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m →Intervals: 36/25 and 25/18 were missing, but also, 53edo is a model of 2.3.5.7.13 JI so it helps to have more accurate descriptors, but also, the cent values feel wrong and this is amplified by stacking them
 
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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 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 distintion to [[41edo]] or [[46edo]]. Like 41 and 46, 53 is distinctly [[consistent]] in the [[9-odd-limit]], 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. Besides, 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 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]]. It shines however in the 2.3.5.19 and [[2.3.5.13 subgroup|2.3.5.13]] subgroups, where it offers excellent approximations with decent complexity.  


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)]], [[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 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 indistinguishable from just in most contexts.


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.


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 ===
{{Harmonics in equal|53|columns=11}}
{{Harmonics in equal|53|columns=12|start=12|collapsed=true|title=Approximation of prime harmonics in 53edo (continued)}}


=== Prime harmonics ===
See [[#Approximation to JI]] for details and a more in-depth discussion on the higher harmonics.
{{Harmonics in equal|53|columns=13}}
{{Harmonics in equal|53|columns=10|start=14|collapsed=true|title=Approximation of prime harmonics in 53edo (continued)}}


See [[#Approximation to JI]] for details and a more in-depth discussion.
=== As a tuning of other temperaments ===
As an equal temperament, 53et 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.  


=== Subsets and supersets ===
=== Subsets and supersets ===
53edo is the 16th [[prime edo]], following [[47edo]] and coming before [[59edo]].
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.
Many of its multiples such as [[159edo]], [[212edo]], [[742edo]], [[901edo]] and the zeta [[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 ==
== Intervals ==
{| class="wikitable center-all right-2 left-3 left-5"
{| class="wikitable center-all right-2 left-3 left-5"
|-
|-
! &#35;
! #
! Cents
! Cents
! 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>
! 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]] ([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>5</sup>A1 and ^d2)
! 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 49: Line 50:
| 1
| 1
| 22.64
| 22.64
| [[81/80]], [[64/63]], [[50/49]]
| ''[[50/49]]'', [[64/63]], [[81/80]]
| ^1
| ^1
| up unison
| up unison
Line 58: Line 59:
| 2
| 2
| 45.28
| 45.28
| [[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 67: Line 68:
| 3
| 3
| 67.92
| 67.92
| [[25/24]], [[28/27]], [[22/21]], [[27/26]], [[26/25]]
| ''[[22/21]]'', [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| vvA1, vm2
| vvA1, vm2
| dudaug 1sn, downminor 2nd
| dudaug 1sn, downminor 2nd
Line 85: Line 86:
| 5
| 5
| 113.21
| 113.21
| [[16/15]], [[15/14]]
| [[15/14]], [[16/15]]
| A1, ^m2
| A1, ^m2
| aug 1sn, upminor 2nd
| aug 1sn, upminor 2nd
Line 94: Line 95:
| 6
| 6
| 135.85
| 135.85
| [[14/13]], [[13/12]], [[27/25]]
| [[13/12]], [[14/13]], [[27/25]]
| ^^m2
| ^^m2
| dupminor 2nd
| dupminor 2nd
Line 103: Line 104:
| 7
| 7
| 158.49
| 158.49
| [[35/32]], [[12/11]], [[11/10]], [[57/52]], [[800/729]]
| [[11/10]], [[12/11]], [[35/32]], [[57/52]], [[800/729]]
| vvM2
| vvM2
| dudmajor 2nd
| dudmajor 2nd
Line 139: Line 140:
| 11
| 11
| 249.06
| 249.06
| [[15/13]], [[144/125]], [[125/108]]
| [[15/13]], [[22/19]], [[125/108]], [[144/125]]
| ^^M2, vvm3
| ^^M2, vvm3
| dupmajor 2nd, dudminor 3rd
| dupmajor 2nd, dudminor 3rd
Line 229: Line 230:
| 21
| 21
| 475.47
| 475.47
| [[21/16]], [[25/19]], [[675/512]], [[320/243]]
| [[21/16]], [[25/19]], [[320/243]], [[675/512]]
| v4
| v4
| down 4th
| down 4th
Line 247: Line 248:
| 23
| 23
| 520.75
| 520.75
| [[27/20]]
| [[19/14]], [[27/20]]
| ^4
| ^4
| up 4th
| up 4th
Line 265: Line 266:
| 25
| 25
| 566.04
| 566.04
| [[18/13]]
| [[18/13]], [[25/18]]
| vvA4, vd5
| vvA4, vd5
| dudaug 4th, downdim 5th
| dudaug 4th, downdim 5th
Line 292: Line 293:
| 28
| 28
| 633.96
| 633.96
| [[13/9]]
| [[13/9]], [[36/25]]
| ^A4, ^^d5
| ^A4, ^^d5
| upaug 4th, dupdim 5th
| upaug 4th, dupdim 5th
Line 310: Line 311:
| 30
| 30
| 679.25
| 679.25
| [[40/27]]
| [[28/19]], [[40/27]]
| v5
| v5
| down 5th
| down 5th
Line 337: Line 338:
| 33
| 33
| 747.17
| 747.17
| [[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 346: Line 347:
| 34
| 34
| 769.81
| 769.81
| [[14/9]], [[25/16]], ''[[11/7]]''
| ''[[11/7]]'', [[14/9]], [[25/16]]
| vm6
| vm6
| downminor 6th
| downminor 6th
Line 418: Line 419:
| 42
| 42
| 950.94
| 950.94
| [[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 454: Line 455:
| 46
| 46
| 1041.51
| 1041.51
| [[64/35]], [[11/6]], [[20/11]], [[729/400]]
| [[11/6]], [[20/11]], [[64/35]], [[729/400]]
| ^^m7
| ^^m7
| dupminor 7th
| dupminor 7th
Line 490: Line 491:
| 50
| 50
| 1132.08
| 1132.08
| [[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 499: Line 500:
| 51
| 51
| 1154.72
| 1154.72
| [[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 508: Line 509:
| 52
| 52
| 1177.36
| 1177.36
| [[160/81]], [[63/32]], [[49/25]]
| ''[[49/25]]'', [[63/32]], [[160/81]]
| v8
| v8
| down 8ve
| down 8ve
Line 516: Line 517:
|-
|-
| 53
| 53
| 1200.00
| 1200.0
| [[2/1]]
| [[2/1]]
| P8
| P8
Line 532: Line 533:
! 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
|-
|-
| dupminor
| dupminor
| ilo
| ilo
| (a, b, 0, 0, 1)
| {{nowrap|(a, b, 0, 0, 1)}}
| 11/9, 11/6
| 11/9, 11/6
|-
|-
| dudmajor
| 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 581: Line 582:
|-
|-
! [[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 596: Line 597:
| 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 603: Line 604:
| ilo
| ilo
| 18:22:27
| 18:22:27
| 0-15-31
| 0–15–31
| C ^^Eb G
| C ^^Eb G
| C^^m
| C^^m
Line 610: Line 611:
| lu
| lu
| 22:27:33
| 22:27:33
| 0-16-31
| 0–16–31
| C vvE G
| C vvE G
| Cvv
| Cvv
Line 617: Line 618:
| yo
| yo
| 4:5:6
| 4:5:6
| 0-17-31
| 0–17–31
| C vE G
| C vE G
| Cv
| Cv
Line 624: Line 625:
| 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 ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
[[Ups and downs notation]] uses free-standing arrows, but the arrows can be connected to conventional accidentals using [[Helmholtz–Ellis notation|Helmholtz–Ellis]] accidentals:
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5}}
{{Sharpness-sharp5-szg}}
Here, a sharp raises by five steps (commas), and a flat lowers by five steps, so single and double arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with triple arrows.
 
