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{{Infobox ET}}
{{Infobox ET}}
{{Wikipedia| 53 equal temperament }}
{{Wikipedia| 53 equal temperament }}
{{EDO intro|53}}
{{ED intro}}


== Theory ==
== Theory ==
53edo is notable as a [[5-limit]] system, a fact apparently first noted by Isaac Newton, notably tempering out [[Mercator's comma]] (3<sup>53</sup>/2<sup>84</sup>), the [[schisma|schisma (32805/32768)]], and the [[15625/15552|kleisma (15625/15552)]]. More complex 5-limit commas tempered out include the [[amity comma|amity comma (1600000/1594323)]], the [[semicomma|semicomma (2109375/2097152)]], and the [[vulture comma|vulture comma ({{monzo| 24 -21 4 }}]]). In the 7-limit it tempers out the [[225/224|marvel comma (225/224)]], [[1728/1715|orwellisma (1728/1715)]], and [[3125/3087|gariboh comma (3125/3087)]]. In the 11-limit, it tempers out [[99/98]] and [[121/120]] (in addition to their difference, [[540/539]]), and is the [[optimal patent val]] for [[big brother]] temperament, which tempers out both, as well as 11-limit [[orwell]] temperament, which also tempers out the 11-limit commas [[176/175]] and [[385/384]]. In the 13-limit, it tempers out [[169/168]], [[275/273]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[2080/2079]], and [[4096/4095]], and gives the optimal patent val for [[Marvel family #Athene|athene]] temperament. It is the seventh [[The Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]]. It can also be treated as a no-elevens, no-seventeens tuning, on which it is [[consistent]] all the way up to the 23-odd-limit.
53edo is notable as an excellent [[5-limit]] system, a fact apparently first noted by {{w|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3–4):223–48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>. It is the seventh [[The Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]]. In the opinion of some, 53edo is the first equal division to deal adequately with the [[13-limit]], while others award that distinction to [[41edo]] or [[46edo]]. Like 41 and 46, 53 is distinctly [[consistent]] in the [[9-odd-limit]] (and if we exclude the most damaged interval pair, 7/5 and 10/7, is [[consistent to distance]] 2), but among them, 53 is the first that finds the [[interseptimal interval]]s [[15/13]] and [[13/10]] distinctly from adjacent [[7-limit|septimal]] intervals [[8/7]] and [[7/6]], and [[9/7]] and [[21/16]], respectively, which is essential to its 13-limit credibility. It also avoids equating [[11/9]] with [[16/13]], so that the former is tuned very flat to equate it with a slightly flat [[~]][[39/32]] – a feature shared by 46edo. It is almost consistent to the entire [[15-odd-limit]], with the only inconsistency occurring at [[14/11]] (and its octave complement), which is mapped inconsistently sharp and equated with [[9/7]], but it has the benefit of doing very well in larger prime/subgroup-limited odd-limits. It can be treated as a no-11's, no-17's tuning, on which it is consistent all the way up to the [[27-odd-limit]].  


53edo has also found a certain dissemination as an edo tuning for [[Arabic, Turkish, Persian music]]. Due to its fifths being almost indistinguishable from just, it can also be used as an extended [[3-limit|Pythagorean tuning]].
As an equal temperament, it notably [[tempering out|tempers out]] [[Mercator's comma]] (3<sup>53</sup>/2<sup>84</sup>), the [[schisma|schisma (32805/32768)]], [[15625/15552|kleisma (15625/15552)]], and [[amity comma|amity comma (1600000/1594323)]]. In the 7-limit it tempers out the [[225/224|marvel comma (225/224)]] for which it is a [[Marvel#Tunings|relatively efficient tuning]], [[1728/1715|orwellisma (1728/1715)]], [[3125/3087|gariboh comma (3125/3087)]], and [[4375/4374|ragisma (4375/4374)]]. In the 11-limit, it tempers out [[99/98]] and [[121/120]] (in addition to their difference, [[540/539]]), and is the [[optimal patent val]] for [[big brother]] temperament, which tempers out both, as well as 11-limit [[orwell]] temperament, which also tempers out the 11-limit commas [[176/175]] and [[385/384]]. In the 13-limit, it tempers out [[169/168]], [[275/273]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[2080/2079]], and [[4096/4095]], and gives the optimal patent val for [[marvel family #Athene|athene]] temperament.  


53edo's step is sometimes called the "Holdrian comma", despite the 53rd root of 2 being an irrational number; the reason it is referred to as a comma is because it approximates the Pythagorean comma and syntonic comma (and is almost exactly equal to the geometric mean of the two).
53edo has also found a certain dissemination as an edo tuning for [[Arabic, Turkish, Persian|Arabic, Turkish, and Persian music]]. It can also be used as an extended [[3-limit|Pythagorean tuning]], since its fifths are almost indistinguishable from just.
 
53edo's step is sometimes called the "Holdrian comma", despite the 53rd root of 2 being an irrational number; the step's role as a "comma" comes from it being an approximation of the Pythagorean comma and syntonic comma.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|53|columns=13}}
{{Harmonics in equal|53|columns=9}}
{{Harmonics in equal|53|columns=10|start=10|collapsed=true|title=Approximation of prime harmonics in 53edo (continued)}}
 
See [[#Approximation to JI]] for details and a more in-depth discussion.


