53edo: Difference between revisions

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→Intervals: added my solfege. Fixed an obvious mistake in the other solfege (Andrew Heathwaite's?) where both vm6 and ^M6 were named Lo. Changed Lo to Li to match his 41edo solfege.
Line 27: Line 27:


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-3 left-4 left-6"
{| class="wikitable center-all right-2 left-3 left-5"
|-
|-
! #
! #
! Solfege
! Cents
! Cents
! Approximate ratios*
! Approximate ratios*
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Ups and downs notation]]
! colspan="2" |[[53edo solfege|Solfeges]]
|-
|-
| 0
| 0
| do
| 0.00
| 0.00
| [[1/1]]
| [[1/1]]
Line 42: Line 41:
| unison
| unison
| D
| D
|Da
| Do
|-
|-
| 1
| 1
| di
| 22.64
| 22.64
| [[81/80]], [[64/63]], [[50/49]]
| [[81/80]], [[64/63]], [[50/49]]
Line 50: Line 50:
| up unison
| up unison
| ^D
| ^D
|Du
| Di
|-
|-
| 2
| 2
| daw
| 45.28
| 45.28
| [[49/48]], [[36/35]], [[33/32]], [[128/125]]
| [[49/48]], [[36/35]], [[33/32]], [[128/125]]
Line 58: Line 59:
| double-up unison, <br>double-down minor 2nd
| double-up unison, <br>double-down minor 2nd
| ^^D, <br>vvEb
| ^^D, <br>vvEb
|De/Fre
| Daw
|-
|-
| 3
| 3
| ro
| 67.92
| 67.92
| [[25/24]], [[28/27]], [[22/21]], [[27/26]], [[26/25]]
| [[25/24]], [[28/27]], [[22/21]], [[27/26]], [[26/25]]
| vm2
| vvA1,<br>vm2
| downminor 2nd
| double-down aug 1sn,<br>downminor 2nd
| vEb
| vvD#,<br>vEb
|Fro
| Ro
|-
|-
| 4
| 4
| rih
| 90.57
| 90.57
| [[19/18]], [[20/19]], [[21/20]], [[256/243]]
| [[19/18]], [[20/19]], [[21/20]], [[256/243]]
| m2
| vA1, m2
| minor 2nd
| downaug 1sn, minor 2nd
| Eb
| vD#, Eb
|Fra
| Rih
|-
|-
| 5
| 5
| ra
| 113.21
| 113.21
| [[16/15]], [[15/14]]
| [[16/15]], [[15/14]]
| ^m2
| A1, ^m2
| upminor 2nd
| aug 1sn, upminor 2nd
| ^Eb
| D#, ^Eb
|Fru
| Ra
|-
|-
| 6
| 6
| ru
| 135.85
| 135.85
| [[14/13]], [[13/12]], [[27/25]]
| [[14/13]], [[13/12]], [[27/25]]
Line 90: Line 95:
| downmid 2nd
| downmid 2nd
| ^^Eb
| ^^Eb
|Fri
| Ru
|-
|-
| 7
| 7
| ruh
| 158.49
| 158.49
| [[35/32]], [[12/11]], [[11/10]], [[57/52]], [[800/729]]
| [[35/32]], [[12/11]], [[11/10]], [[57/52]], [[800/729]]
Line 98: Line 104:
| upmid 2nd
| upmid 2nd
| vvE
| vvE
|Ri
| Ruh
|-
|-
| 8
| 8
| reh
| 181.13
| 181.13
| [[10/9]]
| [[10/9]]
Line 106: Line 113:
| downmajor 2nd
| downmajor 2nd
| vE
| vE
|Ro
| Reh
|-
|-
| 9
| 9
| re
| 203.77
| 203.77
| [[9/8]]
| [[9/8]]
Line 114: Line 122:
| major 2nd
| major 2nd
| E
| E
|Ra
| Re
|-
|-
| 10
| 10
| ri
| 226.42
| 226.42
| [[8/7]], [[256/225]]
| [[8/7]], [[256/225]]
Line 122: Line 131:
| upmajor 2nd
| upmajor 2nd
| ^E
| ^E
|Ru
| Ri
|-
|-
| 11
| 11
| raw
| 249.06
| 249.06
| [[15/13]], [[144/125]]