=== Kite's ups and downs notation ===
53edo can also be notated with [[Kite's ups and downs notation|Kite's 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).
{{Ups and downs sharpness}}
 
=== Sagittal notation ===
53edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal notation#Spartan|Spartan set]], with the apotome equal to 5 edosteps and the limma to 4 edosteps. Here is a simplified table:
 
{| class="wikitable" style="text-align: center;"
! colspan="2" | Steps
!'''0'''
! 1
! 2
! 3
! 4
!'''5'''
|-
! rowspan="2" | Symbol
! Evo
| rowspan="2" | <big>{{sagittal||//|}}</big>
| rowspan="2" | <big>{{sagittal|/|}}</big>
| rowspan="2" | <big>{{sagittal|//|}}</big>
| {{sagittal|\\!}}{{sagittal|#}}
| {{sagittal|\!}}{{sagittal|#}}
| <big>{{sagittal|#}}</big>
|-
! Revo
| <big>{{sagittal|)||(}}</big>
| <big>{{sagittal|||\}}</big>
| <big>{{sagittal|/||\}}</big>
|}
The following enharmonics from the Spartan set are present (comma tempered out):
* {{sagittal|//|}} = {{Sagittal|/|)}} = {{Sagittal|/|\}} ([[325/324]], [[352/351]])
* {{sagittal|/|}} = {{sagittal||)}} ([[225/224]])
* {{sagittal||(}} = {{sagittal||//|}} ([[5120/5103]])
 
See [[Sagittal notation #Revo|apotome complements]] for equivalent accidental pairs.