=== Subsets and supersets ===
=== Subsets and supersets ===
Line 29: Line 34:
! #
! #
! Cents
! Cents
! Approximate Ratios*
! Approximate ratios<ref group="note">{{sg|limit=no-17's [[19-limit]]}} ''Italics'' represent inconsistent intervals which are mapped by the 19-limit [[patent val]] to their second-closest (as opposed to closest) approximation in 53edo. </ref>
! colspan="3" | [[Ups and downs notation|Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]] ([[enharmonic unisons in ups and downs notation|EUs]]: v<sup>5</sup>A1 and ^d2)
! colspan="2" | [[Solfege]]s
! colspan="2" | [[Solfege]]s
|-
|-
| 0
| 0
| 0.00
| 0.0
| [[1/1]]
| [[1/1]]
| P1
| P1
Line 43: Line 48:
|-
|-
| 1
| 1
| 22.64
| 22.6
| [[81/80]], [[64/63]], [[50/49]]
| ''[[50/49]]'', [[64/63]], [[81/80]]
| ^1
| ^1
| up unison
| up unison
Line 52: Line 57:
|-
|-
| 2
| 2
| 45.28
| 45.3
| [[49/48]], [[36/35]], [[33/32]], [[128/125]]
| [[33/32]], [[36/35]], [[49/48]], [[128/125]]
| ^^1, vvm2
| ^^1, vvm2
| dup unison, dudminor 2nd
| dup unison, dudminor 2nd
Line 61: Line 66:
|-
|-
| 3
| 3
| 67.92
| 67.9
| [[25/24]], [[28/27]], [[22/21]], [[27/26]], [[26/25]]
| ''[[22/21]]'', [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| vvA1, vm2
| vvA1, vm2
| dud-aug 1sn, downminor 2nd
| dudaug 1sn, downminor 2nd
| vvD#, vEb
| vvD#, vEb
| Fro
| Fro
Line 70: Line 75:
|-
|-
| 4
| 4
| 90.57
| 90.6
| [[19/18]], [[20/19]], [[21/20]], [[256/243]]
| [[19/18]], [[20/19]], [[21/20]], [[256/243]]
| vA1, m2
| vA1, m2
Line 79: Line 84:
|-
|-
| 5
| 5
| 113.21
| 113.2
| [[16/15]], [[15/14]]
| [[15/14]], [[16/15]]
| A1, ^m2
| A1, ^m2
| aug 1sn, upminor 2nd
| aug 1sn, upminor 2nd
Line 88: Line 93:
|-
|-
| 6
| 6
| 135.85
| 135.8
| [[14/13]], [[13/12]], [[27/25]]
| [[13/12]], [[14/13]], [[27/25]]
| v~2
| ^^m2
| downmid 2nd
| dupminor 2nd
| ^^Eb
| ^^Eb
| Fri
| Fri
Line 97: Line 102:
|-
|-
| 7
| 7
| 158.49
| 158.5
| [[35/32]], [[12/11]], [[11/10]], [[57/52]], [[800/729]]
| [[11/10]], [[12/11]], [[35/32]], [[57/52]], [[800/729]]
| ^~2
| vvM2
| upmid 2nd
| dudmajor 2nd
| vvE
| vvE
| Re
| Re
Line 106: Line 111:
|-
|-
| 8
| 8
| 181.13
| 181.1
| [[10/9]]
| [[10/9]]
| vM2
| vM2
Line 115: Line 120:
|-
|-
| 9
| 9
| 203.77
| 203.8
| [[9/8]]
| [[9/8]]
| M2
| M2
Line 124: Line 129:
|-
|-
| 10
| 10
| 226.42
| 226.4
| [[8/7]], [[256/225]]
| [[8/7]], [[256/225]]
| ^M2
| ^M2
Line 133: Line 138:
|-
|-
| 11
| 11
| 249.06
| 249.1
| [[15/13]], [[144/125]], [[125/108]]
| [[15/13]], [[22/19]], [[125/108]], [[144/125]]
| ^^M2, <br>vvm3
| ^^M2, vvm3
| dupmajor 2nd, <br>dudminor 3rd
| dupmajor 2nd, dudminor 3rd
| ^^E, <br>vvF
| ^^E, vvF
| Ri / Ne
| Ri / Ne
| Raw
| Raw
|-
|-
| 12
| 12
| 271.70
| 271.7
| [[7/6]], [[75/64]]
| [[7/6]], [[75/64]]
| vm3
| vm3
Line 151: Line 156:
|-
|-
| 13
| 13
| 294.34
| 294.3
| [[13/11]], [[19/16]], [[32/27]]
| [[13/11]], [[19/16]], [[32/27]]
| m3
| m3
Line 160: Line 165:
|-
|-
| 14
| 14
| 316.98
| 317.0
| [[6/5]]
| [[6/5]]
| ^m3
| ^m3
Line 169: Line 174:
|-
|-
| 15
| 15
| 339.62
| 339.6
| [[11/9]], [[243/200]]
| [[11/9]], [[243/200]]
| v~3
| ^^m3
| downmid 3rd
| dupminor 3rd
| ^^F
| ^^F
| Ni
| Ni
Line 178: Line 183:
|-
|-
| 16
| 16
| 362.26
| 362.3
| [[16/13]], [[100/81]]
| [[16/13]], [[100/81]]
| ^~3
| vvM3
| upmid 3rd
| dudmajor 3rd
| vvF#
| vvF#
| Me
| Me
Line 187: Line 192:
|-
|-
| 17
| 17
| 384.91
| 384.9
| [[5/4]]
| [[5/4]]
| vM3
| vM3
Line 196: Line 201:
|-
|-
| 18
| 18
| 407.55
| 407.5
| [[19/15]], [[24/19]], [[81/64]]
| [[19/15]], [[24/19]], [[81/64]]
| M3
| M3
Line 205: Line 210:
|-
|-
| 19
| 19
| 430.19
| 430.2
| [[9/7]], ''[[14/11]]''
| [[9/7]], ''[[14/11]]''
| ^M3
| ^M3
Line 214: Line 219:
|-
|-
| 20
| 20
| 452.83
| 452.8
| [[13/10]], [[125/96]], [[162/125]]
| [[13/10]], [[125/96]], [[162/125]]
| ^^M3, vv4
| ^^M3, vv4
Line 223: Line 228:
|-
|-
| 21
| 21
| 475.47
| 475.5
| [[21/16]], [[25/19]], [[675/512]], [[320/243]]
| [[21/16]], [[25/19]], [[320/243]], [[675/512]]
| v4
| v4
| down 4th
| down 4th
Line 232: Line 237:
|-
|-
| 22
| 22
| 498.11
| 498.1
| [[4/3]]
| [[4/3]]
| P4
| P4
Line 241: Line 246:
|-
|-
| 23
| 23
| 520.75
| 520.8
| [[27/20]]
| [[19/14]], [[27/20]]
| ^4
| ^4
| up 4th
| up 4th
Line 250: Line 255:
|-
|-
| 24
| 24
| 543.40
| 543.4
| [[11/8]], [[15/11]], [[26/19]]
| [[11/8]], [[15/11]], [[26/19]]
| v~4
| ^^4
| downmid 4th
| dup 4th
| ^^G
| ^^G
| Fi / She
| Fi / She
Line 259: Line 264:
|-
|-
| 25
| 25
| 566.04
| 566.0
| [[18/13]]
| [[18/13]]
| ^~4, <br>vd5
| vvA4, vd5
| upmid 4th, <br>downdim 5th
| dudaug 4th, downdim 5th
| vvG#, <br>vAb
| vvG#, vAb
| Pe / Sho
| Pe / Sho
| Fuh
| Fuh
|-
|-
| 26
| 26
| 588.68
| 588.7
| [[7/5]], [[45/32]]
| [[7/5]], [[45/32]]
| vA4, <br>d5
| vA4, d5
| downaug 4th, <br>dim 5th
| downaug 4th, dim 5th
| vG#, <br>Ab
| vG#, Ab
| Po / Sha
| Po / Sha
| Fi
| Fi
|-
|-
| 27
| 27
| 611.32
| 611.3
| [[10/7]], [[64/45]]
| [[10/7]], [[64/45]]
| A4, <br>^d5
| A4, ^d5
| aug 4th, <br>updim 5th
| aug 4th, updim 5th
| G#, <br>^Ab
| G#, ^Ab
| Pa / Shu
| Pa / Shu
| Se
| Se
|-
|-
| 28
| 28
| 633.96
| 634.0
| [[13/9]]
| [[13/9]]
| ^A4, <br>v~5
| ^A4, ^^d5
| upaug 4th, <br>downmid 5th
| upaug 4th, dupdim 5th
| ^G#, <br>^^Ab
| ^G#, ^^Ab
| Pu / Shi
| Pu / Shi
| Suh
| Suh
|-
|-
| 29
| 29
| 656.60
| 656.6
| [[16/11]], [[19/13]], [[22/15]]
| [[16/11]], [[19/13]], [[22/15]]
| ^~5
| vv5
| upmid 5th
| dud 5th
| vvA
| vvA
| Pi / Se
| Pi / Se
Line 304: Line 309:
|-
|-
| 30
| 30
| 679.25
| 679.2
| [[40/27]]
| [[28/19]], [[40/27]]
| v5
| v5
| down 5th
| down 5th
Line 313: Line 318:
|-
|-
| 31
| 31
| 701.89
| 701.9
| [[3/2]]
| [[3/2]]
| P5
| P5
Line 322: Line 327:
|-
|-
| 32
| 32
| 724.53
| 724.5
| [[32/21]], [[38/25]], [[243/160]], [[1024/675]]
| [[32/21]], [[38/25]], [[243/160]], [[1024/675]]
| ^5
| ^5
Line 331: Line 336:
|-
|-
| 33
| 33
| 747.17
| 747.2
| [[20/13]], [[192/125]], [[125/81]]
| [[20/13]], [[125/81]], [[192/125]]
| ^^5, vvm6
| ^^5, vvm6
| dup 5th, dudminor 6th
| dup 5th, dudminor 6th
Line 340: Line 345:
|-
|-
| 34
| 34
| 769.81
| 769.8
| [[14/9]], [[25/16]], ''[[11/7]]''
| ''[[11/7]]'', [[14/9]], [[25/16]]
| vm6
| vm6
| downminor 6th
| downminor 6th
Line 349: Line 354:
|-
|-
| 35
| 35
| 792.45
| 792.5
| [[19/12]], [[30/19]], [[128/81]]
| [[19/12]], [[30/19]], [[128/81]]
| m6
| m6
Line 358: Line 363:
|-
|-
| 36
| 36
| 815.09
| 815.1
| [[8/5]]
| [[8/5]]
| ^m6
| ^m6
Line 367: Line 372:
|-
|-
| 37
| 37
| 837.74
| 837.7
| [[13/8]], [[81/50]]
| [[13/8]], [[81/50]]
| v~6
| ^^m6
| downmid 6th
| dupminor 6th
| ^^Bb
| ^^Bb
| Fli
| Fli
Line 376: Line 381:
|-
|-
| 38
| 38
| 860.38
| 860.4
| [[18/11]], [[400/243]]
| [[18/11]], [[400/243]]
| ^~6
| vvM6
| upmid 6th
| dudmajor 6th
| vvB
| vvB
| Le
| Le
Line 385: Line 390:
|-
|-
| 39
| 39
| 883.02
| 883.0
| [[5/3]]
| [[5/3]]
| vM6
| vM6
Line 394: Line 399:
|-
|-
| 40
| 40
| 905.66
| 905.7
| [[22/13]], [[27/16]], [[32/19]]
| [[22/13]], [[27/16]], [[32/19]]
| M6
| M6
Line 403: Line 408:
|-
|-
| 41
| 41
| 928.30
| 928.3
| [[12/7]]
| [[12/7]]
| ^M6
| ^M6
Line 412: Line 417:
|-
|-
| 42
| 42
| 950.94
| 950.9
| [[26/15]], [[125/72]], [[216/125]]
| [[19/11]], [[26/15]], [[125/72]], [[216/125]]
| ^^M6, vvm7
| ^^M6, vvm7
| dupmajor 6th, dudminor 7th
| dupmajor 6th, dudminor 7th
Line 421: Line 426:
|-
|-
| 43
| 43
| 973.58
| 973.6
| [[7/4]]
| [[7/4]]
| vm7
| vm7
Line 430: Line 435:
|-
|-
| 44
| 44
| 996.23
| 996.2
| [[16/9]]
| [[16/9]]
| m7
| m7
Line 439: Line 444:
|-
|-
| 45
| 45
| 1018.87
| 1018.9
| [[9/5]]
| [[9/5]]
| ^m7
| ^m7
Line 448: Line 453:
|-
|-
| 46
| 46
| 1041.51
| 1041.5
| [[64/35]], [[11/6]], [[20/11]], [[729/400]]
| [[11/6]], [[20/11]], [[64/35]], [[729/400]]
| v~7
| ^^m7
| downmid 7th
| dupminor 7th
| ^^C
| ^^C
| Thi
| Thi
Line 457: Line 462:
|-
|-
| 47
| 47
| 1064.15
| 1064.2
| [[13/7]], [[24/13]], [[50/27]]
| [[13/7]], [[24/13]], [[50/27]]
| ^~7
| vvM7
| upmid 7th
| dudmajor 7th
| vvC#
| vvC#
| Te
| Te
Line 466: Line 471:
|-
|-
| 48
| 48
| 1086.79
| 1086.8
| [[15/8]]
| [[15/8]]
| vM7
| vM7
Line 475: Line 480:
|-
|-
| 49
| 49
| 1109.43
| 1109.4
| [[19/10]], [[36/19]], [[40/21]], [[243/128]]
| [[19/10]], [[36/19]], [[40/21]], [[243/128]]
| M7
| M7
Line 484: Line 489:
|-
|-
| 50
| 50
| 1132.08
| 1132.1
| [[48/25]], [[27/14]], [[21/11]], [[52/27]], [[25/13]]
| ''[[21/11]]'', [[25/13]], [[27/14]], [[52/27]], [[48/25]]
| ^M7
| ^M7
| upmajor 7th
| upmajor 7th
Line 493: Line 498:
|-
|-
| 51
| 51
| 1154.72
| 1154.7
| [[96/49]], [[35/18]], [[64/33]], [[125/64]]
| [[35/18]], [[64/33]], [[96/49]], [[125/64]]
| ^^M7, vv8
| ^^M7, vv8
| dupmajor 7th, dud 8ve
| dupmajor 7th, dud 8ve
Line 502: Line 507:
|-
|-
| 52
| 52
| 1177.36
| 1177.4
| [[160/81]], [[63/32]], [[49/25]]
| ''[[49/25]]'', [[63/32]], [[160/81]]
| v8
| v8
| down 8ve
| down 8ve
Line 511: Line 516:
|-
|-
| 53
| 53
| 1200.00
| 1200.0
| [[2/1]]
| [[2/1]]
| P8
| P8
Line 519: Line 524:
| Do
| Do
|}
|}
<nowiki>*</nowiki> based on interpreting 53edo as a no-17's [[19-limit]] temperament. ''Italics'' represent inconsistent intervals which are mapped by the 19-limit [[patent val]] to their second-closest (as opposed to closest) approximation in 53edo.


