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== Theory | <span style="display: block; text-align: right;">Other languages: [[:de:53edo|Deutsch]]</span> | ||
=Theory= | |||
The famous ''53 equal division'' divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a [[5-limit|5-limit]] system, a fact apparently first noted by Isaac Newton, tempering out the schisma, 32805/32768, the kleisma, 15625/15552, the amity comma, 1600000/1594323 and the semicomma, 2109375/2097152. 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, and is the [[Optimal_patent_val|optimal patent val]] for [[Nuwell_family|Big Brother]] temperament, which tempers out both, as well as 11-limit [[Semicomma_family|orwell temperament]], which also tempers out the 11-limit comma 176/175. In the 13-limit, it tempers out 169/168 and 245/243, and gives the optimal patent val for [[Marvel_family|athene temperament]]. It is the eighth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]] and the 16th [[prime_numbers|prime]] edo, following [[47edo|47edo]] and coming before [[59edo|59edo]]. | The famous ''53 equal division'' divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a [[5-limit|5-limit]] system, a fact apparently first noted by Isaac Newton, tempering out the schisma, 32805/32768, the kleisma, 15625/15552, the amity comma, 1600000/1594323 and the semicomma, 2109375/2097152. 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, and is the [[Optimal_patent_val|optimal patent val]] for [[Nuwell_family|Big Brother]] temperament, which tempers out both, as well as 11-limit [[Semicomma_family|orwell temperament]], which also tempers out the 11-limit comma 176/175. In the 13-limit, it tempers out 169/168 and 245/243, and gives the optimal patent val for [[Marvel_family|athene temperament]]. It is the eighth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]] and the 16th [[prime_numbers|prime]] edo, following [[47edo|47edo]] and coming before [[59edo|59edo]]. | ||
| Line 8: | Line 10: | ||
It can also be treated as a no-elevens, no-seventeens tuning, on which it is consistent all the way up to the 21-limit. | It can also be treated as a no-elevens, no-seventeens tuning, on which it is consistent all the way up to the 21-limit. | ||
[http://en.wikipedia.org/wiki/53_equal_temperament Wikipeda article about 53edo] | |||
=Linear temperaments= | |||
[[List_of_edo-distinct_53et_rank_two_temperaments|List of edo-distinct 53et rank two temperaments]] | |||
=Just Approximation= | |||
53edo provides excellent approximations for the classic 5-limit [[just|just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale. | 53edo provides excellent approximations for the classic 5-limit [[just|just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale. | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! interval | ! | interval | ||
! ratio | ! | ratio | ||
! size | ! | size | ||
! difference | ! | difference | ||
|- | |- | ||
| perfect fifth | | | perfect fifth | ||
| 3/2 | | | 3/2 | ||
| style="text-align:center;" | 31 | | style="text-align:center;" | 31 | ||
| −0.07 cents | | | −0.07 cents | ||
|- | |- | ||
| major third | | | major third | ||
| 5/4 | | | 5/4 | ||
| style="text-align:center;" | 17 | | style="text-align:center;" | 17 | ||
| −1.40 cents | | | −1.40 cents | ||
|- | |- | ||
| minor third | | | minor third | ||
| 6/5 | | | 6/5 | ||
| style="text-align:center;" | 14 | | style="text-align:center;" | 14 | ||
| +1.34 cents | | | +1.34 cents | ||
|- | |- | ||
| major tone | | | major tone | ||
| 9/8 | | | 9/8 | ||
| style="text-align:center;" | 9 | | style="text-align:center;" | 9 | ||
| −0.14 cents | | | −0.14 cents | ||
|- | |- | ||
| minor tone | | | minor tone | ||
| 10/9 | | | 10/9 | ||
| style="text-align:center;" | 8 | | style="text-align:center;" | 8 | ||
| −1.27 cents | | | −1.27 cents | ||
|- | |- | ||
| diat. semitone | | | diat. semitone | ||
| 16/15 | | | 16/15 | ||
| style="text-align:center;" | 5 | | style="text-align:center;" | 5 | ||
| +1.48 cents | | | +1.48 cents | ||
|} | |} | ||
One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds. | One notable property of 53EDO is that it offers good approximations for both pure 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. | 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|septimal kleisma]], 225/224. | ||