===Sagittal notation===
Featured below is the 53edo gamut notated using the best accidental approximants; in this case, pai/pao and phai/phao.
In the following diagrams, 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.
====Evo flavor====


<imagemap>
==== Evo flavor ====
File:53-EDO_Evo_Sagittal.svg
{{Sagittal chart|Evo}}
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>


====Revo flavor====
==== Revo flavor ====
{{Sagittal chart}}


<imagemap>
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:53-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 543 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_Revo_Sagittal.svg]]
</imagemap>


== Relationship to 12edo ==
== Relationship to 12edo ==
53edo's [[circle of fifths|circle of 53 fifths]] can be bent into a [[spiral chart|12-spoked "spiral of fifths"]]. This is possible because 31\53 is on the 7\12 kite in the [[scale tree]]. Stated another way, it is possible because the absolute value of 53edo's [[sharpness#dodeca-sharpness|dodeca-sharpness]] (edosteps per [[Pythagorean comma]]) is 1.  
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.
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.
Line 675: Line 694:


== Approximation to JI ==
== Approximation to JI ==
[[File:53ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 53edo]]
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 716: Line 733:
|}
|}


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.
Because the 5th is so incredibly accurate (next edo with a more accurate fifth is [[200edo]]), 53edo also offers a great approximation to Pythagorean tuning. 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 ===
Line 722: Line 739:


=== 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(.79.83.101) 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}}
 