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
Line 528: Line 532:
! Quality
! Quality
! [[Kite's color notation|Color]]
! [[Kite's color notation|Color]]
! Monzo Format
! Monzo format
! Examples
! Examples
|-
|-
| downminor
| downminor
| zo
| zo
| (a, b, 0, 1)
| {{nowrap|(a, b, 0, 1)}}
| 7/6, 7/4
| 7/6, 7/4
|-
|-
| minor
| minor
| fourthward wa
| fourthward wa
| (a, b) with b &lt; -1
| {{nowrap|(a, b)}} with {{nowrap|b &lt; −1}}
| 32/27, 16/9
| 32/27, 16/9
|-
|-
| upminor
| upminor
| gu
| gu
| (a, b, -1)
| {{nowrap|(a, b, −1)}}
| 6/5, 9/5
| 6/5, 9/5
|-
|-
| downmid
| dupminor
| ilo
| ilo
| (a, b, 0, 0, 1)
| {{nowrap|(a, b, 0, 0, 1)}}
| 11/9, 11/6
| 11/9, 11/6
|-
|-
| upmid
| dudmajor
| lu
| lu
| (a, b, 0, 0, -1)
| {{nowrap|(a, b, 0, 0, −1)}}
| 12/11, 18/11
| 12/11, 18/11
|-
|-
| downmajor
| downmajor
| yo
| yo
| (a, b, 1)
| {{nowrap|(a, b, 1)}}
| 5/4, 5/3
| 5/4, 5/3
|-
|-
| major
| major
| fifthward wa
| fifthward wa
| (a, b) with b &gt; 1
| {{nowrap|(a, b)}} with {{nowrap|b &gt; 1}}
| 9/8, 27/16
| 9/8, 27/16
|-
|-
| upmajor
| upmajor
| ru
| ru
| (a, b, 0, -1)
| {{nowrap|(a, b, 0, −1)}}
| 9/7, 12/7
| 9/7, 12/7
|}
|}
All 53edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are.
All 53edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked−3rds chord {{nowrap|{{dash|6, 1, 3, 5, 7, 9, 11, 13}}}}). Alterations are always enclosed in parentheses, additions never are.


Here are the zo, gu, ilo, lu, yo and ru triads:
Here are the zo, gu, ilo, lu, yo and ru triads:
Line 577: Line 581:
|-
|-
! [[Kite's color notation|Color of the 3rd]]
! [[Kite's color notation|Color of the 3rd]]
! JI Chord
! JI chord
! Notes as Edosteps
! Notes as edosteps
! Notes of C Chord
! Notes of C chord
! Written Name
! Written name
! Spoken Name
! Spoken name
|-
|-
| zo
| zo
| 6:7:9
| 6:7:9
| 0-12-31
| 0–12–31
| C vEb G
| C vEb G
| Cvm
| Cvm
Line 592: Line 596:
| gu
| gu
| 10:12:15
| 10:12:15
| 0-14-31
| 0–14–31
| C ^Eb G
| C ^Eb G
| C^m
| C^m
Line 599: Line 603:
| ilo
| ilo
| 18:22:27
| 18:22:27
| 0-15-31
| 0–15–31
| C ^^Eb G
| C ^^Eb G
| Cv~
| C^^m
| C downmid
| C dupminor
|-
|-
| lu
| lu
| 22:27:33
| 22:27:33
| 0-16-31
| 0–16–31
| C vvE G
| C vvE G
| C^~
| Cvv
| C upmid
| C dudmajor or C dud
|-
|-
| yo
| yo
| 4:5:6
| 4:5:6
| 0-17-31
| 0–17–31
| C vE G
| C vE G
| Cv
| Cv
Line 620: Line 624:
| ru
| ru
| 14:18:21
| 14:18:21
| 0-19-31
| 0–19–31
| C ^E G
| C ^E G
| C^
| C^
| C upmajor or C up
| C upmajor or C up
|}
|}
For a more complete list, see [[Ups and downs notation #Chords and Chord Progressions]].
For a more complete list, see [[Ups and downs notation #Chords and chord progressions]].


== Notation ==
== Notation ==
=== Sagittal ===
=== Ups and downs notation ===
The following table shows [[sagittal notation]] accidentals in one apotome for 53edo.  
53edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
{{Sharpness-sharp5a}}
Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp5}}
 
=== Sagittal notation ===
==== Evo flavor ====
<imagemap>
File:53-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 567 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 270 106 [[6561/6400]]
rect 270 80 370 106 [[40/39]]
default [[File:53-EDO_Evo_Sagittal.svg]]
</imagemap>


{| class="wikitable center-all"
==== Revo flavor ====
! Steps
<imagemap>
| 0
File:53-EDO_Revo_Sagittal.svg
| 1
desc none
| 2
rect 80 0 300 50 [[Sagittal_notation]]
| 3
rect 300 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| 4
rect 20 80 120 106 [[81/80]]
| 5
rect 120 80 270 106 [[6561/6400]]
|-
rect 270 80 370 106 [[40/39]]
! Symbol
default [[File:53-EDO_Revo_Sagittal.svg]]
| [[File:Sagittal natural.png]]
</imagemap>
| [[File:Sagittal pai.png]]
 
| [[File:Sagittal phai.png]]
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
| [[File:Sagittal sharp phao.png]]
| [[File:Sagittal sharp pao.png]]
| [[File:Sagittal sharp.png]]
|}


== Relationship to 12edo ==
== Relationship to 12edo ==
Whereas 12edo has a circle of twelve 5ths, 53edo has a spiral of twelve 5ths (since 31\53 is on the 7\12 kite in the scale tree). This shows 53edo in a 12edo-friendly format. Excellent for introducing 53edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.
53edo's [[circle of fifths|circle of 53 fifths]] can be bent into a [[spiral chart|12-spoked "spiral of fifths"]]. This makes sense to do because going up by 12 fifths results in the Pythagorean comma (by definition), which is mapped to one edostep and is thus also the syntonic and septimal comma, introducing a simple second accidental in the form of the arrow to reach useful intervals from the basic 12-chromatic scale. The one-edostep comma is a requirement in Kite's theory, and implies that 31\53 is on the 7\12 kite in the [[scale tree]].  
 
This "spiral of fifths" can be a useful construct for introducing 53edo to musicians unfamiliar with microtonal music. It may help composers and musicians to make visual sense of the notation, and to understand what size of a jump is likely to land them where compared to 12edo.
 
The two innermost and two outermost intervals on the spiral are duplicates, reflecting the fact that it is a repeating circle at heart and the spiral shape is only a helpful illusion.


[[File:53-edo spiral.png|588x588px]]
[[File:53-edo spiral.png|588x588px]]


== Approximation to JI ==
== Approximation to JI ==
53edo provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.
53edo provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy–Zarlino "just major" scale.


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 694: Line 714:
|}
|}


Because the 5th is so very accurate, 53edo also offers good approximations for Pythagorean thirds. In addition, the 43\53 interval is only 4.8 cents wider than the just ratio 7/4, so 53edo can also be used for 7-limit harmony, tempering out the [[septimal kleisma]], 225/224.
Because the 5th is so accurate, 53edo also offers good approximations for Pythagorean thirds. In addition, the 43\53 interval is only 4.8 cents wider than 7/4, so 53edo can also be used for 7-limit harmony, in which it tempers out the [[septimal kleisma]], 225/224.


=== 15-odd-limit interval mappings ===
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 53edo. Octave-reduced prime harmonics are '''bolded'''; inconsistent intervals are in ''italic''.
{{Q-odd-limit intervals|53}}
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct mapping (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| '''[[3/2]], [[4/3]]'''
| '''0.068'''
| '''0.3'''
|-
| [[9/8]], [[16/9]]
| 0.136
| 0.6
|-
| [[9/5]], [[10/9]]
| 1.272
| 5.6
|-
| [[15/13]], [[26/15]]
| 1.316
| 5.8
|-
| [[5/3]], [[6/5]]
| 1.340
| 5.9
|-
| [[13/10]], [[20/13]]
| 1.384
| 6.1
|-
| '''[[5/4]], [[8/5]]'''
| '''1.408'''
| '''6.2'''
|-
| [[15/8]], [[16/15]]
| 1.476
| 6.5
|-
| [[13/9]], [[18/13]]
| 2.655
| 11.7
|-
| [[13/12]], [[24/13]]
| 2.724
| 12.0
|-
| '''[[13/8]], [[16/13]]'''
| '''2.792'''
| '''12.3'''
|-
| '''[[7/4]], [[8/7]]'''
| '''4.759'''
| '''21.0'''
|-
| [[7/6]], [[12/7]]
| 4.827
| 21.3
|-
| [[9/7]], [[14/9]]
| 4.895
| 21.6
|-
| [[13/11]], [[22/13]]
| 5.130
| 22.7
|-
| [[7/5]], [[10/7]]
| 6.167
| 27.2
|-
| [[15/14]], [[28/15]]
| 6.235
| 27.5
|-
| [[15/11]], [[22/15]]
| 6.445
| 28.5
|-
| [[11/10]], [[20/11]]
| 6.514
| 28.8
|-
| [[13/7]], [[14/13]]
| 7.551
| 33.3
|-
| [[11/9]], [[18/11]]
| 7.785
| 34.4
|-
| [[11/6]], [[12/11]]
| 7.854
| 34.7
|-
| '''[[11/8]], [[16/11]]'''
| '''7.922'''
| '''35.0'''
|-
| ''[[11/7]], [[14/11]]''
| ''9.961''
| ''44.0''
|}
{{15-odd-limit|53}}


=== Higher-limit JI ===
=== Higher-limit JI ===
There is also a cluster of usable higher primes starting at 71; even 89 (4.84{{cent}} flat), 97 (4.63{{cent}} sharp) and 101 (2.6{{cent}} sharp) are usable if placed in just the right context.
53edo has only 5 pairs of inconsistent intervals in the full 27-odd-limit: {11/7,&nbsp;14/11}, {[[17/11]],&nbsp;[[22/17]]}, {[[19/17]],&nbsp;[[34/19]]}, {[[21/11]],&nbsp;[[22/21]]}, and {[[23/22]],&nbsp;[[44/23]]}. This is perhaps remarkable compared to 9 pairs in 46edo and 11 pairs in 41edo, because the smallest edo after 53edo to get 5 or less inconsistencies in the 27-odd-limit is [[99edo]] (using the 99[[wart|ef]] [[val]]), followed by [[111edo]] ([[patent val]]), [[130edo]] (patent val) and [[159edo]] (also patent); all of these get 5 inconsistencies as well except 159edo which gets 1 and which is itself a superset of 53edo. However, most interpret the approximation of prime 17 in 53edo as too off for all but the most opportunistic harmonies, and some question the 23 and possibly also 11, so the practical significance of this is debatable.
 