=Intervals= | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! degree | ! | degree | ||
! solfege | ! | solfege | ||
! cents | ! | cents | ||
! approximate ratios | ! | approximate ratios | ||
! colspan="3" | [[ | ! colspan="3" | [[Ups_and_Downs_Notation|ups and downs]] [[Ups_and_Downs_Notation|notation]] | ||
! generator for | ! | generator for | ||
|- | |- | ||
| style="text-align:center;" | 0 | | style="text-align:center;" | 0 | ||
| Line 75: | Line 77: | ||
| style="text-align:center;" | P1 | | style="text-align:center;" | P1 | ||
| style="text-align:center;" | unison | | style="text-align:center;" | unison | ||
| style="text-align:center;" | | | style="text-align:center;" | D | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 1 | | style="text-align:center;" | 1 | ||
| Line 83: | Line 85: | ||
| style="text-align:center;" | 81/80, 64/63, 50/49 | | style="text-align:center;" | 81/80, 64/63, 50/49 | ||
| style="text-align:center;" | ^1 | | style="text-align:center;" | ^1 | ||
| style="text-align:center;" | | | style="text-align:center;" | up unison | ||
| style="text-align:center;" | | | style="text-align:center;" | D^ | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 2 | | style="text-align:center;" | 2 | ||
| Line 91: | Line 93: | ||
| style="text-align:center;" | 45.28 | | style="text-align:center;" | 45.28 | ||
| style="text-align:center;" | 49/48, 36/35, 33/32, 128/125 | | style="text-align:center;" | 49/48, 36/35, 33/32, 128/125 | ||
| style="text-align:center;" | ^^1, | | style="text-align:center;" | ^^1, | ||
| style="text-align:center;" | | |||
| style="text-align:center;" | | vvm2 | ||
| [[Quartonic]] | | style="text-align:center;" | double-up unison, | ||
double-down minor 2nd | |||
| style="text-align:center;" | D^^, | |||
Ebvv | |||
| | [[Quartonic|Quartonic]] | |||
|- | |- | ||
| style="text-align:center;" | 3 | | style="text-align:center;" | 3 | ||
| Line 101: | Line 109: | ||
| style="text-align:center;" | 27/26, 26/25, 25/24, 22/21 | | style="text-align:center;" | 27/26, 26/25, 25/24, 22/21 | ||
| style="text-align:center;" | vm2 | | style="text-align:center;" | vm2 | ||
| style="text-align:center;" | | | style="text-align:center;" | downminor 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | Ebv | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 4 | | style="text-align:center;" | 4 | ||
| Line 110: | Line 118: | ||
| style="text-align:center;" | 21/20, 256/243 | | style="text-align:center;" | 21/20, 256/243 | ||
| style="text-align:center;" | m2 | | style="text-align:center;" | m2 | ||
| style="text-align:center;" | | | style="text-align:center;" | minor 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | Eb | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 5 | | style="text-align:center;" | 5 | ||
| Line 119: | Line 127: | ||
| style="text-align:center;" | 16/15, 15/14 | | style="text-align:center;" | 16/15, 15/14 | ||
| style="text-align:center;" | ^m2 | | style="text-align:center;" | ^m2 | ||
| style="text-align:center;" | | | style="text-align:center;" | upminor 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | Eb^ | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 6 | | style="text-align:center;" | 6 | ||
| Line 128: | Line 136: | ||
| style="text-align:center;" | 14/13, 13/12, 27/25 | | style="text-align:center;" | 14/13, 13/12, 27/25 | ||
| style="text-align:center;" | v~2 | | style="text-align:center;" | v~2 | ||
| style="text-align:center;" | | | style="text-align:center;" | downmid 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | Eb^^ | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 7 | | style="text-align:center;" | 7 | ||
| Line 137: | Line 145: | ||
| style="text-align:center;" | 12/11, 11/10, 800/729 | | style="text-align:center;" | 12/11, 11/10, 800/729 | ||
| style="text-align:center;" | ^~2 | | style="text-align:center;" | ^~2 | ||
| style="text-align:center;" | | | style="text-align:center;" | upmid 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | Evv | ||
| [[Hemikleismic]] | | | [[Hemikleismic|Hemikleismic]] | ||
|- | |- | ||