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]] (/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]].
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]].
Line 731: Line 750:
== 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]]
! rowspan="2" | [[Comma list]]
Line 741: Line 761:
|-
|-
| 2.3
| 2.3
| {{monzo| -84 53 }}
| {{Monzo| -84 53 }}
| {{mapping| 53 84 }}
| {{Mapping| 53 84 }}
| +0.022
| +0.022
| 0.022
| 0.022
Line 749: Line 769:
| 2.3.5
| 2.3.5
| 15625/15552, 32805/32768
| 15625/15552, 32805/32768
| {{mapping| 53 84 123 }}
| {{Mapping| 53 84 123 }}
| +0.216
| +0.216
| 0.276
| 0.276
Line 756: Line 776:
| 2.3.5.7
| 2.3.5.7
| 225/224, 1728/1715, 3125/3087
| 225/224, 1728/1715, 3125/3087
| {{mapping| 53 84 123 149 }}
| {{Mapping| 53 84 123 149 }}
| &minus;0.262
| −0.262
| 0.861
| 0.861
| 3.81
| 3.81
Line 763: Line 783:
| 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
| {{mapping| 53 84 123 149 183 }}
| {{Mapping| 53 84 123 149 183 }}
| +0.248
| +0.248
| 1.279
| 1.279
Line 770: Line 790:
| 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
| {{mapping| 53 84 123 149 183 196 }}
| {{Mapping| 53 84 123 149 183 196 }}
| +0.332
| +0.332
| 1.183
| 1.183
Line 777: Line 797:
| 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
| {{mapping| 53 84 123 149 183 196 225 }}
| {{Mapping| 53 84 123 149 183 196 225 }}
| +0.391
| +0.391
| 1.105
| 1.105
Line 798: Line 818:
| 3
| 3
| <abbr title="19383245667680019896796723/19342813113834066795298816">(52 digits)</abbr>
| <abbr title="19383245667680019896796723/19342813113834066795298816">(52 digits)</abbr>
| {{monzo| -84 53 }}
| {{Monzo| -84 53 }}
| 3.62
| 3.62
| Tribilawa
| Tribilawa
Line 805: Line 825:
| 5
| 5
| [[2109375/2097152|(14 digits)]]
| [[2109375/2097152|(14 digits)]]
| {{monzo| -21 3 7 }}
| {{Monzo| -21 3 7 }}
| 10.06
| 10.06
| Lasepyo
| Lasepyo
Line 812: Line 832:
| 5
| 5
| [[15625/15552]]
| [[15625/15552]]
| {{monzo| -6 -5 6 }}
| {{Monzo| -6 -5 6 }}
| 8.11
| 8.11
| Tribiyo
| Tribiyo
Line 819: Line 839:
| 5
| 5
| <abbr title="1600000/1594323">(14 digits)</abbr>
| <abbr title="1600000/1594323">(14 digits)</abbr>
| {{monzo| 9 -13 5 }}
| {{Monzo| 9 -13 5 }}
| 6.15
| 6.15
| Saquinyo
| Saquinyo
Line 826: Line 846:
| 5
| 5
| <abbr title="10485760000/10460353203">(22 digits)</abbr>
| <abbr title="10485760000/10460353203">(22 digits)</abbr>
| {{monzo| 24 -21 4 }}
| {{Monzo| 24 -21 4 }}
| 4.20
| 4.20
| Sasaquadyo
| Sasaquadyo
Line 833: Line 853:
| 5
| 5
| [[32805/32768]]
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| {{Monzo| -15 8 1 }}
| 1.95
| 1.95
| Layo
| Layo
Line 840: Line 860:
| 7
| 7
| [[3125/3087]]
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| 21.18
| Triru-aquinyo
| Triru-aquinyo
Line 847: Line 867:
| 7
| 7
| [[1728/1715]]
| [[1728/1715]]
| {{monzo| 6 3 -1 -3 }}
| {{Monzo| 6 3 -1 -3 }}
| 13.07
| 13.07
| Triru-agu
| Triru-agu
Line 854: Line 874:
| 7
| 7
| [[225/224]]
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| {{Monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Ruyoyo
| Ruyoyo
Line 861: Line 881:
| 7
| 7
| [[4375/4374]]
| [[4375/4374]]
| {{monzo| -1 -7 4 1 }}
| {{Monzo| -1 -7 4 1 }}
| 0.40
| 0.40
| Zoquadyo
| Zoquadyo
Line 868: Line 888:
| 11
| 11
| [[99/98]]
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| 17.58
| Loruru
| Loruru
Line 875: Line 895:
| 11
| 11
| [[121/120]]
| [[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| 14.37
| Lologu
| Lologu
Line 882: Line 902:
| 11
| 11
| [[176/175]]
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Lorugugu
Line 889: Line 909:
| 11
| 11
| <abbr title="94489280512/94143178827">(22 digits)</abbr>
| <abbr title="94489280512/94143178827">(22 digits)</abbr>
| {{monzo| 33 -23 0 0 1 }}
| {{Monzo| 33 -23 0 0 1 }}
| 6.35
| 6.35
| Trisalo
| Trisalo
Line 896: Line 916:
| 11
| 11
| [[385/384]]
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| 4.50
| Lozoyo
| Lozoyo
Line 903: Line 923:
| 11
| 11
| [[540/539]]
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| 3.21
| Lururuyo
| Lururuyo
Line 910: Line 930:
| 13
| 13
| [[275/273]]
| [[275/273]]
| {{monzo| 0 -1 2 -1 1 -1 }}
| {{Monzo| 0 -1 2 -1 1 -1 }}
| 12.64
| 12.64
| Thuloruyoyo
| Thuloruyoyo
Line 917: Line 937:
| 13
| 13
| [[169/168]]
| [[169/168]]
| {{monzo| -3 -1 0 -1 0 2 }}
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.27
| 10.27
| Thothoru
| Thothoru
Line 924: Line 944:
| 13
| 13
| [[625/624]]
| [[625/624]]
| {{monzo| -4 -1 4 0 0 -1 }}
| {{Monzo| -4 -1 4 0 0 -1 }}
| 2.77
| 2.77
| Thuquadyo
| Thuquadyo
Line 931: Line 951:
| 13
| 13
| [[676/675]]
| [[676/675]]
| {{monzo| 2 -3 -2 0 0 2 }}
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| 2.56
| Bithogu
| Bithogu
Line 938: Line 958:
| 13
| 13
| [[1001/1000]]
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| {{Monzo| -3 0 -3 1 1 1 }}
| 1.73
| 1.73
| Tholozotrigu
| Tholozotrigu
Line 945: Line 965:
| 13
| 13
| [[2080/2079]]
| [[2080/2079]]
| {{monzo| 5 -3 1 -1 -1 1 }}
| {{Monzo| 5 -3 1 -1 -1 1 }}
| 0.83
| 0.83
| Tholuruyo
| Tholuruyo
Line 952: Line 972:
| 13
| 13
| [[4096/4095]]
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| 0.42
| Sathurugu
| Sathurugu
| Schismina
| Minisma
|}
|}


