53edo: Difference between revisions

(40 intermediate revisions by 14 users not shown)

Line 7:

== 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]] (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 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]].

As an equal temperament, it notably [[tempering out|tempers out]] [[Mercator's comma]] (353/284), 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.

Line 537:

| downminor

| zo

| (a, b, 0, 1)

| {{nowrap|(a, b, 0, 1)}}

| 7/6, 7/4

|-

| minor

| fourthward wa

| (a, b) with b < -1

| {{nowrap|(a, b)}} with {{nowrap|b < −1}}

| 32/27, 16/9

|-

| upminor

| gu

| (a, b, -1)

| {{nowrap|(a, b, −1)}}

| 6/5, 9/5

|-

| dupminor

| ilo

| (a, b, 0, 0, 1)

| {{nowrap|(a, b, 0, 0, 1)}}

| 11/9, 11/6

|-

| dudmajor

| lu

| (a, b, 0, 0, -1)

| {{nowrap|(a, b, 0, 0, −1)}}

| 12/11, 18/11

|-

| downmajor

| yo

| (a, b, 1)

| {{nowrap|(a, b, 1)}}

| 5/4, 5/3

|-

| major

| fifthward wa

| (a, b) with b > 1

| {{nowrap|(a, b)}} with {{nowrap|b > 1}}

| 9/8, 27/16

|-

| upmajor

| ru

| (a, b, 0, -1)

| {{nowrap|(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.

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:

Line 589:

| zo

| 6:7:9

| ~~0-12-31~~

| 0–12–31

| C vEb G

| Cvm

Line 596:

| gu

| 10:12:15

| ~~0-14-31~~

| 0–14–31

| C ^Eb G

| C^m

Line 603:

| ilo

| 18:22:27

| ~~0-15-31~~

| 0–15–31

| C ^^Eb G

| C^^m

Line 610:

| lu

| 22:27:33

| ~~0-16-31~~

| 0–16–31

| C vvE G

| Cvv

Line 617:

| yo

| 4:5:6

| ~~0-17-31~~

| 0–17–31

| C vE G

| Cv

Line 624:

| ru

| 14:18:21

| ~~0-19-31~~

| 0–19–31

| C ^E G

| C^

Line 633:

== Notation ==

=== Ups and downs 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~~:

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).

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]]:

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.

=== Sagittal notation ===

Line 665:

Line 666:

== 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.

Line 674:

Line 675:

== Approximation to JI ==

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

Line 721:

Line 720:

=== 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 8 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.

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 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.

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 ~~perfect~~ off.)

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=5|start=19|title=Approximation of large prime harmonics in 53edo}}

{{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''n'') 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]].

~~=== Zeta peak index ===~~

~~{| class="wikitable center-all"~~

|-

~~! colspan="3" | Tuning~~

~~! colspan="3" | Strength~~

~~! colspan="2" | Closest edo~~

~~! colspan="2" | Integer limit~~

|-

~~! ZPI~~

~~! Steps per octave~~

~~! Step size (cents)~~

~~! Height~~

~~! Integral~~

~~! Gap~~

~~! Edo~~

~~! Octave (cents)~~

~~! Consistent~~

~~! Distinct~~

|-

~~| [[257zpi]]~~

~~| 52.9968291550147~~

~~| 22.6428640945673~~

~~| 8.249774~~

~~| 1.486620~~

~~| 18.069918~~

~~| 53edo~~

~~| 1200.07179701207~~

~~| 10~~

|}

== Regular temperament properties ==

Line 774:

Line 742:

|-

| 2.3

| {{~~monzo~~| -84 53 }}

| {{Monzo| -84 53 }}

| {{~~mapping~~| 53 84 }}

| {{Mapping| 53 84 }}

| +0.022

| 0.022

Line 782:

Line 750:

| 2.3.5

| 15625/15552, 32805/32768

| {{~~mapping~~| 53 84 123 }}

| {{Mapping| 53 84 123 }}

| +0.216

| 0.276

Line 789:

Line 757:

| 2.3.5.7

| 225/224, 1728/1715, 3125/3087

| {{~~mapping~~| 53 84 123 149 }}

| {{Mapping| 53 84 123 149 }}

| −0.262

| 0.861

Line 796:

Line 764:

| 2.3.5.7.11

| 99/98, 121/120, 176/175, 2200/2187

| {{~~mapping~~| 53 84 123 149 183 }}

| {{Mapping| 53 84 123 149 183 }}

| +0.248

| 1.279

Line 803:

Line 771:

| 2.3.5.7.11.13

| 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

| 1.183

Line 810:

Line 778:

| 2.3.5.7.11.13.19

| 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

| 1.105

Line 831:

Line 799:

| 3

| <abbr title="19383245667680019896796723/19342813113834066795298816">(52 digits)</abbr>

| {{~~monzo~~| -84 53 }}

| {{Monzo| -84 53 }}

| 3.62

| Tribilawa

Line 838:

Line 806:

| 5

| [[2109375/2097152|(14 digits)]]

| {{~~monzo~~| -21 3 7 }}

| {{Monzo| -21 3 7 }}

| 10.06

| Lasepyo

Line 845:

Line 813:

| 5

| [[15625/15552]]

| {{~~monzo~~| -6 -5 6 }}

| {{Monzo| -6 -5 6 }}

| 8.11

| Tribiyo

Line 852:

Line 820:

| 5

| <abbr title="1600000/1594323">(14 digits)</abbr>

| {{~~monzo~~| 9 -13 5 }}

| {{Monzo| 9 -13 5 }}

| 6.15

| Saquinyo

Line 859:

Line 827:

| 5

| <abbr title="10485760000/10460353203">(22 digits)</abbr>

| {{~~monzo~~| 24 -21 4 }}

| {{Monzo| 24 -21 4 }}

| 4.20

| Sasaquadyo

Line 866:

Line 834:

| 5

| [[32805/32768]]

| {{~~monzo~~| -15 8 1 }}

| {{Monzo| -15 8 1 }}

| 1.95

| Layo

Line 873:

Line 841:

| 7

| [[3125/3087]]

| {{~~monzo~~| 0 -2 5 -3 }}

| {{Monzo| 0 -2 5 -3 }}

| 21.18

| Triru-aquinyo

Line 880:

Line 848:

| 7

| [[1728/1715]]

| {{~~monzo~~| 6 3 -1 -3 }}

| {{Monzo| 6 3 -1 -3 }}

| 13.07

| Triru-agu

Line 887:

Line 855:

| 7

| [[225/224]]

| {{~~monzo~~| -5 2 2 -1 }}

| {{Monzo| -5 2 2 -1 }}

| 7.71

| Ruyoyo

Line 894:

Line 862:

| 7

| [[4375/4374]]

| {{~~monzo~~| -1 -7 4 1 }}

| {{Monzo| -1 -7 4 1 }}

| 0.40

| Zoquadyo

Line 901:

Line 869:

| 11

| [[99/98]]

| {{~~monzo~~| -1 2 0 -2 1 }}

| {{Monzo| -1 2 0 -2 1 }}

| 17.58

| Loruru

Line 908:

Line 876:

| 11

| [[121/120]]

| {{~~monzo~~| -3 -1 -1 0 2 }}

| {{Monzo| -3 -1 -1 0 2 }}

| 14.37

| Lologu

Line 915:

Line 883:

| 11

| [[176/175]]

| {{~~monzo~~| 4 0 -2 -1 1 }}

| {{Monzo| 4 0 -2 -1 1 }}

| 9.86

| Lorugugu

Line 922:

Line 890:

| 11

| <abbr title="94489280512/94143178827">(22 digits)</abbr>

| {{~~monzo~~| 33 -23 0 0 1 }}

| {{Monzo| 33 -23 0 0 1 }}

| 6.35

| Trisalo

Line 929:

Line 897:

| 11

| [[385/384]]

| {{~~monzo~~| -7 -1 1 1 1 }}

| {{Monzo| -7 -1 1 1 1 }}

| 4.50

| Lozoyo

Line 936:

Line 904:

| 11

| [[540/539]]

| {{~~monzo~~| 2 3 1 -2 -1 }}

| {{Monzo| 2 3 1 -2 -1 }}

| 3.21

| Lururuyo

Line 943:

Line 911:

| 13

| [[275/273]]

| {{~~monzo~~| 0 -1 2 -1 1 -1 }}

| {{Monzo| 0 -1 2 -1 1 -1 }}

| 12.64

| Thuloruyoyo

Line 950:

Line 918:

| 13

| [[169/168]]

| {{~~monzo~~| -3 -1 0 -1 0 2 }}

| {{Monzo| -3 -1 0 -1 0 2 }}

| 10.27

| Thothoru

Line 957:

Line 925:

| 13

| [[625/624]]

| {{~~monzo~~| -4 -1 4 0 0 -1 }}

| {{Monzo| -4 -1 4 0 0 -1 }}

| 2.77

| Thuquadyo

Line 964:

Line 932:

| 13

| [[676/675]]

| {{~~monzo~~| 2 -3 -2 0 0 2 }}

| {{Monzo| 2 -3 -2 0 0 2 }}

| 2.56

| Bithogu

Line 971:

Line 939:

| 13

| [[1001/1000]]

| {{~~monzo~~| -3 0 -3 1 1 1 }}

| {{Monzo| -3 0 -3 1 1 1 }}

| 1.73

| Tholozotrigu

Line 978:

Line 946:

| 13

| [[2080/2079]]

| {{~~monzo~~| 5 -3 1 -1 -1 1 }}

| {{Monzo| 5 -3 1 -1 -1 1 }}

| 0.83

| Tholuruyo

Line 985:

Line 953:

| 13

| [[4096/4095]]

| {{~~monzo~~| 12 -2 -1 -1 0 -1 }}

| {{Monzo| 12 -2 -1 -1 0 -1 }}

| 0.42

| Sathurugu

Line 1,015:

Line 983:

| 16/15

| [[Misneb]]

|-

| 1

| 6\53

| 135.8

| [[13/12]]~[[14/13]]

| [[Doublethink]]

|-

| 1

Line 1,110:

Line 1,084:

| 566.0

| 18/13

| [[~~Tricot~~]]

| [[Alphatrimot]]

|-

| 1

Line 1,121:

Line 1,095:

== 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.

* [[List of MOS scales in 53edo]]

Line 1,130:

Line 1,104:

* 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 [[AFDO]]s

* Composite Cliffedge (approximated from [[60afdo]]): 12 10 9 19 3

* Composite Deja Vu (approximated from [[101afdo]]): 14 17 5 9 8

Line 1,147:

Line 1,124:

=== Other scales ===

* [[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 ==

Line 1,158:

Line 1,140:

** [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=vcinR7nUthA "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)

Line 1,177:

Line 1,161:

=== 21st century ===

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/r-Tzq33OGM4 ''microtonal improvisation in 53edo''] (2025)

; [[Francium]]

* [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] – 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]]

Line 1,199:

Line 1,190:

; [[Mundoworld]]

* ~~"No Explanations"~~ from ''No Fun House'' (2025) – [https://mundoworld.bandcamp.com/track/no-explanations Bandcamp] | [https://www.youtube.com/watch?v=WPlxi22rf0I YouTube] – in Gorgo[11], 53edo tuning

* 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]]

Line 1,218:

Line 1,211:

; [[Cam Taylor]]

* [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]]

Line 1,231:

Line 1,229:

* "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

~~== Instruments ==~~

* [[Lumatone mapping for 53edo]]

* [[Skip fretting system 53 3 14]]

== Notes ==

Line 1,242:

Line 1,236:

[[Category:3-limit record edos|##]]

[[Category:Amity]]

~~[[Category:Hanson]]~~

[[Category:Kleismic]]

[[Category:Island]]

Line 1,250:

Line 1,244:

[[Category:Orwell]]

[[Category:Schismic]]

~~[[Category:3-limit]]~~

[[Category:Listen]]