| style="text-align:center;" | 8 | | style="text-align:center;" | 8 | ||
| Line 146: | Line 154: | ||
| style="text-align:center;" | 10/9 | | style="text-align:center;" | 10/9 | ||
| style="text-align:center;" | vM2 | | style="text-align:center;" | vM2 | ||
| style="text-align:center;" | downmajor | | style="text-align:center;" | downmajor 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | Ev | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 9 | | style="text-align:center;" | 9 | ||
| Line 155: | Line 163: | ||
| style="text-align:center;" | 9/8 | | style="text-align:center;" | 9/8 | ||
| style="text-align:center;" | M2 | | style="text-align:center;" | M2 | ||
| style="text-align:center;" | major | | style="text-align:center;" | major 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | E | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 10 | | style="text-align:center;" | 10 | ||
| Line 164: | Line 172: | ||
| style="text-align:center;" | 8/7, 256/225 | | style="text-align:center;" | 8/7, 256/225 | ||
| style="text-align:center;" | ^M2 | | style="text-align:center;" | ^M2 | ||
| style="text-align:center;" | upmajor | | style="text-align:center;" | upmajor 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | E^ | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 11 | | style="text-align:center;" | 11 | ||
| Line 172: | Line 180: | ||
| style="text-align:center;" | 249.06 | | style="text-align:center;" | 249.06 | ||
| style="text-align:center;" | 15/13, 144/125 | | style="text-align:center;" | 15/13, 144/125 | ||
| style="text-align:center;" | ^^M2, | | style="text-align:center;" | ^^M2, | ||
| style="text-align:center;" | | |||
| style="text-align:center;" | | vvm3 | ||
| [[Hemischis]] | | style="text-align:center;" | double-up major 2nd, | ||
double-down minor 3rd | |||
| style="text-align:center;" | E^^, | |||
Fvv | |||
| | [[Hemischis|Hemischis]] | |||
|- | |- | ||
| style="text-align:center;" | 12 | | style="text-align:center;" | 12 | ||
| Line 182: | Line 196: | ||
| style="text-align:center;" | 7/6, 75/64 | | style="text-align:center;" | 7/6, 75/64 | ||
| style="text-align:center;" | vm3 | | style="text-align:center;" | vm3 | ||
| style="text-align:center;" | downminor | | style="text-align:center;" | downminor 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | Fv | ||
| [[Orwell]] | | | [[Orwell|Orwell]] | ||
|- | |- | ||
| style="text-align:center;" | 13 | | style="text-align:center;" | 13 | ||
| Line 191: | Line 205: | ||
| style="text-align:center;" | 13/11, 32/27 | | style="text-align:center;" | 13/11, 32/27 | ||
| style="text-align:center;" | m3 | | style="text-align:center;" | m3 | ||
| style="text-align:center;" | minor | | style="text-align:center;" | minor 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | F | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 14 | | style="text-align:center;" | 14 | ||
| Line 200: | Line 214: | ||
| style="text-align:center;" | 6/5 | | style="text-align:center;" | 6/5 | ||
| style="text-align:center;" | ^m3 | | style="text-align:center;" | ^m3 | ||
| style="text-align:center;" | upminor | | style="text-align:center;" | upminor 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | F^ | ||
| [[Hanson]]/[[Catakleismic]] | | | [[Hanson|Hanson]]/[[catakleismic|Catakleismic]] | ||
|- | |- | ||
| style="text-align:center;" | 15 | | style="text-align:center;" | 15 | ||
| Line 209: | Line 223: | ||
| style="text-align:center;" | 11/9, 243/200 | | style="text-align:center;" | 11/9, 243/200 | ||
| style="text-align:center;" | v~3 | | style="text-align:center;" | v~3 | ||
| style="text-align:center;" | | | style="text-align:center;" | downmid 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | F^^ | ||
| [[Amity]]/[[Hitchcock]] | | | [[Amity|Amity]]/[[Hitchcock|Hitchcock]] | ||
|- | |- | ||
| style="text-align:center;" | 16 | | style="text-align:center;" | 16 | ||
| Line 218: | Line 232: | ||
| style="text-align:center;" | 16/13, 100/81 | | style="text-align:center;" | 16/13, 100/81 | ||
| style="text-align:center;" | ^~3 | | style="text-align:center;" | ^~3 | ||
| style="text-align:center;" | | | style="text-align:center;" | upmid 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | F#vv | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 17 | | style="text-align:center;" | 17 | ||