Line 973: Line 993:
| 1
| 1
| 2\53
| 2\53
| 45.28
| 45.3
| 36/35
| 36/35
| [[Quartonic]]
| [[Quartonic]]
Line 979: Line 999:
| 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 991: Line 1,017:
| 1
| 1
| 9\53
| 9\53
| 203.77
| 203.8
| 9/8
| 9/8
| [[Baldy]]
| [[Baldy]]
Line 997: Line 1,023:
| 1
| 1
| 10\53
| 10\53
| 226.42
| 226.4
| 8/7
| 8/7
| [[Semaja]]
| [[Semaja]]
Line 1,003: Line 1,029:
| 1
| 1
| 11\53
| 11\53
| 249.06
| 249.1
| 15/13
| 15/13
| [[Hemischis]] / [[hemigari]]
| [[Hemischis]] / [[hemigari]]
Line 1,009: Line 1,035:
| 1
| 1
| 12\53
| 12\53
| 271.70
| 271.7
| 7/6
| 7/6
| [[Orwell]]
| [[Orwell]]
Line 1,015: Line 1,041:
| 1
| 1
| 13\53
| 13\53
| 294.34
| 294.3
| 25/21
| 25/21
| [[Kleiboh]]
| [[Kleiboh]]
Line 1,021: Line 1,047:
| 1
| 1
| 14\53
| 14\53
| 316.98
| 317.0
| 6/5
| 6/5
| [[Hanson]] / [[catakleismic]] / [[countercata]]
| [[Hanson]] / [[catakleismic]] / [[countercata]]
Line 1,027: Line 1,053:
| 1
| 1
| 15\53
| 15\53
| 339.62
| 339.6
| 11/9
| 11/9
| [[Amity]] / [[houborizic]]
| [[Amity]] / [[houborizic]]
Line 1,033: Line 1,059:
| 1
| 1
| 16\53
| 16\53
| 362.26
| 362.3
| 16/13
| 16/13
| [[Submajor]]
| [[Demibuzzard]] / submajor / interpental
|-
|-
| 1
| 1
| 18\53
| 18\53
| 407.55
| 407.5
| 1225/972
| 1225/972
| [[Ditonic]] / [[coditone]]
| [[Ditonic]] / [[coditone]]
Line 1,045: Line 1,071:
| 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 1,057: Line 1,089:
| 1
| 1
| 22\53
| 22\53
| 498.11
| 498.1
| 4/3
| 4/3
| [[Garibaldi]] / [[pontiac]]
| [[Garibaldi]] / [[pontiac]]
Line 1,063: Line 1,095:
| 1
| 1
| 23\53
| 23\53
| 520.75
| 520.8
| 4/3
| 4/3
| [[Mavila]] (53bbcc)
| [[Mavila]] (53bbcc)
Line 1,069: Line 1,101:
| 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
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave


== Scales ==
== Scales ==
 
=== Mos 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]]
* [[List of MOS scales in 53edo]]
Line 1,092: Line 1,123:
* Ptolmey–Zarlino justly-intonated major: 9 8 5 9 8 9 5
* Ptolmey–Zarlino justly-intonated major: 9 8 5 9 8 9 5
* Ptolmey–Zarlino justly-intonated minor: 9 5 8 9 5 9 8
* 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 Cliffedge (approximated from [[60afdo]]): 12 10 9 19 3
* Composite Deja Vu (approximated from [[101afdo]]): 14 17 5 9 8
* Composite Deja Vu (approximated from [[101afdo]]): 14 17 5 9 8
Line 1,108: Line 1,142:


=== Other scales ===
=== Other scales ===
* [[Sequar5m]]: 6 3 6 2 3 6 2 3 6 3 2 6 3 2
* [[cthon5m]]: 6 3 6 2 3 6 2 3 6 3 2 6 3 2
* Palace (approximated from [[Porky]] in [[29edo]]): 7 7 8 9 7 7 8
* 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 ==
Line 1,120: Line 1,159:
** [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=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)
* [https://www.youtube.com/watch?v=vcinR7nUthA "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
Line 1,126: Line 1,167:
* [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)
* [https://www.youtube.com/watch?v=r6R4SsaT8ig ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)
; {{w|Frédéric Chopin}}
* Prelude Op. 28, No. 4 in E minor « Suffocation » (1839), arranged for harpsichord, tuned into 53-edo &mdash; rendered by [[Claudi Meneghin]] (2025)
** [https://www.youtube.com/watch?v=0VB1hv0-AmE Near-Pythagorean version]
** [https://www.youtube.com/shorts/iYtZGBKHcpU Schismatic version]
* [https://www.youtube.com/shorts/4YEHMpaO4bA ''"Waterfall" Étude from 12 Études, op. 10''] (1829–1832) &mdash; rendered by Claudi Meneghin (2025)


; {{W|George Frideric Handel}}
; {{W|George Frideric Handel}}
Line 1,131: Line 1,178:


; {{W|Scott Joplin}}
; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=AKXMuhB3uHQ ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)
* ''Maple Leaf Rag'' (1899) – arranged for harpsichord and rendered by Claudi Meneghin ([https://www.youtube.com/watch?v=AKXMuhB3uHQ 2024 version]; [https://www.youtube.com/shorts/VsOk3az8J40 2025 version]))
* [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)
* ''Maple Leaf Rag'' (1899) – with syntonic comma adjustment, arranged for harpsichord and rendered by Claudi Meneghin ([https://www.youtube.com/watch?v=t-pRqKGX0oo 2024 version]; [https://www.youtube.com/shorts/msBeUJjFlV4 (2025 version)]