| [[15/13]], [[144/125]]
Line 130: Line 140:
| double-up major 2nd, <br>double-down minor 3rd
| double-up major 2nd, <br>double-down minor 3rd
| ^^E, <br>vvF
| ^^E, <br>vvF
|Re / Ne
| Raw
|-
|-
| 12
| 12
| ma
| 271.70
| 271.70
| [[7/6]], [[75/64]]
| [[7/6]], [[75/64]]
Line 138: Line 149:
| downminor 3rd
| downminor 3rd
| vF
| vF
|No
| Ma
|-
|-
| 13
| 13
| meh
| 294.34
| 294.34
| [[13/11]], [[19/16]], [[32/27]]
| [[13/11]], [[19/16]], [[32/27]]
Line 146: Line 158:
| minor 3rd
| minor 3rd
| F
| F
|Na
| Meh
|-
|-
| 14
| 14
| me
| 316.98
| 316.98
| [[6/5]]
| [[6/5]]
Line 154: Line 167:
| upminor 3rd
| upminor 3rd
| ^F
| ^F
|Nu
| Me
|-
|-
| 15
| 15
| mu
| 339.62
| 339.62
| [[11/9]], [[243/200]]
| [[11/9]], [[243/200]]
Line 162: Line 176:
| downmid 3rd
| downmid 3rd
| ^^F
| ^^F
|Ni
| Mu
|-
|-
| 16
| 16
| muh
| 362.26
| 362.26
| [[16/13]], [[100/81]]
| [[16/13]], [[100/81]]
Line 170: Line 185:
| upmid 3rd
| upmid 3rd
| vvF#
| vvF#
|Mi
| Muh
|-
|-
| 17
| 17
| mi
| 384.91
| 384.91
| [[5/4]]
| [[5/4]]
Line 178: Line 194:
| downmajor 3rd
| downmajor 3rd
| vF#
| vF#
|Mo
| Mi
|-
|-
| 18
| 18
| maa
| 407.55
| 407.55
| [[19/15]], [[24/19]], [[81/64]]
| [[19/15]], [[24/19]], [[81/64]]
Line 186: Line 203:
| major 3rd
| major 3rd
| F#
| F#
|Ma
| Maa
|-
|-
| 19
| 19
| mo
| 430.19
| 430.19
| [[9/7]], ''[[14/11]]''
| [[9/7]], ''[[14/11]]''
Line 194: Line 212:
| upmajor 3rd
| upmajor 3rd
| ^F#
| ^F#
|Mu
| Mo
|-
|-
| 20
| 20
| maw
| 452.83
| 452.83
| [[13/10]], [[125/96]]
| [[13/10]], [[125/96]]
Line 202: Line 221:
| double-up major 3rd, <br>double-down 4th
| double-up major 3rd, <br>double-down 4th
| ^^F#, <br>vvG
| ^^F#, <br>vvG
|Me / Fe
| Maw
|-
|-
| 21
| 21
| fe
| 475.47
| 475.47
| [[21/16]], [[25/19]], [[675/512]], [[320/243]]
| [[21/16]], [[25/19]], [[675/512]], [[320/243]]
Line 210: Line 230:
| down 4th
| down 4th
| vG
| vG
|Fo
| Fe
|-
|-
| 22
| 22
| fa
| 498.11
| 498.11
| [[4/3]]
| [[4/3]]
Line 218: Line 239:
| perfect 4th
| perfect 4th
| G
| G
|Fa
| Fa
|-
|-
| 23
| 23
| fih
| 520.75
| 520.75
| [[27/20]]
| [[27/20]]
Line 226: Line 248:
| up 4th
| up 4th
| ^G
| ^G
|Fu
| Fih
|-
|-
| 24
| 24
| fu
| 543.40
| 543.40
| [[11/8]], [[15/11]], [[26/19]]
| [[11/8]], [[15/11]], [[26/19]]
Line 234: Line 257:
| downmid 4th
| downmid 4th
| ^^G
| ^^G
|Fi / She
| Fu
|-
|-
| 25
| 25
| fuh
| 566.04
| 566.04
| [[18/13]]
| [[18/13]]
Line 242: Line 266:
| upmid 4th, <br>downdim 5th
| upmid 4th, <br>downdim 5th
| vvG#, <br>vAb
| vvG#, <br>vAb
|Pi / Sho
| Fuh
|-
|-
| 26
| 26
| fi
| 588.68
| 588.68
| [[7/5]], [[45/32]]
| [[7/5]], [[45/32]]
Line 250: Line 275:
| downaug 4th, <br>dim 5th
| downaug 4th, <br>dim 5th
| vG#, <br>Ab
| vG#, <br>Ab
|Po / Sha
| Fi
|-
