53edo: Difference between revisions

{{ED intro}}

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.  

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'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.

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

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

As an equal temperament, 53et 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.  

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.

! #

! 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]]: v⁵A1 and ^d2)

| 0.0

| ''[[50/49]]'', [[64/63]], [[81/80]]

| [[33/32]], [[36/35]], [[49/48]], [[128/125]]

| ''[[22/21]]'', [[25/24]], [[26/25]], [[27/26]], [[28/27]]

| [[15/14]], [[16/15]]

| [[13/12]], [[14/13]], [[27/25]]

| [[11/10]], [[12/11]], [[35/32]], [[57/52]], [[800/729]]

| [[15/13]], [[22/19]], [[125/108]], [[144/125]]

| [[21/16]], [[25/19]], [[320/243]], [[675/512]]

| [[19/14]], [[27/20]]

| [[18/13]], [[25/18]]

| [[13/9]], [[36/25]]

| [[28/19]], [[40/27]]

| [[20/13]], [[125/81]], [[192/125]]

| ''[[11/7]]'', [[14/9]], [[25/16]]

| [[19/11]], [[26/15]], [[125/72]], [[216/125]]

| [[11/6]], [[20/11]], [[64/35]], [[729/400]]

| ''[[21/11]]'', [[25/13]], [[27/14]], [[52/27]], [[48/25]]

| [[35/18]], [[64/33]], [[96/49]], [[125/64]]

| ''[[49/25]]'', [[63/32]], [[160/81]]

| 1200.0

! Monzo format

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

| {{nowrap|(a, b)}} with {{nowrap|b &lt; −1}}

| {{nowrap|(a, b, −1)}}

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

| {{nowrap|(a, b, 0, 0, −1)}}

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

| {{nowrap|(a, b)}} with {{nowrap|b &gt; 1}}

| {{nowrap|(a, b, 0, −1)}}

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.

! JI chord

! Notes as edosteps

! Notes of C chord

! Written name

! Spoken name

| 0–12–31

| 0–14–31

| 0–15–31

| 0–16–31

| 0–17–31

| 0–19–31

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

=== Stein–Zimmermann–Gould notation ===

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

{{Sharpness-sharp5-szg}}

 

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).

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.

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]].  

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

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.

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.

 

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.

| {{Monzo| -84 53 }}

| {{Mapping| 53 84 }}

| {{Mapping| 53 84 123 }}

| {{Mapping| 53 84 123 149 }}

| −0.262

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

| {{Mapping| 53 84 123 149 183 196 }}

| {{Mapping| 53 84 123 149 183 196 225 }}

| {{Monzo| -84 53 }}

| {{Monzo| -21 3 7 }}

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

| {{Monzo| 9 -13 5 }}

| {{Monzo| 24 -21 4 }}

| {{Monzo| -15 8 1 }}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

| Minisma

| 45.3

| 113.2

| 158.5

| 203.8

| 226.4

| 249.1

| 271.7

| 294.3

| 317.0

| 339.6

| 362.3

| [[Demibuzzard]] / submajor / interpental

| 407.5

| 430.2

| 475.5

| 498.1

| 520.8

| 566.0

| [[Alphatrimot]]

| 588.7

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave

=== Mos scales ===

* [[cthon5m]]: 6 3 6 2 3 6 2 3 6 3 2 6 3 2

* Palace{{idio}} (approximated from [[Porky]] in [[29edo]]): 7 7 8 9 7 7 8

 

* ''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]))

* ''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)]

==== 21st century ====

 

 

 

* "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

* "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]

* ''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

* ''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

<references/>