; {{W|Shirō Sagisu}}
; {{W|Shirō Sagisu}}
Line 1,138: Line 1,185:
* [https://www.youtube.com/watch?v=DCENVnxH6bI ''Bande-announce''] – rendered by MortisTheneRd (2024)
* [https://www.youtube.com/watch?v=DCENVnxH6bI ''Bande-announce''] – rendered by MortisTheneRd (2024)


=== 21st century ===
==== 21st century ====
; [[ALLY195]]
* [https://www.bilibili.com/video/BV1f54y1r7XG/ ''My Soul adaptation''] (2020)
 
; [[Alxeusxiao]]
* [https://www.bilibili.com/video/BV1zM4m1m7Gz/ ''53edo exploration''] (2024)
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/r-Tzq33OGM4 ''microtonal improvisation in 53edo''] (2025)
* [https://www.youtube.com/shorts/8jKjvVw4tvw ''Color & Electricity - muship (microtonal cover in 53edo)''] (2025)
* [https://www.youtube.com/shorts/tIx3PcOyNJo ''53edo improv''] (2025)
* ''Fantasy in 53edo'' ([https://www.youtube.com/shorts/fgsT-1pBw8g abstract version]; [https://www.youtube.com/watch?v=-q-T3HuGehk visualizer version] (2025)
* [https://www.youtube.com/shorts/zCthwbPH2cY ''Finale - Undertale (microtonal cover in 53edo)''] (2026)
* ''Waltz in 53edo'' (2026)
** [https://www.youtube.com/shorts/WtSaDQCyfVc <nowiki>[Short]</nowiki>] (with Lumatone view))
** [https://www.youtube.com/watch?v=_xumSANdf-g <nowiki>[Full version]</nowiki>]
 
; [[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
* [https://www.youtube.com/watch?v=kwELa9kP8YU ''Blasphemous Rumors''] (2024) – Blackdye in 53edo
* "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]
* [https://www.youtube.com/watch?v=efGrW8uSGuE ''Opunish Bathomet''] (2025)


; [[Andrew Heathwaite]]
; [[Andrew Heathwaite]]
Line 1,156: Line 1,224:
; [[Aaron Krister Johnson]] ([http://www.akjmusic.com site]{{dead link}})
; [[Aaron Krister Johnson]] ([http://www.akjmusic.com site]{{dead link}})
* [http://www.akjmusic.com/audio/desert_prayer.mp3 ''Desert Prayer'']{{dead link}}
* [http://www.akjmusic.com/audio/desert_prayer.mp3 ''Desert Prayer'']{{dead link}}
; [[Logan02A4]]
* [https://www.bilibili.com/video/BV1mBCRYmEhg/ ''53edo try''] (2024)
; [[Claudi Meneghin]]
* [https://www.youtube.com/shorts/g7C2OrFd-nk ''Orwell Micro Trio, for Organ (Just: 7 Orwells = 1 Twelfth)''] (2025) &mdash; actually in open-ended Orwell tuning, but with the generator adjusted to be extremely close to 12\53, at 271.71{{c}}
; [[Merct]]
* [https://soundcloud.com/merct/drifting-light ''drifting light''] (2025)
* [https://soundcloud.com/merct/i-dont-want-to-die ''i don't want to die''] (2025)


; [[MortisTheneRd]]
; [[MortisTheneRd]]
* [https://www.youtube.com/watch?v=TWVN8ui48ew ''Psychedelic Inventions in 53edo''] (2024)
* [https://www.youtube.com/watch?v=TWVN8ui48ew ''Psychedelic Inventions in 53edo''] (2024)
* [https://www.youtube.com/watch?v=3kZS6j4N6qg ''Circle/Spiral of Fifth in 53EDO, within human ears spectrum.''] (2025)
; [[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]]
; [[Subhraag Singh]]
Line 1,167: Line 1,254:


; [[Gene Ward Smith]]
; [[Gene Ward Smith]]
* ''Trio in Orwell'' (archived 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]]
; [[Nick Stephens]]
Line 1,174: Line 1,261:
; [[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)
; [[The Evil Doings Of An Intergalactic Skeleton]]
* [https://youtu.be/YalIfCKFkd0 ''Metal Reindeer''] (2025)


; [[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]]
* [https://www.youtube.com/watch?v=eMPQDRTHGhg ''Hero''] (2023)
* [https://www.youtube.com/watch?v=eMPQDRTHGhg ''Hero''] (2023)
; [[VitaminCD]]
* [https://www.youtube.com/watch?v=KCWhecfwlMw ''<nowiki>Orwellian in Nature (Orwell [9] Microtonal Lament)</nowiki>''] (2025)


; [[Randy Wells]]
; [[Randy Wells]]
Line 1,187: Line 1,285:
* "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
* "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
== Instruments ==
* [[Lumatone mapping for 53edo]]
* [[Skip fretting system 53 3 14]]


== Notes ==
== Notes ==
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== References ==
== References ==
<references />
<references/>


[[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 10:36, 8 June 2026

← 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. It shines however in the 2.3.5.19 and 2.3.5.13 subgroups, where it offers excellent approximations with decent complexity.