|-
| 27
| 27
| se
| 611.32
| 611.32
| [[10/7]], [[64/45]]
| [[10/7]], [[64/45]]
Line 258: Line 284:
| aug 4th, <br>updim 5th
| aug 4th, <br>updim 5th
| G#, <br>^Ab
| G#, <br>^Ab
|Pa / Shu
| Se
|-
|-
| 28
| 28
| suh
| 633.96
| 633.96
| [[13/9]]
| [[13/9]]
Line 266: Line 293:
| upaug 4th, <br>downmid 5th
| upaug 4th, <br>downmid 5th
| ^G#, <br>^^Ab
| ^G#, <br>^^Ab
|Pu / Shi
| Suh
|-
|-
| 29
| 29
| su
| 656.60
| 656.60
| [[16/11]], [[19/13]], [[22/15]]
| [[16/11]], [[19/13]], [[22/15]]
Line 274: Line 302:
| upmid 5th
| upmid 5th
| vvA
| vvA
|Pe / Si
| Su
|-
|-
| 30
| 30
| sih
| 679.25
| 679.25
| [[40/27]]
| [[40/27]]
Line 282: Line 311:
| down 5th
| down 5th
| vA
| vA
|So
| Sih
|-
|-
| 31
| 31
| sol
| 701.89
| 701.89
| [[3/2]]
| [[3/2]]
Line 290: Line 320:
| perfect 5th
| perfect 5th
| A
| A
|Sa
| Sol
|-
|-
| 32
| 32
| si
| 724.53
| 724.53
| [[32/21]], [[38/25]], [[243/160]], [[1024/675]]
| [[32/21]], [[38/25]], [[243/160]], [[1024/675]]
Line 298: Line 329:
| up 5th
| up 5th
| ^A
| ^A
|Su
| Si
|-
|-
| 33
| 33
| saw
| 747.17
| 747.17
| [[20/13]], [[192/125]]
| [[20/13]], [[192/125]]
Line 306: Line 338:
| double-up 5th, <br>double-down minor 6th
| double-up 5th, <br>double-down minor 6th
| ^^A, <br>vvBb
| ^^A, <br>vvBb
|Se / Fle
| Saw
|-
|-
| 34
| 34
| lo
| 769.81
| 769.81
| [[14/9]], [[25/16]], ''[[11/7]]''
| [[14/9]], [[25/16]], ''[[11/7]]''
Line 314: Line 347:
| downminor 6th
| downminor 6th
| vBb
| vBb
|Flo
| Lo
|-
|-
| 35
| 35
| leh
| 792.45
| 792.45
| [[19/12]], [[30/19]], [[128/81]]
| [[19/12]], [[30/19]], [[128/81]]
Line 322: Line 356:
| minor 6th
| minor 6th
| Bb
| Bb
|Fla
| Leh
|-
|-
| 36
| 36
| le
| 815.09
| 815.09
| [[8/5]]
| [[8/5]]
Line 330: Line 365:
| upminor 6th
| upminor 6th
| ^Bb
| ^Bb
|Flu
| Le
|-
|-
| 37
| 37
| lu
| 837.74
| 837.74
| [[13/8]], [[81/50]]
| [[13/8]], [[81/50]]
Line 338: Line 374:
| downmid 6th
| downmid 6th
| ^^Bb
| ^^Bb
|Fli
| Lu
|-
|-
| 38
| 38
| luh
| 860.38
| 860.38
| [[18/11]], [[400/243]]
| [[18/11]], [[400/243]]
Line 346: Line 383:
| upmid 6th
| upmid 6th
| vvB
| vvB
|Li
| Luh
|-
|-
| 39
| 39
| la
| 883.02
| 883.02
| [[5/3]]
| [[5/3]]
Line 354: Line 392:
| downmajor 6th
| downmajor 6th
| vB
| vB
|Lo
| La
|-
|-
| 40
| 40
| laa
| 905.66
| 905.66
| [[22/13]], [[27/16]], [[32/19]]
| [[22/13]], [[27/16]], [[32/19]]
Line 362: Line 401:
| major 6th
| major 6th
| B
| B
|La
| Laa
|-
|-
| 41
| 41
| lo
| 928.30
| 928.30
| [[12/7]]
| [[12/7]]
Line 370: Line 410:
| upmajor 6th
| upmajor 6th
| ^B
| ^B
|Lu
| Li
|-
|-
| 42
| 42
| law
| 950.94
| 950.94
| [[26/15]], [[125/72]]
| [[26/15]], [[125/72]]
Line 378: Line 419:
| double-up major 6th, <br>double-down minor 7th
| double-up major 6th, <br>double-down minor 7th
