Chain-of-fifths notation: Difference between revisions

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autopatrolled, Sysops
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clarifications
TallKite (talk | contribs)
autopatrolled, Sysops
4,192 edits
Major rewrite. Removed the opinion stated as fact that perfect, pentatonic and superflat edos are better notated as subsets of larger edos (counter-example: 7edo). Clarified which edos can use quartertone notation, and which can best use it. Other changes as well.
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The '''circle-of-fifths notation''' (aka '''extended Pythagorean notation''') is suitable to open up the 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 later [[12edo]]. It uses seven root notes of the [[diatonic]] scale and accidentals (<span style="font-size:larger">♯, ♭</span> and their multiples) to sharpen and flatten these root notes by the [[chromatic semitone]] (which is an octave-reduced stack of 7 fifths).  
The '''circle-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]].  


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]].  
Circle-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 7-edo and edos of [[sharpness]] 1 or -1. For all other edos, this notation causes the notes to run out of order. For example, 17-edo would run C Db C# D Eb D# E... One can avoid this by using [[ups and downs notation]], or for certain edos by using half-sharps (see below).  


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.  
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]]).  
The '''circle-of-half-fifths notation''' (aka '''circle-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 are useless). Not all even-sharpness edos allow this notation. For example, 34-edo (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, 41-edo (sharp-4) has C Ddb Ct Db C# Dd C#t D.  


== Edos up to 100 ==
Circle-of-third-fifths notation, circle-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 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.
 
==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 [[Sharpness|pentasharpness and sharpness]] respectively.