| Line 227: | Line 241: | ||
| style="text-align:center;" | 5/4 | | style="text-align:center;" | 5/4 | ||
| style="text-align:center;" | vM3 | | style="text-align:center;" | vM3 | ||
| style="text-align:center;" | downmajor | | style="text-align:center;" | downmajor 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | F#v | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 18 | | style="text-align:center;" | 18 | ||
| Line 236: | Line 250: | ||
| style="text-align:center;" | 81/64 | | style="text-align:center;" | 81/64 | ||
| style="text-align:center;" | M3 | | style="text-align:center;" | M3 | ||
| style="text-align:center;" | major | | style="text-align:center;" | major 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | F# | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 19 | | style="text-align:center;" | 19 | ||
| Line 247: | Line 261: | ||
| style="text-align:center;" | upmajor 3rd | | style="text-align:center;" | upmajor 3rd | ||
| style="text-align:center;" | F#^ | | style="text-align:center;" | F#^ | ||
| [[Hamity]] | | | [[Hamity|Hamity]] | ||
|- | |- | ||
| style="text-align:center;" | 20 | | style="text-align:center;" | 20 | ||
| Line 253: | Line 267: | ||
| style="text-align:center;" | 452.83 | | style="text-align:center;" | 452.83 | ||
| style="text-align:center;" | 13/10, 125/96 | | style="text-align:center;" | 13/10, 125/96 | ||
| style="text-align:center;" | ^^M3, | | style="text-align:center;" | ^^M3, | ||
| style="text-align:center;" | double-up major 3rd, | |||
| style="text-align:center;" | F#^^, | vv4 | ||
| | | style="text-align:center;" | double-up major 3rd, | ||
double-down 4th | |||
| style="text-align:center;" | F#^^, | |||
Gvv | |||
| | | |||
|- | |- | ||
| style="text-align:center;" | 21 | | style="text-align:center;" | 21 | ||
| Line 265: | Line 285: | ||
| style="text-align:center;" | down 4th | | style="text-align:center;" | down 4th | ||
| style="text-align:center;" | Gv | | style="text-align:center;" | Gv | ||
| [[Vulture]]/[[Buzzard]] | | | [[Vulture|Vulture]]/[[Buzzard|Buzzard]] | ||
|- | |- | ||
| style="text-align:center;" | 22 | | style="text-align:center;" | 22 | ||
| Line 274: | Line 294: | ||
| style="text-align:center;" | perfect 4th | | style="text-align:center;" | perfect 4th | ||
| style="text-align:center;" | G | | style="text-align:center;" | G | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 23 | | style="text-align:center;" | 23 | ||
| Line 283: | Line 303: | ||
| style="text-align:center;" | up 4th | | style="text-align:center;" | up 4th | ||
| style="text-align:center;" | G^ | | style="text-align:center;" | G^ | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 24 | | style="text-align:center;" | 24 | ||
| Line 292: | Line 312: | ||
| style="text-align:center;" | double-up 4th | | style="text-align:center;" | double-up 4th | ||
| style="text-align:center;" | G^^ | | style="text-align:center;" | G^^ | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 25 | | style="text-align:center;" | 25 | ||
| Line 298: | Line 318: | ||
| style="text-align:center;" | 566.04 | | style="text-align:center;" | 566.04 | ||
| style="text-align:center;" | 18/13 | | style="text-align:center;" | 18/13 | ||
| style="text-align:center;" | vvA4, | | style="text-align:center;" | vvA4, | ||
| style="text-align:center;" | double-down aug 4th, | |||
| style="text-align:center;" | G#vv, | vd5 | ||
| [[Tricot]] | | style="text-align:center;" | double-down aug 4th, | ||
downdim 5th | |||
| style="text-align:center;" | G#vv, | |||
Abv | |||
| | [[Tricot|Tricot]] | |||
|- | |- | ||
| style="text-align:center;" | 26 | | style="text-align:center;" | 26 | ||
| Line 307: | Line 333: | ||
| style="text-align:center;" | 588.68 | | style="text-align:center;" | 588.68 | ||
| style="text-align:center;" | 7/5, 45/32 | | style="text-align:center;" | 7/5, 45/32 | ||
| style="text-align:center;" | vA4, | | style="text-align:center;" | vA4, | ||
| style="text-align:center;" | downaug 4th, | |||
| style="text-align:center;" | G#v, | d5 | ||
| | | style="text-align:center;" | downaug 4th, | ||
dim 5th | |||
| style="text-align:center;" | G#v, | |||
