53edo: Difference between revisions

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 a tuning of other temperaments ===

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.  

 

 

{| class="wikitable center-all right-2 left-3 left-5"

| 1

| 3

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

|-

| 9

| 203.77

| 10

| 226.42

| [[8/7]], [[256/225]]

| upmajor 2nd

| ^E

| Ru

| 11

| 249.06

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

| ^^M2, vvm3

| dupmajor 2nd, dudminor 3rd

| ^^E, vvF

| 12

| 271.70

| [[7/6]], [[75/64]]

| downminor 3rd

| vF

| No

| 13

| [[13/11]], [[19/16]], [[32/27]]

| m3

| minor 3rd

| Na

| 14

| 316.98

| [[6/5]]

| Me

| [[11/9]], [[243/200]]

| 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:

 

{| class="wikitable center-all"

! [[Kite's color notation|Color]]

| downminor

| zo

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

| 7/6, 7/4

| minor

| fourthward wa

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

| 32/27, 16/9

| upminor

| gu

| {{nowrap|(a, b, −1)}}

| 6/5, 9/5

| dupminor

| ilo

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

| 11/9, 11/6

| dudmajor

| lu

| {{nowrap|(a, b, 0, 0, −1)}}

| 12/11, 18/11

| downmajor

| yo

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

| 5/4, 5/3

| major

| fifthward wa

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

| 9/8, 27/16

| upmajor

| ru

| {{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 {{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:

{| class="wikitable center-all"

! [[Kite's color notation|Color of the 3rd]]

| 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

=== 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>

| {{sagittal|\\!}}{{sagittal|#}}

| {{sagittal|\!}}{{sagittal|#}}

* {{sagittal|/|}} = {{sagittal||)}} ([[225/224]])

* {{sagittal||(}} = {{sagittal||//|}} ([[5120/5103]])

 

 

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

 

{{Sagittal chart|Evo}}

 

 

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.

 

== 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 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.

 

{| class="wikitable center-all"

| Perfect fifth

| 3/2

| 31

| −0.07 cents

| Major third

| 5/4

| 17

| −1.40 cents

| Minor third

| 6/5

| 14

| +1.34 cents

| rowspan="2" | 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.

 

 

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.

 

 

 

{| class="wikitable center-4 center-5 center-6"

| 2.3

| 2.3.5

| 15625/15552, 32805/32768

| {{Mapping| 53 84 123 }}

| +0.216

| 0.276

| 1.22

| 2.3.5.7

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

| {{Mapping| 53 84 123 149 }}

| −0.262

| 0.861

| 3.81

| 2.3.5.7.11

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

| {{Mapping| 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

| {{Mapping| 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

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

| +0.391

| 1.105

| 4.88

Commas that 53edo tempers out using its patent val, {{val| 53 84 123 149 183 196 225 }}, include:

 

| [[225/224]]

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

| [[4375/4374]]

| 0.40