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
|+ Edos fit for circle-of-fifths notation
|+Diatonic edos fit for circle-of-fifths notation
|-
|-
! Edo
!Edo
! Fifth
!Fifth
! Fifth-detuning <br> abs(¢), rel(%)
!Fifth-detuning <br> abs(¢), rel(%)
! Whole<br>tone
!Major
! Diatonic<br>semitone
2nd
! Chromatic<br>semitone
!Minor
2nd
!Augmented
unison
|-
|-
! [[12edo|12]]
![[12edo|12]]
| 7 || -2.0 ( -2.0%) || 2 || 1 || 1
|7||-2.0 ( -2.0%) ||2||1||1
|-
|-
! [[17edo|17]]
![[17edo|17]]
| 10 || +3.9 ( +5.6%) || 3 || 1 || 2
|10||+3.9 ( +5.6%) ||3||1||2
|-
|-
! [[19edo|19]]
![[19edo|19]]
| 11 || -7.2 (-11.4%) || 3 || 2 || 1
|11||-7.2 (-11.4%) ||3||2||1
|-
|-
! [[22edo|22]]
![[22edo|22]]
| 13 || +7.1 (+13.1%) || 4 || 1 || 3
|13||+7.1 (+13.1%) ||4||1||3
|-
|-
! [[26edo|26]]
![[26edo|26]]
| 15 || -9.6 (-20.9%) || 4 || 3 || 1
|15||-9.6 (-20.9%) ||4||3||1
|-
|-
! [[27edo|27]]
![[27edo|27]]
| 16 || +9.2 (+20.6%) || 5 || 1 || 4
|16||+9.2 (+20.6%)||5||1||4
|-
|-
! [[29edo|29]]
![[29edo|29]]
| 17 || +1.5 ( +3.6%) || 5 || 2 || 3
|17||+1.5 ( +3.6%)||5||2||3
|-
|-
! [[31edo|31]]
![[31edo|31]]
| 18 || -5.2 (-13.4%) || 5 || 3 || 2
|18||-5.2 (-13.4%)||5||3||2
|-
|-
! [[32edo|32]]
![[32edo|32]]
| 19 || +10.5 (+28.1%) || 6 || 1 || 5
|19||+10.5 (+28.1%) ||6||1||5
|-
|-
! [[33edo|33]]
![[33edo|33]]
| 19 || -11.0 (-30.4%) || 5 || 4 || 1
|19||-11.0 (-30.4%)||5||4||1
|-
|-
! [[37edo|37]]
![[37edo|37]]
| 22 || +11.6 (+35.6%) || 7 || 1 || 6
|22||+11.6 (+35.6%) ||7||1||6
|-
|-
! [[39edo|39]]
![[39edo|39]]
| 23 || +5.7 (+18.6%) || 7 || 2 || 5
|23||+5.7 (+18.6%)||7||2||5
|-
|-
! [[40edo|40]]
![[40edo|40]]
| 23 || -12.0 (-39.9%) || 6 || 5 || 1
|23||-12.0 (-39.9%) ||6||5||1
|-
|-
! [[41edo|41]]
![[41edo|41]]
| 24 || +0.5 ( +1.7%) || 7 || 3 || 4
|24||+0.5 ( +1.7%) ||7||3||4
|-
|-
! [[42edo|42]]
![[42edo|42]]
| 25 || +12.3 (+43.2%) || 8 || 1 || 7
|25||+12.3 (+43.2%) ||8||1||7
|-
|-
! [[43edo|43]]
![[43edo|43]]
| 25 || -4.3 (-15.3%) || 7 || 4 || 3
|25||-4.3 (-15.3%)||7||4||3
|-
|-
! [[45edo|45]]
![[45edo|45]]
| 26 || -8.6 (-32.3%) || 7 || 5 || 2
|26||-8.6 (-32.3%) ||7||5||2
|-
|-
! [[46edo|46]]
![[46edo|46]]
| 27 || +2.4 ( +9.2%) || 8 || 3 || 5
|27||+2.4 ( +9.2%) ||8||3||5
|-
|-
! [[47edo|47]]
![[47edo|47]]
| 27 || -12.6 (-49.3%) || 7 || 6 || 1
|27||-12.6 (-49.3%) ||7||6||1
|-
|-
! [[49edo|49]]
![[49edo|49]]
| 29 || +8.2 (+33.7%) || 9 || 2 || 7
|29||+8.2 (+33.7%) ||9||2||7
|-
|-
! [[50edo|50]]
![[50edo|50]]
| 29 || -6.0 (-24.8%) || 8 || 5 || 3
|29||-6.0 (-24.8%) ||8||5||3
|-
|-
! [[53edo|53]]
![[53edo|53]]
| 31 || -0.1 ( -0.3%) || 9 || 4 || 5
|31||-0.1 ( -0.3%) ||9||4||5
|-
|-
! [[55edo|55]]
![[55edo|55]]
| 32 || -3.8 (-17.3%) || 9 || 5 || 4
|32||-3.8 (-17.3%) ||9||5||4
|-
|-
! [[56edo|56]]
![[56edo|56]]
| 33 || +5.2 (+24.2%) || 10 || 3 || 7