Ab | |||
| | | |||
|- | |- | ||
| style="text-align:center;" | 27 | | style="text-align:center;" | 27 | ||
| Line 316: | Line 348: | ||
| style="text-align:center;" | 611.32 | | style="text-align:center;" | 611.32 | ||
| style="text-align:center;" | 10/7, 64/45 | | style="text-align:center;" | 10/7, 64/45 | ||
| style="text-align:center;" | A4, | | style="text-align:center;" | A4, | ||
| style="text-align:center;" | aug 4th, | |||
| style="text-align:center;" | G#, | ^d5 | ||
| | | style="text-align:center;" | aug 4th, | ||
updim 5th | |||
| style="text-align:center;" | G#, | |||
Ab^ | |||
| | | |||
|- | |- | ||
| style="text-align:center;" | 28 | | style="text-align:center;" | 28 | ||
| Line 325: | Line 363: | ||
| style="text-align:center;" | 633.96 | | style="text-align:center;" | 633.96 | ||
| style="text-align:center;" | 13/9 | | style="text-align:center;" | 13/9 | ||
| style="text-align:center;" | ^A4, | | style="text-align:center;" | ^A4, | ||
| style="text-align:center;" | upaug 4th, | |||
| style="text-align:center;" | G#^, | ^^d5 | ||
| | | style="text-align:center;" | upaug 4th, | ||
double-up dim 5th | |||
| style="text-align:center;" | G#^, | |||
Ab^^ | |||
| | | |||
|- | |- | ||
| style="text-align:center;" | 29 | | style="text-align:center;" | 29 | ||
| Line 337: | Line 381: | ||
| style="text-align:center;" | double-down 5th | | style="text-align:center;" | double-down 5th | ||
| style="text-align:center;" | Avv | | style="text-align:center;" | Avv | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 30 | | style="text-align:center;" | 30 | ||
| Line 346: | Line 390: | ||
| style="text-align:center;" | down 5th | | style="text-align:center;" | down 5th | ||
| style="text-align:center;" | Av | | style="text-align:center;" | Av | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 31 | | style="text-align:center;" | 31 | ||
| Line 355: | Line 399: | ||
| style="text-align:center;" | perfect 5th | | style="text-align:center;" | perfect 5th | ||
| style="text-align:center;" | A | | style="text-align:center;" | A | ||
| [[Helmholtz]]/[[Garibaldi]] | | | [[Helmholtz|Helmholtz]]/[[Garibaldi|Garibaldi]] | ||
|- | |- | ||
| style="text-align:center;" | 32 | | style="text-align:center;" | 32 | ||
| Line 364: | Line 408: | ||
| style="text-align:center;" | up 5th | | style="text-align:center;" | up 5th | ||
| style="text-align:center;" | A^ | | style="text-align:center;" | A^ | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 33 | | style="text-align:center;" | 33 | ||
| Line 370: | Line 414: | ||
| style="text-align:center;" | 747.17 | | style="text-align:center;" | 747.17 | ||
| style="text-align:center;" | 20/13, 192/125 | | style="text-align:center;" | 20/13, 192/125 | ||
| style="text-align:center;" | ^^5, | | style="text-align:center;" | ^^5, | ||
| style="text-align:center;" | double-up 5th, | |||
| style="text-align:center;" | A^^, | vvm6 | ||
| | | style="text-align:center;" | double-up 5th, | ||
double-down minor 6th | |||
| style="text-align:center;" | A^^, | |||
Bbvv | |||
| | | |||
|- | |- | ||
| style="text-align:center;" | 34 | | style="text-align:center;" | 34 | ||
| Line 382: | Line 432: | ||
| style="text-align:center;" | downminor 6th | | style="text-align:center;" | downminor 6th | ||
| style="text-align:center;" | Bbv | | style="text-align:center;" | Bbv | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 35 | | style="text-align:center;" | 35 | ||
| Line 391: | Line 441: | ||
| style="text-align:center;" | minor 6th | | style="text-align:center;" | minor 6th | ||
| style="text-align:center;" | Bb | | style="text-align:center;" | Bb | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 36 | | style="text-align:center;" | 36 | ||
| Line 400: | Line 450: | ||
| style="text-align:center;" | upminor 6th | | style="text-align:center;" | upminor 6th | ||
| style="text-align:center;" | Bb^ | | style="text-align:center;" | Bb^ | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 37 | | style="text-align:center;" | 37 | ||
| Line 409: | Line 459: | ||
| style="text-align:center;" | downmid 6th | | style="text-align:center;" | downmid 6th | ||