| ^^B, <br>vvC
| ^^B, <br>vvC
|Le / The
| Law
|-
|-
| 43
| 43
| ta
| 973.58
| 973.58
| [[7/4]]
| [[7/4]]
Line 386: Line 428:
| downminor 7th
| downminor 7th
| vC
| vC
|Tho
| Ta
|-
|-
| 44
| 44
| teh
| 996.23
| 996.23
| [[16/9]]
| [[16/9]]
Line 394: Line 437:
| minor 7th
| minor 7th
| C
| C
|Tha
| Teh
|-
|-
| 45
| 45
| te
| 1018.87
| 1018.87
| [[9/5]]
| [[9/5]]
Line 402: Line 446:
| upminor 7th
| upminor 7th
| ^C
| ^C
|Thu
| Te
|-
|-
| 46
| 46
| tu
| 1041.51
| 1041.51
| [[64/35]], [[11/6]], [[20/11]], [[729/400]]
| [[64/35]], [[11/6]], [[20/11]], [[729/400]]
Line 410: Line 455:
| downmid 7th
| downmid 7th
| ^^C
| ^^C
|Thi
| Tu
|-
|-
| 47
| 47
| tuh
| 1064.15
| 1064.15
| [[13/7]], [[24/13]], [[50/27]]
| [[13/7]], [[24/13]], [[50/27]]
Line 418: Line 464:
| upmid 7th
| upmid 7th
| vvC#
| vvC#
|Ti
| Tuh
|-
|-
| 48
| 48
| ti
| 1086.79
| 1086.79
| [[15/8]]
| [[15/8]]
Line 426: Line 473:
| downmajor 7th
| downmajor 7th
| vC#
| vC#
|To
| Ti
|-
|-
| 49
| 49
| tih
| 1109.43
| 1109.43
| [[19/10]], [[36/19]], [[40/21]], [[243/128]]
| [[19/10]], [[36/19]], [[40/21]], [[243/128]]
Line 434: Line 482:
| major 7th
| major 7th
| C#
| C#
|Ta
| Tih
|-
|-
| 50
| 50
| to
| 1132.08
| 1132.08
| [[48/25]], [[27/14]], [[21/11]], [[52/27]], [[25/13]]
| [[48/25]], [[27/14]], [[21/11]], [[52/27]], [[25/13]]
Line 442: Line 491:
| upmajor 7th
| upmajor 7th
| ^C#
| ^C#
|Tu
| To
|-
|-
| 51
| 51
| taw
| 1154.72
| 1154.72
| [[96/49]], [[35/18]], [[64/33]], [[125/64]]
| [[96/49]], [[35/18]], [[64/33]], [[125/64]]
Line 450: Line 500:
| double-up major 7th, <br>double-down 8ve
| double-up major 7th, <br>double-down 8ve
| ^^C#, <br>vvD
| ^^C#, <br>vvD
|Te / Di
| Taw
|-
|-
| 52
| 52
| da
| 1177.36
| 1177.36
| [[160/81]], [[63/32]], [[49/25]]
| [[160/81]], [[63/32]], [[49/25]]
Line 458: Line 509:
| down 8ve
| down 8ve
| vD
| vD
|Do
| Da
|-
|-
| 53
| 53
| do
| 1200.00
| 1200.00
| [[2/1]]
| [[2/1]]
Line 466: Line 518:
| perfect 8ve
| perfect 8ve
| D
| D
|Da
| Do
|}
|}
<nowiki>*</nowiki> Based on interpreting 53edo as a no-17's [[19-limit]] temperament. Italics represent inconsistent intervals which are mapped by the 19-limit [[patent val]] to their second-best (as opposed to best) approximation in 53edo.  
<nowiki>*</nowiki> Based on interpreting 53edo as a no-17's [[19-limit]] temperament. Italics represent inconsistent intervals which are mapped by the 19-limit [[patent val]] to their second-best (as opposed to best) approximation in 53edo.  
Line 930: Line 984:
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3 Bach WTC1 Prelude 1 in 53] by Bach and [[Mykhaylo Khramov]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3 Bach WTC1 Prelude 1 in 53] by Bach and [[Mykhaylo Khramov]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/fugue1-53.mp3 Bach WTC1 Fugue 1 in 53] by Bach and Mykhaylo Khramov