|33||+5.2 (+24.2%) ||10||3||7
|-
|-
! [[59edo|59]]
![[59edo|59]]
| 35 || +9.9 (+48.7%) || 11 || 2 || 9
|35||+9.9 (+48.7%) ||11||2||9
|-
|-
! [[61edo|61]]
![[61edo|61]]
| 36 || +6.2 (+31.7%) || 11 || 3 || 8
|36||+6.2 (+31.7%) ||11||3||8
|-
|-
! [[63edo|63]]
![[63edo|63]]
| 37 || +2.8 (+14.7%) || 11 || 4 || 7
|37||+2.8 (+14.7%) ||11||4||7
|-
|-
! [[64edo|64]]
![[64edo|64]]
| 37 || -8.2 (-43.8%) || 10 || 7 || 3
|37||-8.2 (-43.8%) ||10||7||3
|-
|-
! [[65edo|65]]
![[65edo|65]]
| 38 || -0.4 ( -2.3%) || 11 || 5 || 6
|38||-0.4 ( -2.3%) ||11||5||6
|-
|-
! [[67edo|67]]
![[67edo|67]]
| 39 || -3.4 (-19.2%) || 11 || 6 || 5
|39||-3.4 (-19.2%) ||11||6||5
|-
|-
! [[69edo|69]]
![[69edo|69]]
| 40 || -6.3 (-36.2%) || 11 || 7 || 4
|40||-6.3 (-36.2%) ||11||7||4
|-
|-
! [[70edo|70]]
![[70edo|70]]
| 41 || +0.9 ( +5.3%) || 12 || 5 || 7
|41||+0.9 ( +5.3%) ||12||5||7
|-
|-
! [[71edo|71]]
![[71edo|71]]
| 42 || +7.9 (+46.8%) || 13 || 3 || 10
|42||+7.9 (+46.8%) ||13||3||10
|-
|-
! [[73edo|73]]
![[73edo|73]]
| 43 || +4.9 (+29.8%) || 13 || 4 || 9
|43||+4.9 (+29.8%) ||13||4||9
|-
|-
! [[74edo|74]]
![[74edo|74]]
| 43 || -4.7 (-28.7%) || 12 || 7 || 5
|43||-4.7 (-28.7%) ||12||7||5
|-
|-
! [[75edo|75]]
![[75edo|75]]
| 44 || +2.0 (+12.8%) || 13 || 5 || 8
|44||+2.0 (+12.8%) ||13||5||8
|-
|-
! [[77edo|77]]
![[77edo|77]]
| 45 || -0.7 ( -4.2%) || 13 || 6 || 7
|45||-0.7 ( -4.2%) ||13||6||7
|-
|-
! [[79edo|79]]
![[79edo|79]]
| 46 || -3.2 (-21.2%) || 13 || 7 || 6
|46||-3.2 (-21.2%) ||13||7||6
|-
|-
! [[80edo|80]]
![[80edo|80]]
| 47 || +3.0 (+20.3%) || 14 || 5 || 9
|47||+3.0 (+20.3%) ||14||5||9
|-
|-
! [[81edo|81]]
![[81edo|81]]
| 47 || -5.7 (-38.2%) || 13 || 8 || 5
|47||-5.7 (-38.2%) ||13||8||5
|-
|-
! [[83edo|83]]
![[83edo|83]]
| 49 || +6.5 (+44.8%) || 15 || 4 || 11
|49||+6.5 (+44.8%) ||15||4||11
|-
|-
! [[88edo|88]]
![[88edo|88]]
| 51 || -6.5 (-47.7%) || 14 || 9 || 5
|51||-6.5 (-47.7%) ||14||9||5
|-
|-
! [[89edo|89]]
![[89edo|89]]
| 52 || -0.8 ( -6.2%) || 15 || 7 || 8
|52||-0.8 ( -6.2%) ||15||7||8
|-
|-
! [[90edo|90]]
![[90edo|90]]
| 53 || +4.7 (+35.3%) || 16 || 5 || 11
|53||+4.7 (+35.3%) ||16||5||11
|-
|-
! [[91edo|91]]
![[91edo|91]]
| 53 || -3.1 (-23.2%) || 15 || 8 || 7
|53||-3.1 (-23.2%) ||15||8||7
|-
|-
! [[94edo|94]]
![[94edo|94]]
| 55 || +0.2 ( +1.4%) || 16 || 7 || 9
|55||+0.2 ( +1.4%) ||16||7||9
|-
|-
! [[95edo|95]]
![[95edo|95]]
| 56 || +5.4 (+42.9%) || 17 || 5 || 12
|56||+5.4 (+42.9%) ||17||5||12
|-
|-
! [[97edo|97]]
![[97edo|97]]
| 57 || +3.2 (+25.9%) || 17 || 6 || 11
|57||+3.2 (+25.9%)||17||6|| 11
|-
|-
! [[98edo|98]]
![[98edo|98]]
| 57 || -4.0 (-32.6%) || 16 || 9 || 7
|57||-4.0 (-32.6%)||16||9||7
|-
|-
! [[99edo|99]]
![[99edo|99]]
| 58 || +1.1 ( +8.9%) || 17 || 7 || 10
|58||+1.1 ( +8.9%)||17||7||10
|}
|}