| style="text-align:center;" | Bb^^ | | style="text-align:center;" | Bb^^ | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 38 | | style="text-align:center;" | 38 | ||
| Line 418: | Line 468: | ||
| style="text-align:center;" | upmid 6th | | style="text-align:center;" | upmid 6th | ||
| style="text-align:center;" | Bvv | | style="text-align:center;" | Bvv | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 39 | | style="text-align:center;" | 39 | ||
| Line 427: | Line 477: | ||
| style="text-align:center;" | downmajor 6th | | style="text-align:center;" | downmajor 6th | ||
| style="text-align:center;" | Bv | | style="text-align:center;" | Bv | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 40 | | style="text-align:center;" | 40 | ||
| Line 436: | Line 486: | ||
| style="text-align:center;" | major 6th | | style="text-align:center;" | major 6th | ||
| style="text-align:center;" | B | | style="text-align:center;" | B | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 41 | | style="text-align:center;" | 41 | ||
| Line 445: | Line 495: | ||
| style="text-align:center;" | upmajor 6th | | style="text-align:center;" | upmajor 6th | ||
| style="text-align:center;" | B^ | | style="text-align:center;" | B^ | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 42 | | style="text-align:center;" | 42 | ||
| Line 451: | Line 501: | ||
| style="text-align:center;" | 950.94 | | style="text-align:center;" | 950.94 | ||
| style="text-align:center;" | 26/15, 125/72 | | style="text-align:center;" | 26/15, 125/72 | ||
| style="text-align:center;" | ^^M6 | | style="text-align:center;" | ^^M6 | ||
| style="text-align:center;" | double-up major 6th, | |||
| style="text-align:center;" | B^^, | vvm7 | ||
| | | style="text-align:center;" | double-up major 6th, | ||
double-down minor 7th | |||
| style="text-align:center;" | B^^, | |||
Cvv | |||
| | | |||
|- | |- | ||
| style="text-align:center;" | 43 | | style="text-align:center;" | 43 | ||
| Line 463: | Line 519: | ||
| style="text-align:center;" | downminor 7th | | style="text-align:center;" | downminor 7th | ||
| style="text-align:center;" | Cv | | style="text-align:center;" | Cv | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 44 | | style="text-align:center;" | 44 | ||
| Line 472: | Line 528: | ||
| style="text-align:center;" | minor 7th | | style="text-align:center;" | minor 7th | ||
| style="text-align:center;" | C | | style="text-align:center;" | C | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 45 | | style="text-align:center;" | 45 | ||
| Line 481: | Line 537: | ||
| style="text-align:center;" | upminor 7th | | style="text-align:center;" | upminor 7th | ||
| style="text-align:center;" | C^ | | style="text-align:center;" | C^ | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 46 | | style="text-align:center;" | 46 | ||
| Line 490: | Line 546: | ||
| style="text-align:center;" | downmid 7th | | style="text-align:center;" | downmid 7th | ||
| style="text-align:center;" | C^^ | | style="text-align:center;" | C^^ | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 47 | | style="text-align:center;" | 47 | ||
| Line 499: | Line 555: | ||
| style="text-align:center;" | upmid 7th | | style="text-align:center;" | upmid 7th | ||
| style="text-align:center;" | C#vv | | style="text-align:center;" | C#vv | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 48 | | style="text-align:center;" | 48 | ||
| Line 508: | Line 564: | ||
| style="text-align:center;" | downmajor 7th | | style="text-align:center;" | downmajor 7th | ||
| style="text-align:center;" | C#v | | style="text-align:center;" | C#v | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 49 | | style="text-align:center;" | 49 | ||
| Line 517: | Line 573: | ||
| style="text-align:center;" | major 7th | | style="text-align:center;" | major 7th | ||
| style="text-align:center;" | C# | | style="text-align:center;" | C# | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 50 | | style="text-align:center;" | 50 | ||
| Line 526: | Line 582: | ||
| style="text-align:center;" | upmajor 7th | | style="text-align:center;" | upmajor 7th | ||
| style="text-align:center;" | C#^ | | style="text-align:center;" | C#^ | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 51 | | style="text-align:center;" | 51 | ||