@@ Line 7: / Line 7: @@
 {{Infobox ET}}
 {{Wikipedia| 53 equal temperament }}
-{{EDO intro|53}}
+{{ED intro}}
 == Theory ==
-edo 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]] (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]].
+edo 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]].
 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.
@@ Line 537: / Line 537: @@
 | downminor
 | zo
-| (a, b, 0, 1)
+| {{nowrap|(a, b, 0, 1)}}
 | 7/6, 7/4
 |-
 | minor
 | fourthward wa
-| (a, b) with b &lt; -1
+| {{nowrap|(a, b)}} with {{nowrap|b &lt; −1}}
 | 32/27, 16/9
 |-
 | upminor
 | gu
-| (a, b, -1)
+| {{nowrap|(a, b, −1)}}
 | 6/5, 9/5
 |-
 | dupminor
 | ilo
-| (a, b, 0, 0, 1)
+| {{nowrap|(a, b, 0, 0, 1)}}
 | 11/9, 11/6
 |-
 | dudmajor
 | lu
-| (a, b, 0, 0, -1)
+| {{nowrap|(a, b, 0, 0, −1)}}
 | 12/11, 18/11
 |-
 | downmajor
 | yo
-| (a, b, 1)
+| {{nowrap|(a, b, 1)}}
 | 5/4, 5/3
 |-
 | major
 | fifthward wa
-| (a, b) with b &gt; 1
+| {{nowrap|(a, b)}} with {{nowrap|b &gt; 1}}
 | 9/8, 27/16
 |-
 | upmajor
 | ru
-| (a, b, 0, -1)
+| {{nowrap|(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.
+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:
@@ Line 589: / Line 589: @@
 | zo
 | 6:7:9
-| 0-12-31
+| 0–12–31
 | C vEb G
 | Cvm
@@ Line 596: / Line 596: @@
 | gu
 | 10:12:15
-| 0-14-31
+| 0–14–31
 | C ^Eb G
 | C^m
@@ Line 603: / Line 603: @@
 | ilo
 | 18:22:27
-| 0-15-31
+| 0–15–31
 | C ^^Eb G
 | C^^m
@@ Line 610: / Line 610: @@
 | lu
 | 22:27:33
-| 0-16-31
+| 0–16–31
 | C vvE G
 | Cvv
@@ Line 617: / Line 617: @@
 | yo
 | 4:5:6
-| 0-17-31
+| 0–17–31
 | C vE G
 | Cv
@@ Line 624: / Line 624: @@
 | ru
 | 14:18:21
-| 0-19-31
+| 0–19–31
 | C ^E G
 | C^
@@ Line 633: / Line 633: @@
 == Notation ==
 === Ups and downs 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:
+edo 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}}
-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.
 === Sagittal notation ===
@@ Line 665: / Line 666: @@
 == Relationship to 12edo ==
-edo'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.
+edo'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.
@@ Line 674: / Line 675: @@
 == Approximation to JI ==
-[[File:53ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 53edo]]
 edo provides excellent approximations for the classic 5-limit [[just]] chords and scales, such as the Ptolemy–Zarlino "just major" scale.
@@ Line 721: / Line 720: @@
 === Higher-limit JI ===
-edo 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 8 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.
+edo 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.
-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 perfect off.)
+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=5|start=19|title=Approximation of large prime harmonics in 53edo}}
+{{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]].
-=== Zeta peak index ===
-{| class="wikitable center-all"
-|-
-! colspan="3" | Tuning
-! colspan="3" | Strength
-! colspan="2" | Closest edo
-! colspan="2" | Integer limit
-|-
-! ZPI
-! Steps per octave