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/fugue1-53.mp3 Bach WTC1 Fugue 1 in 53] by Bach and Mykhaylo Khramov
* [http://bumpermusic.blogspot.com/2007/05/whisper-song-in-53-edo-now-526-slower.html Whisper Song in 53EDO] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prent/sing53-c5-slow.mp3 play] by [[Prent Rodgers]]
* [https://bumpermusic.blogspot.com/2007/05/whisper-song-in-53-edo-now-526-slower.html Whisper Song in 53EDO] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prent/sing53-c5-slow.mp3 play] by [[Prent Rodgers]]
* [http://www.archive.org/details/TrioInOrwell Trio in Orwell] [http://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3 play] by [[Gene Ward Smith]]
* [https://www.archive.org/details/TrioInOrwell Trio in Orwell] [https://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3 play] by [[Gene Ward Smith]]
* [http://www.akjmusic.com/audio/desert_prayer.mp3 Desert Prayer] by [http://www.akjmusic.com/ Aaron Krister Johnson]
* [http://www.akjmusic.com/audio/desert_prayer.mp3 Desert Prayer] by [http://www.akjmusic.com/ Aaron Krister Johnson]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/sing53-c5-slow.mp3 Whisper Song in 53 EDO] by [[Prent Rodgers]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/sing53-c5-slow.mp3 Whisper Song in 53 EDO] by [[Prent Rodgers]]
* [http://andrewheathwaite.bandcamp.com/track/elf-dine-on-ho-ho Elf Dine on Ho Ho] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2005%20Elf%20Dine%20on%20Ho%20Ho.mp3 play] and [http://andrewheathwaite.bandcamp.com/track/spun Spun] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2008%20Spun.mp3 play] by [[Andrew Heathwaite]]
* [https://andrewheathwaite.bandcamp.com/track/elf-dine-on-ho-ho Elf Dine on Ho Ho] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2005%20Elf%20Dine%20on%20Ho%20Ho.mp3 play] and [https://andrewheathwaite.bandcamp.com/track/spun Spun] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2008%20Spun.mp3 play] by [[Andrew Heathwaite]]
* [http://chrisvaisvil.com/the-fallen-of-kleismic15/ The Fallen of Kleismic15][http://micro.soonlabel.com/53edo/20130903_Kleismic%5b15%5d.mp3 play] by [[Chris Vaisvil]]
* [http://chrisvaisvil.com/the-fallen-of-kleismic15/ The Fallen of Kleismic15][http://micro.soonlabel.com/53edo/20130903_Kleismic%5b15%5d.mp3 play] by [[Chris Vaisvil]]
* [https://soundcloud.com/cam-taylor-2-1/mothers mothers] by [[Cam Taylor]]
* [https://soundcloud.com/cam-taylor-2-1/mothers mothers] by [[Cam Taylor]]