{{Harmonics in equal|53|columns=4|start=20}}


This make 53edo excellent (for its size) in the 2.3.5.7.11.13.19.23.37.41.71.73.79.83 subgroup, although some higher error primes like 11 and 23 require the right context to be convincing.
As shown below, there is also a cluster of usable higher primes starting at 71; even 89 (4.84{{c}} flat), 97 (4.63{{c}} sharp) and 101 (2.6{{c}} sharp) are usable if placed in just the right context. (Note that prime 67 is almost perfectly off.)
{{Harmonics in equal|53|columns=4|start=20|title=Approximation of large prime harmonics in 53edo}}


Note that the high primes, in rooted (/2<sup>n</sup>) position, essentially act as alternate interpretations of [[LCJI]] intervals, if you want to force a rooted interpretation; namely:
This makes 53edo excellent (for its size) in the 2.3.5.7.11.13.19.23.37.41.71.73(.79.83.101) subgroup, although some higher error primes like 11 and 23 require the right context to be convincing.


[[71/64]] as ~[[10/9]], [[73/64]] as ~[[8/7]], [[79/64]] as ~[[16/13]], and perhaps most questionably in the context of 53edo, [[83/64]] as ~[[13/10]]. (Note that [[8edo]] offers a very good approximation of [[83/64]], so if you are working with a system that maps 13/10 to 3\8 = 450.000{{cent}} it makes more sense to think of 83/64 as the rooted approximation of 13/10 in that context.)
Note that the high primes, in [[rooted]] (/2<sup>''n''</sup>) position, essentially act as alternate interpretations of [[LCJI]] intervals, if you want to force a rooted interpretation; namely: [[71/64]] as ~[[10/9]], [[73/64]] as ~[[8/7]], [[79/64]] as ~[[16/13]], and perhaps most questionably in the context of 53edo, [[83/64]] as ~[[13/10]].


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 827: Line 742:
|-
|-
| 2.3
| 2.3
| {{monzo| -84 53 }}
| {{Monzo| -84 53 }}
| [{{val| 53 84 }}]
| {{Mapping| 53 84 }}
| +0.022
| +0.022
| 0.022
| 0.022
Line 835: Line 750:
| 2.3.5
| 2.3.5
| 15625/15552, 32805/32768
| 15625/15552, 32805/32768
| [{{val| 53 84 123 }}]
| {{Mapping| 53 84 123 }}
| +0.216
| +0.216
| 0.276
| 0.276
Line 842: Line 757:
| 2.3.5.7
| 2.3.5.7
| 225/224, 1728/1715, 3125/3087
| 225/224, 1728/1715, 3125/3087
| [{{val| 53 84 123 149 }}]
| {{Mapping| 53 84 123 149 }}
| -0.262
| −0.262
| 0.861
| 0.861
| 3.81
| 3.81
Line 849: Line 764:
| 2.3.5.7.11
| 2.3.5.7.11
| 99/98, 121/120, 176/175, 2200/2187
| 99/98, 121/120, 176/175, 2200/2187
| [{{val| 53 84 123 149 183 }}]
| {{Mapping| 53 84 123 149 183 }}
| +0.248
| +0.248
| 1.279
| 1.279
Line 856: Line 771:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 99/98, 121/120, 169/168, 176/175, 275/273
| 99/98, 121/120, 169/168, 176/175, 275/273
| [{{val| 53 84 123 149 183 196 }}]
| {{Mapping| 53 84 123 149 183 196 }}
| +0.332
| +0.332
| 1.183
| 1.183
Line 863: Line 778:
| 2.3.5.7.11.13.19
| 2.3.5.7.11.13.19
| 99/98, 121/120, 169/168, 176/175, 209/208, 275/273
| 99/98, 121/120, 169/168, 176/175, 209/208, 275/273
| [{{val| 53 84 123 149 183 196 225 }}]
| {{Mapping| 53 84 123 149 183 196 225 }}
| +0.391
| +0.391
| 1.105
| 1.105
| 4.88
| 4.88
|}
|}
* 53et is lower in relative error than any previous equal temperaments in the 3-, 5-, and 13-limit. The next equal temperaments doing better in these subgroups are 306, 118, and 58, respectively. It is even more prominent in the 2.3.5.7.13.19 and 2.3.5.7.13.19.23 subgroups, and the next equal temperament doing better in either subgroup is 130.


53et is lower in relative error than any previous equal temperaments in the 3-, 5-, and 13-limit. The next equal temperaments doing better in these subgroups are 306, 118, and 58, respectively. It is even more prominent in the 2.3.5.7.13.19 and 2.3.5.7.13.19.23 subgroups, and the next equal temperament doing better in either subgroup is 130.  
=== Commas ===
Commas that 53edo tempers out using its patent val, {{val| 53 84 123 149 183 196 225 }}, include:
 
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| <abbr title="19383245667680019896796723/19342813113834066795298816">(52 digits)</abbr>
| {{Monzo| -84 53 }}
| 3.62
| Tribilawa
| 53-comma, [[Mercator's comma]]
|-
| 5
| [[2109375/2097152|(14 digits)]]
| {{Monzo| -21 3 7 }}
| 10.06
| Lasepyo
| [[Semicomma]]
|-
| 5
| [[15625/15552]]
| {{Monzo| -6 -5 6 }}
| 8.11
| Tribiyo
| Kleisma
|-
| 5
| <abbr title="1600000/1594323">(14 digits)</abbr>
| {{Monzo| 9 -13 5 }}
| 6.15
| Saquinyo
| [[Amity comma]]
|-
| 5
| <abbr title="10485760000/10460353203">(22 digits)</abbr>
| {{Monzo| 24 -21 4 }}
| 4.20
| Sasaquadyo
| [[Vulture comma]]
|-
| 5
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 7
| [[3125/3087]]
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[1728/1715]]
| {{Monzo| 6 3 -1 -3 }}
| 13.07
| Triru-agu
| Orwellisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[4375/4374]]
| {{Monzo| -1 -7 4 1 }}
| 0.40
| Zoquadyo
| Ragisma
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| <abbr title="94489280512/94143178827">(22 digits)</abbr>
| {{Monzo| 33 -23 0 0 1 }}
| 6.35
| Trisalo
| [[Pythrabian comma]]
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 13
| [[275/273]]
| {{Monzo| 0 -1 2 -1 1 -1 }}
| 12.64
| Thuloruyoyo
| Gassorma
|-
| 13
| [[169/168]]
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.27
| Thothoru
| Buzurgisma, dhanvantarisma
|-
| 13
| [[625/624]]
| {{Monzo| -4 -1 4 0 0 -1 }}
| 2.77
| Thuquadyo
| Tunbarsma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|-
| 13
| [[1001/1000]]
| {{Monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[2080/2079]]
| {{Monzo| 5 -3 1 -1 -1 1 }}
| 0.83
| Tholuruyo
| Ibnsinma, sinaisma
|-
| 13
| [[4096/4095]]
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|}


=== Linear temperaments ===
=== Linear temperaments ===
* [[List of edo-distinct 53et rank two temperaments]]
* [[List of edo-distinct 53et rank two temperaments]]
* [[Schismic-Mercator equivalence continuum]]
* [[Schismic–Mercator equivalence continuum]]


{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+ Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator
! Generator*
! Cents
! Cents*
! Associated<br>Ratio
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
| 1
| 1
| 2\53
| 2\53
| 45.28
| 45.3
| 36/35
| 36/35
| [[Quartonic]]
| [[Quartonic]]
Line 892: Line 980:
| 1
| 1
| 5\53
| 5\53
| 113.21
| 113.2
| 16/15
| 16/15
| [[Misneb]]
| [[Misneb]]
|-
| 1
| 6\53
| 135.8
| [[13/12]]~[[14/13]]
| [[Doublethink]]
|-
|-
| 1
| 1
| 7\53
| 7\53
| 158.49
| 158.5
| 11/10
| 11/10
| [[Hemikleismic]]
| [[Hemikleismic]]
Line 904: Line 998:
| 1
| 1
| 9\53
| 9\53
| 203.77
| 203.8
| 9/8
| 9/8
| [[Baldy]]
| [[Baldy]]
Line 910: Line 1,004:
| 1
| 1
| 10\53
| 10\53
| 226.42
| 226.4
| 8/7
| 8/7
| [[Semaja]]
| [[Semaja]]
Line 916: Line 1,010:
| 1
| 1
| 11\53
| 11\53
| 249.06
| 249.1
| 15/13
| 15/13
| [[Hemischis]] / [[hemigari]]
| [[Hemischis]] / [[hemigari]]
Line 922: Line 1,016:
| 1
| 1
| 12\53
| 12\53
| 271.70
| 271.7
| 7/6
| 7/6
| [[Orson]] / [[orwell]]
| [[Orwell]]
|-
|-
| 1
| 1
| 13\53
| 13\53
| 294.34
| 294.3
| 25/21
| 25/21
| [[Kleiboh]]
| [[Kleiboh]]
Line 934: Line 1,028:
| 1
| 1
| 14\53
| 14\53
| 316.98
| 317.0
| 6/5
| 6/5
| [[Hanson]] / [[catakleismic]] / [[countercata]]
| [[Hanson]] / [[catakleismic]] / [[countercata]]
Line 940: Line 1,034:
| 1
| 1
| 15\53
| 15\53
| 339.62
| 339.6
| 11/9
| 11/9
| [[Amity]] / [[houborizic]]
| [[Amity]] / [[houborizic]]
Line 946: Line 1,040:
| 1
| 1
| 16\53
| 16\53
| 362.26
| 362.3
| 16/13
| 16/13
| [[Submajor]]
| [[Submajor]]
Line 952: Line 1,046:
| 1
| 1
| 18\53
| 18\53
| 407.55
| 407.5
| 1225/972
| 1225/972
| [[Ditonic]] / [[coditone]]
| [[Ditonic]] / [[coditone]]
Line 958: Line 1,052:
| 1
| 1
| 19\53
| 19\53
| 430.19
| 430.2
| 9/7
| 9/7
| [[Hamity]]
| [[Hamity]]
|-
| 1
| 20\53
| 452.8
| 13/10
| [[Maja]]
|-
|-
| 1
| 1
| 21\53
| 21\53
| 475.47
| 475.5
| 21/16
| 21/16
| [[Vulture]] / [[buzzard]]
| [[Vulture]] / [[buzzard]]
Line 970: Line 1,070:
| 1
| 1
| 22\53
| 22\53
| 498.11
| 498.1
| 4/3
| [[Garibaldi]] / [[pontiac]]
|-
| 1
| 23\53
| 520.8
| 4/3
| 4/3
| [[Helmholtz]] / [[garibaldi]] / [[pontiac]]
| [[Mavila]] (53bbcc)
|-
|-
| 1
| 1
| 25\53
| 25\53
| 566.04
| 566.0
| 18/13
| 18/13
| [[Tricot]]
| [[Alphatrimot]]
|-
|-
| 1
| 1
| 26\53
| 26\53
| 588.68
| 588.7
| 45/32
| 45/32
| [[Untriton]] / [[aufo]]
| [[Untriton]] / [[aufo]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave


== Scales ==
== Scales ==
=== Mos scales ===
While there is only one possible generator for the [[5L 2s|diatonic]] [[mos scale]] supported by this edo, there are a greater number of generators for other mosses such as the [[2L 5s|antidiatonic]] scale.
While there is only one possible generator for the [[5L 2s|diatonic]] [[mos scale]] supported by this edo, there are a greater number of generators for other mosses such as the [[2L 5s|antidiatonic]] scale.
* [[List of MOS scales in 53edo]]
* [[1953 scale]]


The following pages document various kinds of scales in 53edo:
=== Scales approximated from JI ===
* [[1953 scale]]
* The [[eagle 53]] scale described by [[John O'Sullivan]]
* [[5- to 10-tone scales in 53edo]]
* Ptolmey–Zarlino justly-intonated major: 9 8 5 9 8 9 5
* [[List of MOS scales in 53edo]]
* Ptolmey–Zarlino justly-intonated minor: 9 5 8 9 5 9 8
 
; From [[AFDO]]s
{{Idiosyncratic terms}}
* Composite Cliffedge (approximated from [[60afdo]]): 12 10 9 19 3
* Composite Deja Vu (approximated from [[101afdo]]): 14 17 5 9 8
* Composite Dungeon (approximated from [[30afdo]]): 17 5 9 4 18
* Composite Freeway (approximated from [[6afdo]]): 12 10 9 8 7 7
* Composite Geode (approximated from [[6afdo]]): 12 10 9 15 7
* Composite Labyrinth (approximated from [[30afdo]]): 7 7 17 5 17
* Composite Mushroom (approximated from [[30afdo]]): 12 10 9 3 19
* Composite Underpass (approximated from [[10afdo]]): 14 17 10 4 8
* Spectral Arcade (approximated from [[32afdo]]): 17 4 10 12 10
* Spectral Mechanical (approximated from [[16afdo]]): 13 4 14 12 10
* Spectral Moonbeam (approximated from [[16afdo]]): 9 4 18 17 5
* Spectral Springwater (approximated from [[8afdo]]): 9 8 14 12 10
* Spectral Starship (approximated from [[68ifdo]]): 4 13 4 10 12 10
* Spectral Volcanic (approximated from [[16afdo]]): 5 12 14 12 10
 
=== Other scales ===
* [[cthon5m]]: 6 3 6 2 3 6 2 3 6 3 2 6 3 2
* Palace{{idio}} (approximated from [[Porky]] in [[29edo]]): 7 7 8 9 7 7 8
 
== Instruments ==
* [[Lumatone mapping for 53edo]]
* [[Skip fretting system 53 3 14]]
* [[Skip fretting system 53 3 17]]


== Music ==
== Music ==
{{Catrel|53edo tracks}}
{{Catrel| 53edo tracks }}


=== Modern renderings ===
=== Modern renderings ===
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* ''Prelude and Fugue in C Major, No. 1'', BWV 846, from ''The Well-Tempered Clavier'', Book I (1722) – rendered by [[Mykhaylo Khramov]]
* ''Prelude and Fugue in C Major, No. 1'', BWV 846, from ''The Well-Tempered Clavier'', Book I (1722) – rendered by [[Mykhaylo Khramov]]
** [https://web.archive.org/web/20201127013408/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3 Prelude] · [https://web.archive.org/web/20201127012701/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/fugue1-53.mp3 Fugue]
** [https://web.archive.org/web/20201127013408/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3 Prelude] · [https://web.archive.org/web/20201127012701/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/fugue1-53.mp3 Fugue]
* [https://www.youtube.com/watch?v=WyLDjrLa94Y "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=GK9YwSphw5Y "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2025)
* [https://www.youtube.com/watch?v=daWx5-iegW0 "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – with syntonic-comma adjustment, rendered by Claudi Meneghin (2025)
* [https://www.youtube.com/watch?v=dZyrIOMEmzo "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=dZyrIOMEmzo "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=vcinR7nUthA "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)


; {{W|Nicolaus Bruhns}}
; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=aprEqsCAP6Q ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
* [https://www.youtube.com/watch?v=aprEqsCAP6Q ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
* [https://www.youtube.com/watch?v=r6R4SsaT8ig ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)
; {{W|George Frideric Handel}}
* [https://www.youtube.com/watch?v=7I7mD-DzfIo ''Suite in D minor HWV 428 for Harpsichord - Allemande''] (1720) – rendered by Claudi Meneghin (2024)
; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=AKXMuhB3uHQ ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=t-pRqKGX0oo ''Maple Leaf Rag''] (1899) – with syntonic comma adjustment, arranged for harpsichord and rendered by Claudi Meneghin (2024)
; {{W|Shirō Sagisu}}
* [https://www.youtube.com/watch?v=DiPB__AOXdk ''Qui veut faire l'ange fait la bete''] – rendered by [[MortisTheneRd]] (2024)
* [https://www.youtube.com/watch?v=DCENVnxH6bI ''Bande-announce''] – rendered by MortisTheneRd (2024)


=== 21st century ===
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/r-Tzq33OGM4 ''microtonal improvisation in 53edo''] (2025)
; [[Francium]]
; [[Francium]]
* [https://www.youtube.com/watch?v=GLQ1gD4bshY ''Space Race''] (2022)
* [https://www.youtube.com/watch?v=GLQ1gD4bshY ''Space Race''] (2022)
* "strange worlds" from ''hope in dark times'' (2024) [https://open.spotify.com/track/6mjYGHlW7lSoez8NsDz021 Spotify] | [https://francium223.bandcamp.com/track/strange-worlds Bandcamp] | [https://www.youtube.com/watch?v=tPwRWVjeKA8 YouTube] – hanson[11] in 53edo
* "strange worlds" from ''hope in dark times'' (2024) – [https://open.spotify.com/track/6mjYGHlW7lSoez8NsDz021 Spotify] | [https://francium223.bandcamp.com/track/strange-worlds Bandcamp] | [https://www.youtube.com/watch?v=tPwRWVjeKA8 YouTube] – in Hanson[11], 53edo tuning
* "Blasphemous Rumors" from ''TOTMC September to December 2024'' (2024) – [https://open.spotify.com/track/7nOrawE5wLqllqMAApHadh Spotify] | [https://francium223.bandcamp.com/track/blasphemous-rumours Bandcamp] | [https://www.youtube.com/watch?v=kwELa9kP8YU YouTube] – in Blackdye, 53edo tuning
* "It's a Good Idea to Have a Good Time." from ''Random Sentences'' (2025) – [https://open.spotify.com/track/3rYiNMcOQ5Oxz7F6mQZsfw Spotify] | [https://francium223.bandcamp.com/track/its-a-good-idea-to-have-a-good-time Bandcamp] | [https://www.youtube.com/watch?v=D-i-4Sv-vqE YouTube]
* "Decearing Egg" from ''Eggs'' (2025) – [https://open.spotify.com/track/2KfOutrIDfbk4S9kxYi8sL Spotify] | [https://francium223.bandcamp.com/track/decearing-egg Bandcamp] | [https://www.youtube.com/watch?v=_CJ5MgIRKnM YouTube]
* "Husband Head Void" from ''Void'' (2025) – [https://open.spotify.com/track/4yvyDZv8dBjOiurzoTjpBj Spotify] | [https://francium223.bandcamp.com/track/husband-head-void Bandcamp] | [https://www.youtube.com/watch?v=HMnklwjEdF0 YouTube]
* "Lasagna Cat" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/6sJil69QOqxNWrWrkgm3rl Spotify] | [https://francium223.bandcamp.com/track/lasagna-cat Bandcamp] | [https://www.youtube.com/watch?v=Ay2zhVnTlxw YouTube]


; [[Andrew Heathwaite]]
; [[Andrew Heathwaite]]
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; [[Hideya]]
; [[Hideya]]
* [https://www.youtube.com/watch?v=QpeCMz9kJ-s ''Like Uminari''] (2021)
* [https://www.youtube.com/watch?v=QpeCMz9kJ-s ''Like Uminari''] (2021)
; [[Nathan Ho]]
* [https://www.youtube.com/watch?v=lGfV9LB-01U ''53edo microtonal algorithmic IDM in SuperCollider''] (2023)


; [[Aaron Krister Johnson]] ([http://www.akjmusic.com site]{{dead link}})
; [[Aaron Krister Johnson]] ([http://www.akjmusic.com site]{{dead link}})
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; [[MortisTheneRd]]
; [[MortisTheneRd]]
* [https://www.youtube.com/watch?v=DiPB__AOXdk Shiro Sagisu - ''Qui veut faire l'ange fait la bete''] (2024)
* [https://www.youtube.com/watch?v=TWVN8ui48ew ''Psychedelic Inventions in 53edo''] (2024)
 