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
|+ Edos fit for neutral circle-of-fifths notation
|+Diatonic edos fit for circle-of-half-fifths notation
|-
|-
! Edo
!Edo
! Fifth
! Fifth
! Fifth-detuning <br> abs(¢), rel(%)
!Fifth-detuning <br> abs(¢), rel(%)
! Whole<br>tone
!Major
! Diatonic<br>semitone
2nd
! Chromatic<br>semitone
! Minor
2nd
!Augmented
unison
|-
|-
! [[17edo|17]]
![[17edo|17]]
| 10 || +3.9 ( +5.6%) || 3 || 1 || 2
|10 ||+3.9 ( +5.6%)||3||1||2
|-
|-
! [[24edo|24]]
![[24edo|24]]
| 14 || -4.0 (-4.0%) || 4 || 2 || 2
|14 ||-4.0 (-4.0%)||4||2||2
|-
|-
! [[27edo|27]]
![[27edo|27]]
| 16 || +9.2 (+20.6%) || 5 || 1 || 4
| 16 ||+9.2 (+20.6%)||5||1||4
|-
|-
! [[31edo|31]]
![[31edo|31]]
| 18 || -5.2 (-13.4%) || 5 || 3 || 2
|18 ||-5.2 (-13.4%)||5||3||2
|-
|-
! [[37edo|37]]
![[37edo|37]]
| 22 || +11.6 (+35.6%) || 7 || 1 || 6
|22 ||+11.6 (+35.6%)||7||1||6
|-
|-
! [[38edo|38]]
![[38edo|38]]
| 22 || -7.2 (-22.9%) || 6 || 4 || 2
|22 ||-7.2 (-22.9%)||6||4 ||2
|-
|-
! [[41edo|41]]
![[41edo|41]]
| 24 || +0.5 ( +1.7%) || 7 || 3 || 4
| 24 ||+0.5 ( +1.7%)||7||3 ||4
|-
|-
! [[44edo|44]]
![[44edo|44]]
| 26 || +7.1 (+26.2%) || 8 || 2 || 6
| 26 ||+7.1 (+26.2%)||8||2 ||6
|-
|-
! [[45edo|45]]
![[45edo|45]]
| 26 || -8.6 (-32.3%) || 7 || 5 || 2
| 26 ||-8.6 (-32.3%)||7||5 ||2
|-
|-
! [[52edo|52]]
![[52edo|52]]
| 30 || -9.6 (-41.8%) || 8 || 6 || 2
| 30 ||-9.6 (-41.8%)||8||6 ||2
|-
|-
! [[55edo|55]]
![[55edo|55]]
| 32 || -3.8 (-17.3%) || 9 || 5 || 4
| 32 ||-3.8 (-17.3%)||9||5 ||4
|-
|-
! [[58edo|58]]
![[58edo|58]]
| 34 || +1.5 ( +3.6%) || 10 || 4 || 6
| 34 ||+1.5 ( +3.6%)||10||4 ||6
|-
|-
! [[61edo|61]]
![[61edo|61]]
| 36 || +6.2 (+31.7%) || 11 || 3 || 8
| 36 ||+6.2 (+31.7%)||11||3 ||8
|-
|-
! [[65edo|65]]
![[65edo|65]]
| 38 || -0.4 ( -2.3%) || 11 || 5 || 6
| 38 ||-0.4 ( -2.3%)||11||5 ||6
|-
|-
! [[69edo|69]]
![[69edo|69]]
| 40 || -6.3 (-36.2%) || 11 || 7 || 4
| 40 ||-6.3 (-36.2%)||11||7 ||4
|-
|-
! [[71edo|71]]
![[71edo|71]]
| 42 || +7.9 (+46.8%) || 13 || 3 || 10
| 42 ||+7.9 (+46.8%)||13||3 ||10
|-
|-
! [[75edo|75]]
![[75edo|75]]
| 44 || +2.0 (+12.8%) || 13 || 5 || 8
| 44 ||+2.0 (+12.8%)||13||5 ||8
|-
|-
! [[78edo|78]]
![[78edo|78]]
| 46 || +5.7 (+37.3%) || 14 || 4 || 10
| 46 ||+5.7 (+37.3%)||14||4 ||10
|-
|-
! [[79edo|79]]
![[79edo|79]]
| 46 || -3.2 (-21.2%) || 13 || 7 || 6
| 46 ||-3.2 (-21.2%)||13||7 ||6
|-
|-
! [[86edo|86]]
![[86edo|86]]
| 50 || -4.3 (-30.7%) || 14 || 8 || 6
|50||-4.3 (-30.7%)||14 || 8 ||6
|-
|-
! [[89edo|89]]
![[89edo|89]]
| 52 || -0.8 ( -6.2%) || 15 || 7 || 8
|52||-0.8 ( -6.2%)||15 || 7 ||8
|-
|-
! [[92edo|92]]
![[92edo|92]]
| 54 || +2.4 ( +18.3%) || 16 || 6 || 10
|54||+2.4 ( +18.3%)|| 16||6||10
|-
|-
! [[95edo|95]]
![[95edo|95]]
| 56 || +5.4 (+42.9%) || 17 || 5 || 12
|56||+5.4 (+42.9%)||17||5||12
|-
|-
! [[99edo|99]]
![[99edo|99]]
| 58 || +1.1 ( +8.9%) || 17 || 7 || 10
| 58|| +1.1 ( +8.9%)||17 ||7||10
|}
|}