Revision as of 09:09, 26 August 2022

← 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:

The famous 53 equal divisions of the octave (53edo), or 53(-tone) equal temperament (53tet, 53et) when viewed from a regular temperament perspective, divides the octave into 53 equal comma-sized parts of around 22.6 cents each.

Theory

53edo is notable as a 5-limit system, a fact apparently first noted by Isaac Newton, notably tempering out the schisma (32805/32768), the kleisma (15625/15552), the amity comma (1600000/1594323), the semicomma (2109375/2097152), and the vulture comma (10485760000/10460353203). In the 7-limit it tempers out 225/224, 1728/1715 and 3125/3087, the marvel comma, the gariboh, and the orwell comma. 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, and 2080/2079, and gives the optimal patent val for athene temperament. It is the seventh strict zeta edo. It can also be treated as a no-elevens, no-seventeens tuning, on which it is consistent all the way up to the 23-odd-limit.

53edo has also found a certain dissemination as an EDO tuning for Arabic, Turkish, Persian music.

53edo is the 16th prime edo, following 47edo and coming before 59edo.

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)

Intervals

# Cents Approximate ratios* Ups and downs notation Solfeges
0 0.00 1/1 P1 unison D Da Do
1 22.64 81/80, 64/63, 50/49 ^1 up unison ^D Du Di
2 45.28 49/48, 36/35, 33/32, 128/125 ^^1,
vvm2 		double-up unison,
double-down minor 2nd 		^^D,
vvEb 		De/Fre Daw
3 67.92 25/24, 28/27, 22/21, 27/26, 26/25 vvA1,
vm2 		double-down aug 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 16/15, 15/14 A1, ^m2 aug 1sn, upminor 2nd D#, ^Eb Fru Ra
6 135.85 14/13, 13/12, 27/25 v~2 downmid 2nd ^^Eb Fri Ru
7 158.49 35/32, 12/11, 11/10, 57/52, 800/729 ^~2 upmid 2nd vvE Ri 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, 144/125 ^^M2,
vvm3 		double-up major 2nd,
double-down minor 3rd 		^^E,
vvF 		Re / 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 v~3 downmid 3rd ^^F Ni Mu
16 362.26 16/13, 100/81 ^~3 upmid 3rd vvF# Mi 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 ^^M3,
vv4 		double-up major 3rd,
double-down 4th 		^^F#,
vvG 		Me / Fe Maw
21 475.47 21/16, 25/19, 675/512, 320/243 v4 down 4th vG Fo Fe
22 498.11 4/3 P4 perfect 4th G Fa Fa
23 520.75 27/20 ^4 up 4th ^G Fu Fih
24 543.40 11/8, 15/11, 26/19 v~4 downmid 4th ^^G Fi / She Fu
25 566.04 18/13 ^~4,
vd5 		upmid 4th,
downdim 5th 		vvG#,
vAb 		Pi / 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 ^A4,
v~5 		upaug 4th,
downmid 5th 		^G#,
^^Ab 		Pu / Shi Suh
29 656.60 16/11, 19/13, 22/15 ^~5 upmid 5th vvA Pe / Si Su
30 679.25 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, 192/125 ^^5,
vvm6 		double-up 5th,
double-down minor 6th 		^^A,
vvBb 		Se / Fle Saw
34 769.81 14/9, 25/16, 11/7 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 v~6 downmid 6th ^^Bb Fli Lu
38 860.38 18/11, 400/243 ^~6 upmid 6th vvB Li 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 26/15, 125/72 ^^M6,
vvm7 		double-up major 6th,
double-down minor 7th 		^^B,
vvC 		Le / 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 64/35, 11/6, 20/11, 729/400 v~7 downmid 7th ^^C Thi Tu
47 1064.15 13/7, 24/13, 50/27 ^~7 upmid 7th vvC# Ti 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 48/25, 27/14, 21/11, 52/27, 25/13 ^M7 upmajor 7th ^C# Tu To
51 1154.72 96/49, 35/18, 64/33, 125/64 ^^M7,
vv8 		double-up major 7th,
double-down 8ve 		^^C#,
vvD 		Te / Di Taw
52 1177.36 160/81, 63/32, 49/25 v8 down 8ve vD Do Da
53 1200.00 2/1 P8 perfect 8ve D Da Do