; [[Mundoworld]]
* from ''No Fun House'' (2025)
** "No Explanations" – [https://open.spotify.com/track/4IM4RoS9BrkgsFXEbAOenQ Spotify] | [https://mundoworld.bandcamp.com/track/no-explanations Bandcamp] | [https://www.youtube.com/watch?v=WPlxi22rf0I YouTube] – in Gorgo[11], 53edo tuning
** "Liminal" – [https://open.spotify.com/track/6ouYOGwv6Vm1hbEC9QxFMc Spotify] | [https://mundoworld.bandcamp.com/track/liminal Bandcamp] | [https://www.youtube.com/watch?v=yKKZ_x8sIjg YouTube] – in Gorgo[11], 53edo tuning


; [[Prent Rodgers]]
; [[Prent Rodgers]]
* ''Whisper Song'' (2007) – [https://bumpermusic.blogspot.com/2007/05/whisper-song-in-53-edo-now-526-slower.html blog] | [https://web.archive.org/web/20201127013644/http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/sing53-c5-slow.mp3 play] | [https://soundcloud.com/prent-rodgers/whisper-song-in-53-edo SoundCloud]
* ''Whisper Song'' (2007) – [https://bumpermusic.blogspot.com/2007/05/whisper-song-in-53-edo-now-526-slower.html blog] | [https://web.archive.org/web/20201127013644/http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/sing53-c5-slow.mp3 play] | [https://soundcloud.com/prent-rodgers/whisper-song-in-53-edo SoundCloud]
; [[Sevish]]
* "[[Droplet]]", from ''[[Rhythm and Xen]]'' (2015) – [https://sevish.bandcamp.com/track/droplet Bandcamp] | [https://soundcloud.com/sevish/droplet?in=sevish/sets/rhythm-and-xen SoundCloud] | [https://www.youtube.com/watch?v=xVZy9GUeMqY YouTube] – drum and bass in Orwell[9], 53edo tuning
; [[Subhraag Singh]]
* [https://soundcloud.com/user-215518655-72150190/stranges-53edo-inspired-by ''"Stranges"''] (2021)


; [[Gene Ward Smith]]
; [[Gene Ward Smith]]
* ''Trio in Orwell'' (2010) – [https://www.archive.org/details/TrioInOrwell detail] | [https://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3 play] – orwell[9] in 53edo
* ''Trio in Orwell'' (archived 2010) – [https://www.archive.org/details/TrioInOrwell detail] | [https://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3 play] – in Orwell[9], 53edo tuning
 
; [[Nick Stephens]]
* "Initialising", from ''Microwave'' (2019) – [https://microwave64.bandcamp.com/track/initialising Bandcamp] | [https://soundcloud.com/nick-stephens-8/initialising SoundCloud]


; [[Cam Taylor]]
; [[Cam Taylor]]
* [https://soundcloud.com/cam-taylor-2-1/mothers ''mothers''] (2014)
* [https://soundcloud.com/cam-taylor-2-1/mothers ''mothers''] (2014)
* [https://www.youtube.com/watch?v=xIy8I0XfUDI ''Schumann: The Poet Speaks in 53-equal (5-limit) on the Lumatone''] (2022)
* [https://www.youtube.com/watch?v=vpgbnACq1rA ''53-equal: lydian/aeolian pentatonic''] (2023)
* [https://www.youtube.com/watch?v=LyWW3w7PhlE ''53-equal Luma MKI: around a drone on middle C''] (2023)
* [https://www.youtube.com/watch?v=l9Y8NEqIkug ''22 shrutis as Schismatic[22] in A446Hz (53-equal)''] (2024)
* [https://www.youtube.com/watch?v=IdiMNP4MSx8&t=2s&pp=0gcJCbIJAYcqIYzv ''A meander around 53-equal on the Lumatone''] (2025) (this is actually a keyboard mapping guide)


; [[Chris Vaisvil]]
; [[Chris Vaisvil]]
* ''The Fallen of Kleismic15'' (2013) – [http://chrisvaisvil.com/the-fallen-of-kleismic15/ blog] | [http://micro.soonlabel.com/53edo/20130903_Kleismic%5b15%5d.mp3 play] – kleismic[15] in 53edo
* ''The Fallen of Kleismic15'' (2013) – [http://chrisvaisvil.com/the-fallen-of-kleismic15/ blog] | [http://micro.soonlabel.com/53edo/20130903_Kleismic%5b15%5d.mp3 play] – in Kleismic[15], 53edo tuning


; [[Valeriana of the Night]]
; [[Valeriana of the Night]]
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; [[Xotla]]
; [[Xotla]]
* "Taking Flight" from ''Nano Particular'' (2019) – [https://open.spotify.com/track/2zp6oM57m6BvQgyOZ5kmuZ Spotify] | [https://xotla.bandcamp.com/track/taking-flight-53edo Bandcamp] | [https://www.youtube.com/watch?v=sIsfYQATouc YouTube]
* "Taking Flight" from ''Nano Particular'' (2019) – [https://open.spotify.com/track/2zp6oM57m6BvQgyOZ5kmuZ Spotify] | [https://xotla.bandcamp.com/track/taking-flight-53edo Bandcamp] | [https://www.youtube.com/watch?v=sIsfYQATouc YouTube]
* "Detective Duckweed" from ''Jazzbeetle'' (2023) – [https://open.spotify.com/track/77iDGy7hRx8az3ODrDm5Kl Spotify] | [https://xotla.bandcamp.com/track/detective-duckweed-53edo Bandcamp] | [https://youtu.be/FNXEPB4Gm54 YouTube] – jazzy big band electronic hybrid
== Notes ==
<references group="note" />


== Instruments ==
== References ==
* [[Lumatone mapping for 53edo]]
<references/>
* [[Skip fretting system 53 3 14]]


[[Category:3-limit record edos|##]] <!-- 2-digit number -->
[[Category:Amity]]
[[Category:Amity]]
[[Category:Hanson]]
[[Category:Kleismic]]
[[Category:Kleismic]]
[[Category:Island]]
[[Category:Island]]
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[[Category:Orwell]]
[[Category:Orwell]]
[[Category:Schismic]]
[[Category:Schismic]]
[[Category:3-limit]]
[[Category:Listen]]
[[Category:Listen]]

Latest revision as of 00:43, 19 August 2025

← 52edo 53edo 54edo →
Prime factorization 53 (prime)
Step size 22.6415 ¢ 
Fifth 31\53 (701.887 ¢)
(convergent)
Semitones (A1:m2) 5:4 (113.2 ¢ : 90.57 ¢)
Consistency limit 9
Distinct consistency limit 9
English Wikipedia has an article on:

53 equal divisions of the octave (abbreviated 53edo or 53ed2), also called 53-tone equal temperament (53tet) or 53 equal temperament (53et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 53 equal parts of about 22.6 ¢ each. Each step represents a frequency ratio of 21/53, or the 53rd root of 2.

Theory

53edo is notable as an excellent 5-limit system, a fact apparently first noted by Isaac Newton[1]. It is the seventh strict zeta edo. In the opinion of some, 53edo is the first equal division to deal adequately with the 13-limit, while others award that distinction to 41edo or 46edo. Like 41 and 46, 53 is distinctly consistent in the 9-odd-limit (and if we exclude the most damaged interval pair, 7/5 and 10/7, is consistent to distance 2), but among them, 53 is the first that finds the interseptimal intervals 15/13 and 13/10 distinctly from adjacent septimal intervals 8/7 and 7/6, and 9/7 and 21/16, respectively, which is essential to its 13-limit credibility. It also avoids equating 11/9 with 16/13, so that the former is tuned very flat to equate it with a slightly flat ~39/32 – a feature shared by 46edo. It is almost consistent to the entire 15-odd-limit, with the only inconsistency occurring at 14/11 (and its octave complement), which is mapped inconsistently sharp and equated with 9/7, but it has the benefit of doing very well in larger prime/subgroup-limited odd-limits. It can be treated as a no-11's, no-17's tuning, on which it is consistent all the way up to the 27-odd-limit.

As an equal temperament, it notably tempers out Mercator's comma (353/284), the schisma (32805/32768), kleisma (15625/15552), and amity comma (1600000/1594323). In the 7-limit it tempers out the marvel comma (225/224) for which it is a relatively efficient tuning, orwellisma (1728/1715), gariboh comma (3125/3087), and ragisma (4375/4374). In the 11-limit, it tempers out 99/98 and 121/120 (in addition to their difference, 540/539), and is the optimal patent val for big brother temperament, which tempers out both, as well as 11-limit orwell temperament, which also tempers out the 11-limit commas 176/175 and 385/384. In the 13-limit, it tempers out 169/168, 275/273, 325/324, 625/624, 676/675, 1001/1000, 2080/2079, and 4096/4095, and gives the optimal patent val for athene temperament.

53edo has also found a certain dissemination as an edo tuning for Arabic, Turkish, and Persian music. It can also be used as an extended Pythagorean tuning, since its fifths are almost indistinguishable from just.

53edo's step is sometimes called the "Holdrian comma", despite the 53rd root of 2 being an irrational number; the step's role as a "comma" comes from it being an approximation of the Pythagorean comma and syntonic comma.

Prime harmonics

Approximation of prime harmonics in 53edo
Harmonic 2 3 5 7 11 13 17 19 23
Error Absolute (¢) +0.00 -0.07 -1.41 +4.76 -7.92 -2.79 +8.25 -3.17 +5.69
Relative (%) +0.0 -0.3 -6.2 +21.0 -35.0 -12.3 +36.4 -14.0 +25.1
Steps
(reduced) 		53
(0) 		84
(31) 		123
(17) 		149
(43) 		183
(24) 		196
(37) 		217
(5) 		225
(13) 		240
(28)
Approximation of prime harmonics in 53edo (continued)
Harmonic 29 31 37 41 43 47 53 59 61 67
Error Absolute (¢) -10.71 +9.68 -2.29 +1.13 +9.24 -8.90 +9.51 +4.98 -7.45 +11.26
Relative (%) -47.3 +42.8 -10.1 +5.0 +40.8 -39.3 +42.0 +22.0 -32.9 +49.7
Steps
(reduced) 		257
(45) 		263
(51) 		276
(11) 		284
(19) 		288
(23) 		294
(29) 		304
(39) 		312
(47) 		314
(49) 		322
(4)

See #Approximation to JI for details and a more in-depth discussion.

Subsets and supersets

53edo is the 16th prime edo, following 47edo and coming before 59edo.

Many of its multiples such as 159edo, 212edo, 742edo, 901edo and 954edo have good consistency limits and are each of their own interest. The mercator family comprises rank-2 temperaments with 1/53-octave periods.