== Expansions ==
==Expansions ==
* [[Syntonic-rastmic subchroma notation]] – built on neutral circle-of-fifths notation
*[[Syntonic-rastmic subchroma notation]] – built on neutral circle-of-fifths notation
* [[Ups and downs notation]] – built on circle-of-fifths notation
*[[Ups and downs notation]] – built on circle-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 circle-of-fifths notation or neutral circle-of-fifths notation
*[[Sagittal notation]] (''evo flavor'') – built on circle-of-fifths notation or neutral circle-of-fifths notation


== See also ==
==See also==
* [[Nominal-accidental chain]]
*[[Nominal-accidental chain]]
* [[Circle of fifths]]
*[[Circle 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]]")


[[Category:Notation]]
[[Category:Notation]]
[[Category:Method]]
[[Category:Method]]
[[Category:Fifth]]
[[Category:Fifth]]

Revision as of 10:11, 1 June 2023

The circle-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.

Circle-of-fifths notation only works for single-ring edos. A counter-example is 24edo, which is double-ring. This notation works best for 7-edo and edos of sharpness 1 or -1. For all other edos, this notation causes the notes to run out of order. For example, 17-edo would run C Db C# D Eb D# E... One can avoid this by using ups and downs notation, or for certain edos by using half-sharps (see below).

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 circle-of-half-fifths notation (aka circle-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 are useless). Not all even-sharpness edos allow this notation. For example, 34-edo (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, 41-edo (sharp-4) has C Ddb Ct Db C# Dd C#t D.

Circle-of-third-fifths notation, circle-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.

Diatonic edos fit for circle-of-fifths notation
Edo Fifth Fifth-detuning
abs(¢), rel(%) 		Major

2nd

Minor

2nd

Augmented

unison

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
Diatonic edos fit for circle-of-half-fifths notation
Edo Fifth Fifth-detuning
abs(¢), rel(%) 		Major

2nd

Minor

2nd

Augmented

unison

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

See also

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