* Based on interpreting 53edo as a no-17's 19-limit temperament. Italics represent inconsistent intervals which are mapped by the 19-limit patent val to their second-best (as opposed to best) approximation in 53edo.

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
downmid ilo (a, b, 0, 0, 1) 11/9, 11/6
upmid 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 Cv~ C downmid
lu 22:27:33 0-16-31 C vvE G C^~ C upmid
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

Sagittal

The following table shows sagittal notation accidentals in one apotome for 53edo.

Steps 0 1 2 3 4 5
Symbol

Relationship to 12edo

Whereas 12edo has a circle of twelve 5ths, 53edo has a spiral of twelve 5ths (since 31\53 is on the 7\12 kite in the scale tree). This shows 53edo in a 12edo-friendly format. Excellent for introducing 53edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.

JI approximation

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 tone 9/8 9 −0.14 cents
Minor tone 10/9 8 −1.27 cents
Diat. semitone 16/15 5 +1.48 cents

One notable property of 53edo is that it offers good approximations for both just and Pythagorean major thirds.

The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.8 cents away from the just ratio 7/4, so 53edo can also be used for 7-limit harmony, tempering out the septimal kleisma, 225/224.

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 53edo. Octave-reduced prime harmonics are bolded; inconsistent intervals are in italic.

15-odd-limit intervals by direct mapping (even if inconsistent)
Interval, complement Error (abs, ¢) Error (rel, %)
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

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

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 ETs 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 ET doing better in either subgroup is 130.

Linear temperaments

Table of rank-2 temperaments by generator
Periods
per Octave 		Generator Cents Associated
Ratio 		Temperament
1 2\53 45.28 36/35 Quartonic
1 5\53 113.21 16/15 Misneb
1 7\53 158.49 11/10 Hemikleismic
1 9\53 203.77 9/8 Baldy
1 10\53 226.42 8/7 Semaja
1 11\53 249.06 15/13 Hemischis / hemigari
1 12\53 271.70 7/6 Orson / orwell
1 13\53 294.34 25/21 Kleiboh
1 14\53 316.98 6/5 Hanson / catakleismic / countercata
1 15\53 339.62 11/9 Amity / houborizic
1 16\53 362.26 16/13 Submajor
1 18\53 407.55 1225/972 Ditonic / coditone
1 19\53 430.19 9/7 Hamity
1 21\53 475.47 21/16 Vulture / buzzard
1 22\53 498.11 4/3 Helmholtz / garibaldi / pontiac
1 25\53 566.04 18/13 Tricot
1 26\53 588.68 45/32 Untriton / aufo

Music

See also: Category:53edo tracks

See also

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