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 indistinguishable from just in most contexts.

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 29 31
Error Absolute (¢) +0.00 -0.07 -1.41 +4.76 -7.92 -2.79 +8.25 -3.17 +5.69 -10.71 +9.68
Relative (%) +0.0 -0.3 -6.2 +21.0 -35.0 -12.3 +36.4 -14.0 +25.1 -47.3 +42.8
Steps
(reduced) 		53
(0) 		84
(31) 		123
(17) 		149
(43) 		183
(24) 		196
(37) 		217
(5) 		225
(13) 		240
(28) 		257
(45) 		263
(51)
Approximation of prime harmonics in 53edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79 83
Error Absolute (¢) -2.29 +1.13 +9.24 -8.90 +9.51 +4.98 -7.45 +11.26 +1.44 -1.37 -2.27 +2.78
Relative (%) -10.1 +5.0 +40.8 -39.3 +42.0 +22.0 -32.9 +49.7 +6.3 -6.1 -10.0 +12.3
Steps
(reduced) 		276
(11) 		284
(19) 		288
(23) 		294
(29) 		304
(39) 		312
(47) 		314
(49) 		322
(4) 		326
(8) 		328
(10) 		334
(16) 		338
(20)

See #Approximation to JI for details and a more in-depth discussion on the higher harmonics.

As a tuning of other temperaments

As an equal temperament, 53et 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.

