53edo: Difference between revisions

| ja = 53平均律

{{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 [[Riemann zeta function #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=11}}

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

 

 

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]]

| dudaug 1sn, downminor 2nd

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

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

| ^^m2

| dupminor 2nd

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

| vvM2

| dudmajor 2nd

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

| ^^M2, vvm3

| dupmajor 2nd, dudminor 3rd

| ^^E, vvF

| ^^m3

| dupminor 3rd

| vvM3

| dudmajor 3rd

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

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

| ^^4

| dup 4th

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

| vvA4, vd5

| dudaug 4th, downdim 5th

| vvG#, vAb

| vA4, d5

| downaug 4th, dim 5th

| vG#, Ab

| A4, ^d5

| aug 4th, updim 5th

| G#, ^Ab

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

| ^A4, ^^d5

| upaug 4th, dupdim 5th

| ^G#, ^^Ab

| vv5

| dud 5th

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

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

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

| ^^m6

| dupminor 6th

| vvM6

| dudmajor 6th

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

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

| ^^m7

| dupminor 7th

| vvM7

| dudmajor 7th

| ''[[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)}}

| dupminor

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

| dudmajor

| {{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

| C^^m

| C dupminor

| 0–16–31

| Cvv

| C dudmajor or C dud

| 0–17–31

| 0–19–31

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

 

 

=== 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;"

|-

! 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>

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]].  

 

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.

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.

{{Q-odd-limit intervals|53}}

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.

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.)

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]].

! rowspan="2" | [[Comma list]]

! rowspan="2" | Optimal<br>8ve stretch (¢)

! colspan="2" | Tuning error

| {{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 }}

* 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.  

! [[Harmonic limit|Prime<br>limit]]

! [[Ratio]]<ref group="note">{{rd}}</ref>

| {{Monzo| -84 53 }}

| {{Monzo| -21 3 7 }}

| 10.06

| {{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 }}

| 7.71

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

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

| 17.58

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

| {{Monzo| 4 0 -2 -1 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

* [[Schismic–Mercator equivalence continuum]]

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Generator*

! Cents*

! Associated<br>ratio*

| 45.3

| 113.2

| 158.5

| 203.8

| 226.4

| 249.1

| 271.7

| [[Orwell]]

| 294.3

| 317.0

| 339.6

| 362.3

| [[Demibuzzard]] / submajor / interpental

| 407.5

| 430.2

| 475.5

| 498.1

| [[Garibaldi]] / [[pontiac]]

| 1

| 520.8

| [[Mavila]] (53bbcc)

| 566.0

| [[Alphatrimot]]

| 588.7

* The [[eagle 53]] scale described by [[John O'Sullivan]]

 

* 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 ===

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

 

* [[Lumatone mapping for 53edo]]

* [[Skip fretting system 53 3 14]]

* [[Skip fretting system 53 3 17]]

* "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

* ''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

== Notes ==