroma notation]] – built on neutral chain-of-fifths notation | ||
* [[Ups and downs notation]] – built on | * [[Ups and downs notation]] – built on chain-of-fifths notation | ||
** Neutral ups and downs notation (→ [[Alternative symbols for ups and downs notation]]) | ** Neutral ups and downs notation (→ [[Alternative symbols for ups and downs notation]]) | ||
* [[Sagittal notation]] (''evo flavor'') – built on | * [[Sagittal notation]] (''evo flavor'') – built on chain-of-fifths notation or neutral chain-of-fifths notation | ||
== See also == | == See also == | ||
* [[Nominal-accidental chain]] | * [[Nominal-accidental chain]] | ||
* [[ | * [[Chain of fifths]] | ||
* [[Fifthspan]] | * [[Fifthspan]] | ||
* [[User:Xenwolf/cofn]] – sortable table with more intervals (all fifths within the interval [4\7, 3\5], the "[[diatonic range]]") | * [[User:Xenwolf/cofn]] – sortable table with more intervals (all fifths within the interval [4\7, 3\5], the "[[diatonic range]]") | ||
Revision as of 08:53, 18 December 2023
The chain-of-fifths notation (aka extended Pythagorean notation) is suitable for a variety of tuning systems which are octave repeating and generated by the fifth. A good number of edos and regular temperaments can be notated this way, as it generalizes the traditional classical notation system for the Pythagorean tuning, the meantone tunings, and 12edo. It uses the seven natural notes of the diatonic scale and accidentals (♯, ♭ and their multiples) to sharpen and flatten these seven notes by the augmented unison aka the chromatic semitone. Any regular rank-2 temperament generated by the 8ve and the 5th (i.e. one with the unsplit pergen) can be notated this way.
Chain-of-fifths notation only works for single-ring edos. A counter-example is 24edo, which is double-ring. This notation works best for edos of sharpness 1, and for 7edo, where accidentals have no effects. For any multi-sharpness edos, this notation causes the notes to run out of order. For example, 17edo would run C Db C# D Eb D# E… For negative sharpness edos the accidentals will be inverse. One can avoid these by using ups and downs notation, or for certain edos by using half-sharps (see below). Edos whose fifth has a high relative error makes more sense considered as dual-fifth, and notated as subsets. For example, 13edo can be notated as a subset of 26edo.
The neutral chain-of-fifths notation (aka chain-of-half-fifths notation, chain-of-neutral-thirds notation, or less accurately, quartertone notation) uses an extended accidental set including half-sharps and half-flats. 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), such as the mohaha temperament. It also works for certain edos of even sharpness (except sharp-0 edos, in which sharps and flats have no effects). Not all even-sharpness edos allow this notation. For example, 34edo (sharp-4) does not, because its half-fifth is 10\34, and 10 and 34 are not coprime. The GCD is 2, thus there are two rings of half-fifths. In other words, the edo must be single-ring with respect to the half-fifth. All edos with sharpness 2 or -2 qualify. If a qualifying edo's sharpness is not ±2, the notes will run out of order. For example, 41edo (sharp-4) has C Ddb Ct Db C# Dd C#t D.
Chain-of-third-fifths notation, chain-of-quarter-fifths notation, etc., are theoretical possibilities. In practice, ups and downs are usually used for third-sharps or quarter-sharps.
Edos up to 100
Edos up to 100 are listed in the following tables. The unit (if not stated otherwise) is edosteps of the corresponding edo which is given in the first column of each row. The tables contain only diatonic edos (i.e. A1 and m2 have edostepspans > 0). The last two columns are the edo's pentasharpness and sharpness respectively.