-! Step size (cents)
-! Height
-! Integral
-! Gap
-! Edo
-! Octave (cents)
-! Consistent
-! Distinct
-|-
-| [[257zpi]]
-| 52.9968291550147
-| 22.6428640945673
-| 8.249774
-| 1.486620
-| 18.069918
-| 53edo
-| 1200.07179701207
-| 10
-| 10
-|}
 == Regular temperament properties ==
@@ Line 774: / Line 742: @@
 |-
 | 2.3
-| {{monzo| -84 53 }}
+| {{Monzo| -84 53 }}
-| {{mapping| 53 84 }}
+| {{Mapping| 53 84 }}
 | +0.022
 | 0.022
@@ Line 782: / Line 750: @@
 | 2.3.5
 | 15625/15552, 32805/32768
-| {{mapping| 53 84 123 }}
+| {{Mapping| 53 84 123 }}
 | +0.216
 | 0.276
@@ Line 789: / Line 757: @@
 | 2.3.5.7
 | 225/224, 1728/1715, 3125/3087
-| {{mapping| 53 84 123 149 }}
+| {{Mapping| 53 84 123 149 }}
 | −0.262
 | 0.861
@@ Line 796: / Line 764: @@
 | 2.3.5.7.11
 | 99/98, 121/120, 176/175, 2200/2187
-| {{mapping| 53 84 123 149 183 }}
+| {{Mapping| 53 84 123 149 183 }}
 | +0.248
 | 1.279
@@ Line 803: / Line 771: @@
 | 2.3.5.7.11.13
 | 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
 | 1.183
@@ Line 810: / Line 778: @@
 | 2.3.5.7.11.13.19
 | 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
 | 1.105
@@ Line 831: / Line 799: @@
 | 3
 | <abbr title="19383245667680019896796723/19342813113834066795298816">(52 digits)</abbr>
-| {{monzo| -84 53 }}
+| {{Monzo| -84 53 }}
 | 3.62
 | Tribilawa
@@ Line 838: / Line 806: @@
 | 5
 | [[2109375/2097152|(14 digits)]]
-| {{monzo| -21 3 7 }}
+| {{Monzo| -21 3 7 }}
 | 10.06
 | Lasepyo
@@ Line 845: / Line 813: @@
 | 5
 | [[15625/15552]]
-| {{monzo| -6 -5 6 }}
+| {{Monzo| -6 -5 6 }}
 | 8.11
 | Tribiyo
@@ Line 852: / Line 820: @@
 | 5
 | <abbr title="1600000/1594323">(14 digits)</abbr>
-| {{monzo| 9 -13 5 }}
+| {{Monzo| 9 -13 5 }}
 | 6.15
 | Saquinyo
@@ Line 859: / Line 827: @@
 | 5
 | <abbr title="10485760000/10460353203">(22 digits)</abbr>
-| {{monzo| 24 -21 4 }}
+| {{Monzo| 24 -21 4 }}
 | 4.20
 | Sasaquadyo
@@ Line 866: / Line 834: @@
 | 5
 | [[32805/32768]]
-| {{monzo| -15 8 1 }}
+| {{Monzo| -15 8 1 }}
 | 1.95
 | Layo
@@ Line 873: / Line 841: @@
 | 7
 | [[3125/3087]]
-| {{monzo| 0 -2 5 -3 }}
+| {{Monzo| 0 -2 5 -3 }}
 | 21.18
 | Triru-aquinyo
@@ Line 880: / Line 848: @@
 | 7
 | [[1728/1715]]
-| {{monzo| 6 3 -1 -3 }}
+| {{Monzo| 6 3 -1 -3 }}
 | 13.07
 | Triru-agu
@@ Line 887: / Line 855: @@
 | 7
 | [[225/224]]
-| {{monzo| -5 2 2 -1 }}
+| {{Monzo| -5 2 2 -1 }}
 | 7.71
 | Ruyoyo
@@ Line 894: / Line 862: @@
 | 7
 | [[4375/4374]]
-| {{monzo| -1 -7 4 1 }}
+| {{Monzo| -1 -7 4 1 }}
 | 0.40
 | Zoquadyo
@@ Line 901: / Line 869: @@
 | 11
 | [[99/98]]
-| {{monzo| -1 2 0 -2 1 }}
+| {{Monzo| -1 2 0 -2 1 }}
 | 17.58
 | Loruru
@@ Line 908: / Line 876: @@
 | 11
 | [[121/120]]
-| {{monzo| -3 -1 -1 0 2 }}
+| {{Monzo| -3 -1 -1 0 2 }}
 | 14.37
 | Lologu
@@ Line 915: / Line 883: @@
 | 11
 | [[176/175]]
-| {{monzo| 4 0 -2 -1 1 }}
+| {{Monzo| 4 0 -2 -1 1 }}
 | 9.86
 | Lorugugu
@@ Line 922: / Line 890: @@
 | 11
 | <abbr title="94489280512/94143178827">(22 digits)</abbr>
-| {{monzo| 33 -23 0 0 1 }}
+| {{Monzo| 33 -23 0 0 1 }}
 | 6.35
 | Trisalo
@@ Line 929: / Line 897: @@
 | 11
 | [[385/384]]
-| {{monzo| -7 -1 1 1 1 }}