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 the zeta 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 22.64 50/49, 64/63, 81/80 ^1 up unison ^D Du Di
2 45.28 33/32, 36/35, 49/48, 128/125 ^^1, vvm2 dup unison, dudminor 2nd ^^D, vvEb Di / Fre Daw
3 67.92 22/21, 25/24, 26/25, 27/26, 28/27 vvA1, vm2 dudaug 1sn, downminor 2nd vvD#, vEb Fro Ro
4 90.57 19/18, 20/19, 21/20, 256/243 vA1, m2 downaug 1sn, minor 2nd vD#, Eb Fra Rih
5 113.21 15/14, 16/15 A1, ^m2 aug 1sn, upminor 2nd D#, ^Eb Fru Ra
6 135.85 13/12, 14/13, 27/25 ^^m2 dupminor 2nd ^^Eb Fri Ru
7 158.49 11/10, 12/11, 35/32, 57/52, 800/729 vvM2 dudmajor 2nd vvE Re Ruh
8 181.13 10/9 vM2 downmajor 2nd vE Ro Reh
9 203.77 9/8 M2 major 2nd E Ra Re
10 226.42 8/7, 256/225 ^M2 upmajor 2nd ^E Ru Ri
11 249.06 15/13, 22/19, 125/108, 144/125 ^^M2, vvm3 dupmajor 2nd, dudminor 3rd ^^E, vvF Ri / Ne Raw
12 271.70 7/6, 75/64 vm3 downminor 3rd vF No Ma
13 294.34 13/11, 19/16, 32/27 m3 minor 3rd F Na Meh
14 316.98 6/5 ^m3 upminor 3rd ^F Nu Me
15 339.62 11/9, 243/200 ^^m3 dupminor 3rd ^^F Ni Mu
16 362.26 16/13, 100/81 vvM3 dudmajor 3rd vvF# Me Muh
17 384.91 5/4 vM3 downmajor 3rd vF# Mo Mi
18 407.55 19/15, 24/19, 81/64 M3 major 3rd F# Ma Maa
19 430.19 9/7, 14/11 ^M3 upmajor 3rd ^F# Mu Mo
20 452.83 13/10, 125/96, 162/125 ^^M3, vv4 dupmajor 3rd, dud 4th ^^F#, vvG Mi / Fe Maw
21 475.47 21/16, 25/19, 320/243, 675/512 v4 down 4th vG Fo Fe
22 498.11 4/3 P4 perfect 4th G Fa Fa
23 520.75 19/14, 27/20 ^4 up 4th ^G Fu Fih
24 543.40 11/8, 15/11, 26/19 ^^4 dup 4th ^^G Fi / She Fu
25 566.04 18/13, 25/18 vvA4, vd5 dudaug 4th, downdim 5th vvG#, vAb Pe / Sho Fuh
26 588.68 7/5, 45/32 vA4, d5 downaug 4th, dim 5th vG#, Ab Po / Sha Fi
27 611.32 10/7, 64/45 A4, ^d5 aug 4th, updim 5th G#, ^Ab Pa / Shu Se
28 633.96 13/9, 36/25 ^A4, ^^d5 upaug 4th, dupdim 5th ^G#, ^^Ab Pu / Shi Suh
29 656.60 16/11, 19/13, 22/15 vv5 dud 5th vvA Pi / Se Su
30 679.25 28/19, 40/27 v5 down 5th vA So Sih
31 701.89 3/2 P5 perfect 5th A Sa Sol
32 724.53 32/21, 38/25, 243/160, 1024/675 ^5 up 5th ^A Su Si
33 747.17 20/13, 125/81, 192/125 ^^5, vvm6 dup 5th, dudminor 6th ^^A, vvBb Si / Fle Saw
34 769.81 11/7, 14/9, 25/16 vm6 downminor 6th vBb Flo Lo
35 792.45 19/12, 30/19, 128/81 m6 minor 6th Bb Fla Leh
36 815.09 8/5 ^m6 upminor 6th ^Bb Flu Le
37 837.74 13/8, 81/50 ^^m6 dupminor 6th ^^Bb Fli Lu
38 860.38 18/11, 400/243 vvM6 dudmajor 6th vvB Le Luh
39 883.02 5/3 vM6 downmajor 6th vB Lo La
40 905.66 22/13, 27/16, 32/19 M6 major 6th B La Laa
41 928.30 12/7 ^M6 upmajor 6th ^B Lu Li
42 950.94 19/11, 26/15, 125/72, 216/125 ^^M6, vvm7 dupmajor 6th, dudminor 7th ^^B, vvC Li / The Law
43 973.58 7/4 vm7 downminor 7th vC Tho Ta
44 996.23 16/9 m7 minor 7th C Tha Teh
45 1018.87 9/5 ^m7 upminor 7th ^C Thu Te
46 1041.51 11/6, 20/11, 64/35, 729/400 ^^m7 dupminor 7th ^^C Thi Tu
47 1064.15 13/7, 24/13, 50/27 vvM7 dudmajor 7th vvC# Te Tuh
48 1086.79 15/8 vM7 downmajor 7th vC# To Ti
49 1109.43 19/10, 36/19, 40/21, 243/128 M7 major 7th C# Ta Tih
50 1132.08 21/11, 25/13, 27/14, 52/27, 48/25 ^M7 upmajor 7th ^C# Tu To
51 1154.72 35/18, 64/33, 96/49, 125/64 ^^M7, vv8 dupmajor 7th, dud 8ve ^^C#, vvD Ti / De Taw
52 1177.36 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

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats with arrows:

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

Kite's ups and downs notation

53edo can also be notated with Kite's 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
Sharp symbol   
 
  
    
   
  
   
    
    
   
  
   
Flat symbol
 
  
    
   
  
   
    
    
   
  
   

Sagittal notation

53edo can be notated in Sagittal using the Spartan set, with the apotome equal to 5 edosteps and the limma to 4 edosteps. Here is a simplified table:

Steps 0 1 2 3 4 5
Symbol Evo
Revo

The following enharmonics from the Spartan set are present (comma tempered out):

See apotome complements for equivalent accidental pairs.

Featured below is the 53edo gamut notated using the best accidental approximants; in this case, pai/pao and phai/phao.

Evo flavor

53-EDO_Evo_Sagittal.svg

Revo flavor

53-EDO_Revo_Sagittal.svg

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 incredibly accurate (next edo with a more accurate fifth is 200edo), 53edo also offers a great approximation to Pythagorean tuning. 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 Minisma

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 Demibuzzard / submajor / interpental
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
Frédéric Chopin
George Frideric Handel
Scott Joplin
  • Maple Leaf Rag (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024 version; 2025 version))
  • Maple Leaf Rag (1899) – with syntonic comma adjustment, arranged for harpsichord and rendered by Claudi Meneghin (2024 version; (2025 version)
Shirō Sagisu

21st century

ALLY195
Alxeusxiao
Bryan Deister
Francium
Andrew Heathwaite
Hideya
Nathan Ho
Aaron Krister Johnson (site[dead link])
Logan02A4
Claudi Meneghin
Merct
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
The Evil Doings Of An Intergalactic Skeleton
Chris Vaisvil
  • The Fallen of Kleismic15 (2013) – blog | play – in Kleismic[15], 53edo tuning
Valeriana of the Night
VitaminCD
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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