Intervals

# Cents Approximate ratios[note 1] Ups and downs notation (EUs: v5A1 and ^d2) Solfeges
0 0.0 1/1 P1 unison D Da Do
1 22.6 50/49, 64/63, 81/80 ^1 up unison ^D Du Di
2 45.3 33/32, 36/35, 49/48, 128/125 ^^1, vvm2 dup unison, dudminor 2nd ^^D, vvEb Di / Fre Daw
3 67.9 22/21, 25/24, 26/25, 27/26, 28/27 vvA1, vm2 dudaug 1sn, downminor 2nd vvD#, vEb Fro Ro
4 90.6 19/18, 20/19, 21/20, 256/243 vA1, m2 downaug 1sn, minor 2nd vD#, Eb Fra Rih
5 113.2 15/14, 16/15 A1, ^m2 aug 1sn, upminor 2nd D#, ^Eb Fru Ra
6 135.8 13/12, 14/13, 27/25 ^^m2 dupminor 2nd ^^Eb Fri Ru
7 158.5 11/10, 12/11, 35/32, 57/52, 800/729 vvM2 dudmajor 2nd vvE Re Ruh
8 181.1 10/9 vM2 downmajor 2nd vE Ro Reh
9 203.8 9/8 M2 major 2nd E Ra Re
10 226.4 8/7, 256/225 ^M2 upmajor 2nd ^E Ru Ri
11 249.1 15/13, 22/19, 125/108, 144/125 ^^M2, vvm3 dupmajor 2nd, dudminor 3rd ^^E, vvF Ri / Ne Raw
12 271.7 7/6, 75/64 vm3 downminor 3rd vF No Ma
13 294.3 13/11, 19/16, 32/27 m3 minor 3rd F Na Meh
14 317.0 6/5 ^m3 upminor 3rd ^F Nu Me
15 339.6 11/9, 243/200 ^^m3 dupminor 3rd ^^F Ni Mu
16 362.3 16/13, 100/81 vvM3 dudmajor 3rd vvF# Me Muh
17 384.9 5/4 vM3 downmajor 3rd vF# Mo Mi
18 407.5 19/15, 24/19, 81/64 M3 major 3rd F# Ma Maa
19 430.2 9/7, 14/11 ^M3 upmajor 3rd ^F# Mu Mo
20 452.8 13/10, 125/96, 162/125 ^^M3, vv4 dupmajor 3rd, dud 4th ^^F#, vvG Mi / Fe Maw
21 475.5 21/16, 25/19, 320/243, 675/512 v4 down 4th vG Fo Fe
22 498.1 4/3 P4 perfect 4th G Fa Fa
23 520.8 19/14, 27/20 ^4 up 4th ^G Fu Fih
24 543.4 11/8, 15/11, 26/19 ^^4 dup 4th ^^G Fi / She Fu
25 566.0 18/13 vvA4, vd5 dudaug 4th, downdim 5th vvG#, vAb Pe / Sho Fuh
26 588.7 7/5, 45/32 vA4, d5 downaug 4th, dim 5th vG#, Ab Po / Sha Fi
27 611.3 10/7, 64/45 A4, ^d5 aug 4th, updim 5th G#, ^Ab Pa / Shu Se
28 634.0 13/9 ^A4, ^^d5 upaug 4th, dupdim 5th ^G#, ^^Ab Pu / Shi Suh
29 656.6 16/11, 19/13, 22/15 vv5 dud 5th vvA Pi / Se Su
30 679.2 28/19, 40/27 v5 down 5th vA So Sih
31 701.9 3/2 P5 perfect 5th A Sa Sol
32 724.5 32/21, 38/25, 243/160, 1024/675 ^5 up 5th ^A Su Si
33 747.2 20/13, 125/81, 192/125 ^^5, vvm6 dup 5th, dudminor 6th ^^A, vvBb Si / Fle Saw
34 769.8 11/7, 14/9, 25/16 vm6 downminor 6th vBb Flo Lo
35 792.5 19/12, 30/19, 128/81 m6 minor 6th Bb Fla Leh
36 815.1 8/5 ^m6 upminor 6th ^Bb Flu Le
37 837.7 13/8, 81/50 ^^m6 dupminor 6th ^^Bb Fli Lu
38 860.4 18/11, 400/243 vvM6 dudmajor 6th vvB Le Luh
39 883.0 5/3 vM6 downmajor 6th vB Lo La
40 905.7 22/13, 27/16, 32/19 M6 major 6th B La Laa
41 928.3 12/7 ^M6 upmajor 6th ^B Lu Li
42 950.9 19/11, 26/15, 125/72, 216/125 ^^M6, vvm7 dupmajor 6th, dudminor 7th ^^B, vvC Li / The Law
43 973.6 7/4 vm7 downminor 7th vC Tho Ta
44 996.2 16/9 m7 minor 7th C Tha Teh
45 1018.9 9/5 ^m7 upminor 7th ^C Thu Te
46 1041.5 11/6, 20/11, 64/35, 729/400 ^^m7 dupminor 7th ^^C Thi Tu
47 1064.2 13/7, 24/13, 50/27 vvM7 dudmajor 7th vvC# Te Tuh
48 1086.8 15/8 vM7 downmajor 7th vC# To Ti
49 1109.4 19/10, 36/19, 40/21, 243/128 M7 major 7th C# Ta Tih
50 1132.1 21/11, 25/13, 27/14, 52/27, 48/25 ^M7 upmajor 7th ^C# Tu To
51 1154.7 35/18, 64/33, 96/49, 125/64 ^^M7, vv8 dupmajor 7th, dud 8ve ^^C#, vvD Ti / De Taw
52 1177.4 49/25, 63/32, 160/81 v8 down 8ve vD Do Da
53 1200.0 2/1 P8 perfect 8ve D Da Do

Interval quality and chord names in color notation

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

Quality Color Monzo format Examples
downminor zo (a, b, 0, 1) 7/6, 7/4
minor fourthward wa (a, b) with b < −1 32/27, 16/9
upminor gu (a, b, −1) 6/5, 9/5
dupminor ilo (a, b, 0, 0, 1) 11/9, 11/6
dudmajor lu (a, b, 0, 0, −1) 12/11, 18/11
downmajor yo (a, b, 1) 5/4, 5/3
major fifthward wa (a, b) with b > 1 9/8, 27/16
upmajor ru (a, b, 0, −1) 9/7, 12/7

All 53edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked−3rds chord 6 – 1 – 3 – 5 – 7 – 9 – 11 – 13). Alterations are always enclosed in parentheses, additions never are.

Here are the zo, gu, ilo, lu, yo and ru triads:

Color of the 3rd JI chord Notes as edosteps Notes of C chord Written name Spoken name
zo 6:7:9 0–12–31 C vEb G Cvm C downminor
gu 10:12:15 0–14–31 C ^Eb G C^m C upminor
ilo 18:22:27 0–15–31 C ^^Eb G C^^m C dupminor
lu 22:27:33 0–16–31 C vvE G Cvv C dudmajor or C dud
yo 4:5:6 0–17–31 C vE G Cv C downmajor or C down
ru 14:18:21 0–19–31 C ^E G C^ C upmajor or C up

For a more complete list, see Ups and downs notation #Chords and chord progressions.

Notation

Ups and downs notation

53edo can be notated with ups and downs, spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
      
         
    
Flat symbol       
              

Another notation uses alternative ups and downs. Here, this can be done using sharps and flats with arrows, borrowed from extended Helmholtz–Ellis notation:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
Flat symbol

Sagittal notation

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/806561/640040/39

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/806561/640040/39

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

Relationship to 12edo

53edo's circle of 53 fifths can be bent into a 12-spoked "spiral of fifths". This makes sense to do because going up by 12 fifths results in the Pythagorean comma (by definition), which is mapped to one edostep and is thus also the syntonic and septimal comma, introducing a simple second accidental in the form of the arrow to reach useful intervals from the basic 12-chromatic scale. The one-edostep comma is a requirement in Kite's theory, and implies that 31\53 is on the 7\12 kite in the scale tree.

This "spiral of fifths" can be a useful construct for introducing 53edo to musicians unfamiliar with microtonal music. It may help composers and musicians to make visual sense of the notation, and to understand what size of a jump is likely to land them where compared to 12edo.

The two innermost and two outermost intervals on the spiral are duplicates, reflecting the fact that it is a repeating circle at heart and the spiral shape is only a helpful illusion.

Approximation to JI

53edo provides excellent approximations for the classic 5-limit just chords and scales, such as the Ptolemy–Zarlino "just major" scale.

Interval Ratio Size Difference
Perfect fifth 3/2 31 −0.07 cents
Major third 5/4 17 −1.40 cents
Minor third 6/5 14 +1.34 cents
Major second 9/8 9 −0.14 cents
10/9 8 −1.27 cents
Minor second 16/15 5 +1.48 cents

Because the 5th is so accurate, 53edo also offers good approximations for Pythagorean thirds. In addition, the 43\53 interval is only 4.8 cents wider than 7/4, so 53edo can also be used for 7-limit harmony, in which it tempers out the septimal kleisma, 225/224.