+| {{Monzo| -7 -1 1 1 1 }}
 | 4.50
 | Lozoyo
@@ Line 936: / Line 904: @@
 | 11
 | [[540/539]]
-| {{monzo| 2 3 1 -2 -1 }}
+| {{Monzo| 2 3 1 -2 -1 }}
 | 3.21
 | Lururuyo
@@ Line 943: / Line 911: @@
 | 13
 | [[275/273]]
-| {{monzo| 0 -1 2 -1 1 -1 }}
+| {{Monzo| 0 -1 2 -1 1 -1 }}
 | 12.64
 | Thuloruyoyo
@@ Line 950: / Line 918: @@
 | 13
 | [[169/168]]
-| {{monzo| -3 -1 0 -1 0 2 }}
+| {{Monzo| -3 -1 0 -1 0 2 }}
 | 10.27
 | Thothoru
@@ Line 957: / Line 925: @@
 | 13
 | [[625/624]]
-| {{monzo| -4 -1 4 0 0 -1 }}
+| {{Monzo| -4 -1 4 0 0 -1 }}
 | 2.77
 | Thuquadyo
@@ Line 964: / Line 932: @@
 | 13
 | [[676/675]]
-| {{monzo| 2 -3 -2 0 0 2 }}
+| {{Monzo| 2 -3 -2 0 0 2 }}
 | 2.56
 | Bithogu
@@ Line 971: / Line 939: @@
 | 13
 | [[1001/1000]]
-| {{monzo| -3 0 -3 1 1 1 }}
+| {{Monzo| -3 0 -3 1 1 1 }}
 | 1.73
 | Tholozotrigu
@@ Line 978: / Line 946: @@
 | 13
 | [[2080/2079]]
-| {{monzo| 5 -3 1 -1 -1 1 }}
+| {{Monzo| 5 -3 1 -1 -1 1 }}
 | 0.83
 | Tholuruyo
@@ Line 985: / Line 953: @@
 | 13
 | [[4096/4095]]
-| {{monzo| 12 -2 -1 -1 0 -1 }}
+| {{Monzo| 12 -2 -1 -1 0 -1 }}
 | 0.42
 | Sathurugu
@@ Line 1,015: / Line 983: @@
 | 16/15
 | [[Misneb]]
+|-
+| 1
+| 6\53
+| 135.8
+| [[13/12]]~[[14/13]]
+| [[Doublethink]]
 |-
 | 1
@@ Line 1,110: / Line 1,084: @@
 | 566.0
 | 18/13
-| [[Tricot]]
+| [[Alphatrimot]]
 |-
 | 1
@@ Line 1,121: / Line 1,095: @@
 == 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.
 * [[List of MOS scales in 53edo]]
@@ Line 1,130: / Line 1,104: @@
 * 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 [[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
@@ Line 1,147: / Line 1,124: @@
 === Other scales ===
 * [[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 ==
@@ Line 1,158: / Line 1,140: @@
 ** [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=vcinR7nUthA "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
@@ Line 1,177: / Line 1,161: @@
 === 21st century ===
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/r-Tzq33OGM4 ''microtonal improvisation in 53edo''] (2025)
 ; [[Francium]]
 * [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] – 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]]
@@ Line 1,199: / Line 1,190: @@
 ; [[Mundoworld]]
-* "No Explanations" from ''No Fun House'' (2025) – [https://mundoworld.bandcamp.com/track/no-explanations Bandcamp] | [https://www.youtube.com/watch?v=WPlxi22rf0I YouTube] – in Gorgo[11], 53edo tuning
+* 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]]
@@ Line 1,218: / Line 1,211: @@
 ; [[Cam Taylor]]
 * [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]]
@@ Line 1,231: / Line 1,229: @@
 * "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
-== Instruments ==
-* [[Lumatone mapping for 53edo]]
-* [[Skip fretting system 53 3 14]]
 == Notes ==
@@ Line 1,242: / Line 1,236: @@
 <references/>
+[[Category:3-limit record edos|##]] <!-- 2-digit number -->
 [[Category:Amity]]
-[[Category:Hanson]]
 [[Category:Kleismic]]
 [[Category:Island]]
@@ Line 1,250: / Line 1,244: @@
 [[Category:Orwell]]
 [[Category:Schismic]]
-[[Category:3-limit]]
 [[Category:Listen]]