| Edo | Fifth | Fifth-detuning abs (¢), rel (%) |
Major 2nd |
Minor 2nd |
Augmented 1sn |
|---|---|---|---|---|---|
| 12 | 7 | -2.0 ( -2.0%) | 2 | 1 | 1 |
| 17 | 10 | +3.9 ( +5.6%) | 3 | 1 | 2 |
| 19 | 11 | -7.2 (-11.4%) | 3 | 2 | 1 |
| 22 | 13 | +7.1 (+13.1%) | 4 | 1 | 3 |
| 26 | 15 | -9.6 (-20.9%) | 4 | 3 | 1 |
| 27 | 16 | +9.2 (+20.6%) | 5 | 1 | 4 |
| 29 | 17 | +1.5 ( +3.6%) | 5 | 2 | 3 |
| 31 | 18 | -5.2 (-13.4%) | 5 | 3 | 2 |
| 32 | 19 | +10.5 (+28.1%) | 6 | 1 | 5 |
| 33 | 19 | -11.0 (-30.4%) | 5 | 4 | 1 |
| 37 | 22 | +11.6 (+35.6%) | 7 | 1 | 6 |
| 39 | 23 | +5.7 (+18.6%) | 7 | 2 | 5 |
| 40 | 23 | -12.0 (-39.9%) | 6 | 5 | 1 |
| 41 | 24 | +0.5 ( +1.7%) | 7 | 3 | 4 |
| 42 | 25 | +12.3 (+43.2%) | 8 | 1 | 7 |
| 43 | 25 | -4.3 (-15.3%) | 7 | 4 | 3 |
| 45 | 26 | -8.6 (-32.3%) | 7 | 5 | 2 |
| 46 | 27 | +2.4 ( +9.2%) | 8 | 3 | 5 |
| 47 | 27 | -12.6 (-49.3%) | 7 | 6 | 1 |
| 49 | 29 | +8.2 (+33.7%) | 9 | 2 | 7 |
| 50 | 29 | -6.0 (-24.8%) | 8 | 5 | 3 |
| 53 | 31 | -0.1 ( -0.3%) | 9 | 4 | 5 |
| 55 | 32 | -3.8 (-17.3%) | 9 | 5 | 4 |
| 56 | 33 | +5.2 (+24.2%) | 10 | 3 | 7 |
| 59 | 35 | +9.9 (+48.7%) | 11 | 2 | 9 |
| 61 | 36 | +6.2 (+31.7%) | 11 | 3 | 8 |
| 63 | 37 | +2.8 (+14.7%) | 11 | 4 | 7 |
| 64 | 37 | -8.2 (-43.8%) | 10 | 7 | 3 |
| 65 | 38 | -0.4 ( -2.3%) | 11 | 5 | 6 |
| 67 | 39 | -3.4 (-19.2%) | 11 | 6 | 5 |
| 69 | 40 | -6.3 (-36.2%) | 11 | 7 | 4 |
| 70 | 41 | +0.9 ( +5.3%) | 12 | 5 | 7 |
| 71 | 42 | +7.9 (+46.8%) | 13 | 3 | 10 |
| 73 | 43 | +4.9 (+29.8%) | 13 | 4 | 9 |
| 74 | 43 | -4.7 (-28.7%) | 12 | 7 | 5 |
| 75 | 44 | +2.0 (+12.8%) | 13 | 5 | 8 |
| 77 | 45 | -0.7 ( -4.2%) | 13 | 6 | 7 |
| 79 | 46 | -3.2 (-21.2%) | 13 | 7 | 6 |
| 80 | 47 | +3.0 (+20.3%) | 14 | 5 | 9 |
| 81 | 47 | -5.7 (-38.2%) | 13 | 8 | 5 |
| 83 | 49 | +6.5 (+44.8%) | 15 | 4 | 11 |
| 88 | 51 | -6.5 (-47.7%) | 14 | 9 | 5 |
| 89 | 52 | -0.8 ( -6.2%) | 15 | 7 | 8 |
| 90 | 53 | +4.7 (+35.3%) | 16 | 5 | 11 |
| 91 | 53 | -3.1 (-23.2%) | 15 | 8 | 7 |
| 94 | 55 | +0.2 ( +1.4%) | 16 | 7 | 9 |
| 95 | 56 | +5.4 (+42.9%) | 17 | 5 | 12 |
| 97 | 57 | +3.2 (+25.9%) | 17 | 6 | 11 |
| 98 | 57 | -4.0 (-32.6%) | 16 | 9 | 7 |
| 99 | 58 | +1.1 ( +8.9%) | 17 | 7 | 10 |
| Edo | Fifth | Fifth-detuning abs (¢), rel (%) |
Major 2nd |
Minor 2nd |
Augmented 1sn |
|---|---|---|---|---|---|
| 17 | 10 | +3.9 ( +5.6%) | 3 | 1 | 2 |
| 24 | 14 | -4.0 (-4.0%) | 4 | 2 | 2 |
| 27 | 16 | +9.2 (+20.6%) | 5 | 1 | 4 |
| 31 | 18 | -5.2 (-13.4%) | 5 | 3 | 2 |
| 37 | 22 | +11.6 (+35.6%) | 7 | 1 | 6 |
| 38 | 22 | -7.2 (-22.9%) | 6 | 4 | 2 |
| 41 | 24 | +0.5 ( +1.7%) | 7 | 3 | 4 |
| 44 | 26 | +7.1 (+26.2%) | 8 | 2 | 6 |
| 45 | 26 | -8.6 (-32.3%) | 7 | 5 | 2 |
| 52 | 30 | -9.6 (-41.8%) | 8 | 6 | 2 |
| 55 | 32 | -3.8 (-17.3%) | 9 | 5 | 4 |
| 58 | 34 | +1.5 ( +3.6%) | 10 | 4 | 6 |
| 61 | 36 | +6.2 (+31.7%) | 11 | 3 | 8 |
| 65 | 38 | -0.4 ( -2.3%) | 11 | 5 | 6 |
| 69 | 40 | -6.3 (-36.2%) | 11 | 7 | 4 |
| 71 | 42 | +7.9 (+46.8%) | 13 | 3 | 10 |
| 75 | 44 | +2.0 (+12.8%) | 13 | 5 | 8 |
| 78 | 46 | +5.7 (+37.3%) | 14 | 4 | 10 |
| 79 | 46 | -3.2 (-21.2%) | 13 | 7 | 6 |
| 86 | 50 | -4.3 (-30.7%) | 14 | 8 | 6 |
| 89 | 52 | -0.8 ( -6.2%) | 15 | 7 | 8 |
| 92 | 54 | +2.4 ( +18.3%) | 16 | 6 | 10 |
| 95 | 56 | +5.4 (+42.9%) | 17 | 5 | 12 |
| 99 | 58 | +1.1 ( +8.9%) | 17 | 7 | 10 |
Expansions
- Syntonic-rastmic subchroma notation – built on neutral chain-of-fifths notation
- Ups and downs notation – built on chain-of-fifths notation
- Neutral ups and downs notation (→ Alternative symbols for ups and downs notation)
- Sagittal notation (evo flavor) – built on chain-of-fifths notation or neutral chain-of-fifths notation
See also
- Nominal-accidental chain
- Chain of fifths
- Fifthspan
- User:Xenwolf/cofn – sortable table with more intervals (all fifths within the interval [4\7, 3\5], the "diatonic range")