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 53edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 53edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 0.068 0.3
9/8, 16/9 0.136 0.6
9/5, 10/9 1.272 5.6
15/13, 26/15 1.316 5.8
5/3, 6/5 1.340 5.9
13/10, 20/13 1.384 6.1
5/4, 8/5 1.408 6.2
15/8, 16/15 1.476 6.5
13/9, 18/13 2.655 11.7
13/12, 24/13 2.724 12.0
13/8, 16/13 2.792 12.3
7/4, 8/7 4.759 21.0
7/6, 12/7 4.827 21.3
9/7, 14/9 4.895 21.6
13/11, 22/13 5.130 22.7
7/5, 10/7 6.167 27.2
15/14, 28/15 6.235 27.5
15/11, 22/15 6.445 28.5
11/10, 20/11 6.514 28.8
13/7, 14/13 7.551 33.3
11/9, 18/11 7.785 34.4
11/6, 12/11 7.854 34.7
11/8, 16/11 7.922 35.0
11/7, 14/11 9.961 44.0
15-odd-limit intervals in 53edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 0.068 0.3
9/8, 16/9 0.136 0.6
9/5, 10/9 1.272 5.6
15/13, 26/15 1.316 5.8
5/3, 6/5 1.340 5.9
13/10, 20/13 1.384 6.1
5/4, 8/5 1.408 6.2
15/8, 16/15 1.476 6.5
13/9, 18/13 2.655 11.7
13/12, 24/13 2.724 12.0
13/8, 16/13 2.792 12.3
7/4, 8/7 4.759 21.0
7/6, 12/7 4.827 21.3
9/7, 14/9 4.895 21.6
13/11, 22/13 5.130 22.7
7/5, 10/7 6.167 27.2
15/14, 28/15 6.235 27.5
15/11, 22/15 6.445 28.5
11/10, 20/11 6.514 28.8
13/7, 14/13 7.551 33.3
11/9, 18/11 7.785 34.4
11/6, 12/11 7.854 34.7
11/8, 16/11 7.922 35.0
11/7, 14/11 12.681 56.0

Higher-limit JI

53edo has only 5 pairs of inconsistent intervals in the full 27-odd-limit: {11/7, 14/11}, {17/11, 22/17}, {19/17, 34/19}, {21/11, 22/21}, and {23/22, 44/23}. This is perhaps remarkable compared to 9 pairs in 46edo and 11 pairs in 41edo, because the smallest edo after 53edo to get 5 or less inconsistencies in the 27-odd-limit is 99edo (using the 99ef val), followed by 111edo (patent val), 130edo (patent val) and 159edo (also patent); all of these get 5 inconsistencies as well except 159edo which gets 1 and which is itself a superset of 53edo. However, most interpret the approximation of prime 17 in 53edo as too off for all but the most opportunistic harmonies, and some question the 23 and possibly also 11, so the practical significance of this is debatable.

As shown below, there is also a cluster of usable higher primes starting at 71; even 89 (4.84 ¢ flat), 97 (4.63 ¢ sharp) and 101 (2.6 ¢ sharp) are usable if placed in just the right context. (Note that prime 67 is almost perfectly off.)

Approximation of large prime harmonics in 53edo
Harmonic 71 73 79 83
Error Absolute (¢) +1.44 -1.37 -2.27 +2.78
Relative (%) +6.3 -6.1 -10.0 +12.3
Steps
(reduced) 		326
(8) 		328
(10) 		334
(16) 		338
(20)

This makes 53edo excellent (for its size) in the 2.3.5.7.11.13.19.23.37.41.71.73(.79.83.101) subgroup, although some higher error primes like 11 and 23 require the right context to be convincing.

Note that the high primes, in rooted (/2n) position, essentially act as alternate interpretations of LCJI intervals, if you want to force a rooted interpretation; namely: 71/64 as ~10/9, 73/64 as ~8/7, 79/64 as ~16/13, and perhaps most questionably in the context of 53edo, 83/64 as ~13/10.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-84 53⟩ [⟨53 84]] +0.022 0.022 0.10
2.3.5 15625/15552, 32805/32768 [⟨53 84 123]] +0.216 0.276 1.22
2.3.5.7 225/224, 1728/1715, 3125/3087 [⟨53 84 123 149]] −0.262 0.861 3.81
2.3.5.7.11 99/98, 121/120, 176/175, 2200/2187 [⟨53 84 123 149 183]] +0.248 1.279 5.64
2.3.5.7.11.13 99/98, 121/120, 169/168, 176/175, 275/273 [⟨53 84 123 149 183 196]] +0.332 1.183 5.22
2.3.5.7.11.13.19 99/98, 121/120, 169/168, 176/175, 209/208, 275/273 [⟨53 84 123 149 183 196 225]] +0.391 1.105 4.88
  • 53et is lower in relative error than any previous equal temperaments in the 3-, 5-, and 13-limit. The next equal temperaments doing better in these subgroups are 306, 118, and 58, respectively. It is even more prominent in the 2.3.5.7.13.19 and 2.3.5.7.13.19.23 subgroups, and the next equal temperament doing better in either subgroup is 130.

Commas

Commas that 53edo tempers out using its patent val, ⟨53 84 123 149 183 196 225], include:

Prime
limit 		Ratio[note 2] Monzo Cents Color name Name(s)
3 (52 digits) [-84 53⟩ 3.62 Tribilawa 53-comma, Mercator's comma
5 (14 digits) [-21 3 7⟩ 10.06 Lasepyo Semicomma
5 15625/15552 [-6 -5 6⟩ 8.11 Tribiyo Kleisma
5 (14 digits) [9 -13 5⟩ 6.15 Saquinyo Amity comma
5 (22 digits) [24 -21 4⟩ 4.20 Sasaquadyo Vulture comma
5 32805/32768 [-15 8 1⟩ 1.95 Layo Schisma
7 3125/3087 [0 -2 5 -3⟩ 21.18 Triru-aquinyo Gariboh comma
7 1728/1715 [6 3 -1 -3⟩ 13.07 Triru-agu Orwellisma
7 225/224 [-5 2 2 -1⟩ 7.71 Ruyoyo Marvel comma, septimal kleisma
7 4375/4374 [-1 -7 4 1⟩ 0.40 Zoquadyo Ragisma
11 99/98 [-1 2 0 -2 1⟩ 17.58 Loruru Mothwellsma
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 (22 digits) [33 -23 0 0 1⟩ 6.35 Trisalo Pythrabian comma
11 385/384 [-7 -1 1 1 1⟩ 4.50 Lozoyo Keenanisma
11 540/539 [2 3 1 -2 -1⟩ 3.21 Lururuyo Swetisma
13 275/273 [0 -1 2 -1 1 -1⟩ 12.64 Thuloruyoyo Gassorma
13 169/168 [-3 -1 0 -1 0 2⟩ 10.27 Thothoru Buzurgisma, dhanvantarisma
13 625/624 [-4 -1 4 0 0 -1⟩ 2.77 Thuquadyo Tunbarsma
13 676/675 [2 -3 -2 0 0 2⟩ 2.56 Bithogu Island comma, parizeksma
13 1001/1000 [-3 0 -3 1 1 1⟩ 1.73 Tholozotrigu Fairytale comma, sinbadma
13 2080/2079 [5 -3 1 -1 -1 1⟩ 0.83 Tholuruyo Ibnsinma, sinaisma
13 4096/4095 [12 -2 -1 -1 0 -1⟩ 0.42 Sathurugu Schismina

Linear temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 2\53 45.3 36/35 Quartonic
1 5\53 113.2 16/15 Misneb
1 6\53 135.8 13/12~14/13 Doublethink
1 7\53 158.5 11/10 Hemikleismic
1 9\53 203.8 9/8 Baldy
1 10\53 226.4 8/7 Semaja
1 11\53 249.1 15/13 Hemischis / hemigari
1 12\53 271.7 7/6 Orwell
1 13\53 294.3 25/21 Kleiboh
1 14\53 317.0 6/5 Hanson / catakleismic / countercata
1 15\53 339.6 11/9 Amity / houborizic
1 16\53 362.3 16/13 Submajor
1 18\53 407.5 1225/972 Ditonic / coditone
1 19\53 430.2 9/7 Hamity
1 20\53 452.8 13/10 Maja
1 21\53 475.5 21/16 Vulture / buzzard
1 22\53 498.1 4/3 Garibaldi / pontiac
1 23\53 520.8 4/3 Mavila (53bbcc)
1 25\53 566.0 18/13 Alphatrimot
1 26\53 588.7 45/32 Untriton / aufo

* Octave-reduced form, reduced to the first half-octave

Scales

Mos scales

While there is only one possible generator for the diatonic mos scale supported by this edo, there are a greater number of generators for other mosses such as the antidiatonic scale.

Scales approximated from JI

  • The eagle 53 scale described by John O'Sullivan
  • Ptolmey–Zarlino justly-intonated major: 9 8 5 9 8 9 5
  • Ptolmey–Zarlino justly-intonated minor: 9 5 8 9 5 9 8
From AFDOs
This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.
  • Composite Cliffedge (approximated from 60afdo): 12 10 9 19 3
  • Composite Deja Vu (approximated from 101afdo): 14 17 5 9 8
  • Composite Dungeon (approximated from 30afdo): 17 5 9 4 18
  • Composite Freeway (approximated from 6afdo): 12 10 9 8 7 7
  • Composite Geode (approximated from 6afdo): 12 10 9 15 7
  • Composite Labyrinth (approximated from 30afdo): 7 7 17 5 17
  • Composite Mushroom (approximated from 30afdo): 12 10 9 3 19
  • Composite Underpass (approximated from 10afdo): 14 17 10 4 8
  • Spectral Arcade (approximated from 32afdo): 17 4 10 12 10
  • Spectral Mechanical (approximated from 16afdo): 13 4 14 12 10
  • Spectral Moonbeam (approximated from 16afdo): 9 4 18 17 5
  • Spectral Springwater (approximated from 8afdo): 9 8 14 12 10
  • Spectral Starship (approximated from 68ifdo): 4 13 4 10 12 10
  • Spectral Volcanic (approximated from 16afdo): 5 12 14 12 10

Other scales

Instruments

Music

See also: Category:53edo tracks

Modern renderings

Johann Sebastian Bach
Nicolaus Bruhns
George Frideric Handel
Scott Joplin
  • Maple Leaf Rag (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)
  • Maple Leaf Rag (1899) – with syntonic comma adjustment, arranged for harpsichord and rendered by Claudi Meneghin (2024)
Shirō Sagisu

21st century

Bryan Deister
Francium
Andrew Heathwaite
Hideya
Nathan Ho
Aaron Krister Johnson (site[dead link])
MortisTheneRd
Mundoworld
Prent Rodgers
Sevish
Subhraag Singh
Gene Ward Smith
  • Trio in Orwell (archived 2010) – detail | play – in Orwell[9], 53edo tuning
Nick Stephens
Cam Taylor
Chris Vaisvil
  • The Fallen of Kleismic15 (2013) – blog | play – in Kleismic[15], 53edo tuning
Valeriana of the Night
Randy Wells
Xotla

Notes

  1. ↑ Based on treating 53edo as a no-17's 19-limit temperament; other approaches are also possible. Italics represent inconsistent intervals which are mapped by the 19-limit patent val to their second-closest (as opposed to closest) approximation in 53edo.
  2. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

References

  1. ↑ Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". Empirical Musicology Review 15 (3–4):223–48. https://doi.org/10.18061/emr.v15i3-4.7647.
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