| Line 532: | Line 588: | ||
| style="text-align:center;" | 1154.72 | | style="text-align:center;" | 1154.72 | ||
| style="text-align:center;" | 125/64 | | style="text-align:center;" | 125/64 | ||
| style="text-align:center;" | ^^M7, | | style="text-align:center;" | ^^M7, | ||
| style="text-align:center;" | double-up major 7th, | |||
| style="text-align:center;" | C#^^, | vv8 | ||
| | | style="text-align:center;" | double-up major 7th, | ||
double-down 8ve | |||
| style="text-align:center;" | C#^^, | |||
Dvv | |||
| | | |||
|- | |- | ||
| style="text-align:center;" | 52 | | style="text-align:center;" | 52 | ||
| Line 544: | Line 606: | ||
| style="text-align:center;" | down 8ve | | style="text-align:center;" | down 8ve | ||
| style="text-align:center;" | Dv | | style="text-align:center;" | Dv | ||
| | | | | ||
|- | |- | ||
| style="text-align:center;" | 53 | | style="text-align:center;" | 53 | ||
| Line 553: | Line 615: | ||
| style="text-align:center;" | perfect 8ve | | style="text-align:center;" | perfect 8ve | ||
| style="text-align:center;" | D | | style="text-align:center;" | D | ||
| | | | | ||
|} | |} | ||
Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors: | Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors: | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
!quality | ! | quality | ||
! [[Kite's color notation|color]] | ! | [[Kite's color notation|color]] | ||
! monzo format | ! | monzo format | ||
! examples | ! | examples | ||
|- | |- | ||
| style="text-align:center;" | downminor | | style="text-align:center;" | downminor | ||
| Line 605: | Line 666: | ||
| style="text-align:center;" | 9/7, 12/7 | | style="text-align:center;" | 9/7, 12/7 | ||
|} | |} | ||
All 53edo chords can be named using ups and downs. Here are the zo, gu, lova, yo and ru triads: | All 53edo chords can be named using ups and downs. Here are the zo, gu, lova, yo and ru triads: | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! [[Kite's color notation|color of the 3rd]] | ! | [[Kite's color notation|color of the 3rd]] | ||
! JI chord | ! | JI chord | ||
! notes as edosteps | ! | notes as edosteps | ||
! notes of C chord | ! | notes of C chord | ||
! written name | ! | written name | ||
! spoken name | ! | spoken name | ||
|- | |- | ||
| style="text-align:center;" | zo | | style="text-align:center;" | zo | ||
| Line 652: | Line 712: | ||
| style="text-align:center;" | C upmajor or C dot up | | style="text-align:center;" | C upmajor or C dot up | ||
|} | |} | ||
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]]. | |||
=Compositions= | |||
[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|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|Prent Rodgers]] | |||
[http://www.archive.org/details/TrioInOrwell Trio in Orwell] [http://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3 play] by [[Gene_Ward_Smith|Gene Ward Smith]] | |||
[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|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|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|Chris Vaisvil]] | |||
[https://soundcloud.com/cam-taylor-2-1/mothers mothers] by Cam Taylor | |||
[[Category:amity]] | [[Category:amity]] | ||
[[Category:athene]] | [[Category:athene]] | ||
| Line 762: | Line 746: | ||
[[Category:semicomma]] | [[Category:semicomma]] | ||
[[Category:zeta]] | [[Category:zeta]] | ||
Revision as of 20:08, 16 October 2018
Other languages: Deutsch
Theory
The famous 53 equal division divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a 5-limit system, a fact apparently first noted by Isaac Newton, tempering out the schisma, 32805/32768, the kleisma, 15625/15552, the amity comma, 1600000/1594323 and the semicomma, 2109375/2097152. 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, 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 comma 176/175. In the 13-limit, it tempers out 169/168 and 245/243, and gives the optimal patent val for athene temperament. It is the eighth zeta integral edo and the 16th prime edo, following 47edo and coming before 59edo.
53EDO has also found a certain dissemination as an EDO tuning for Arabic/Turkish/Persian music.
It can also be treated as a no-elevens, no-seventeens tuning, on which it is consistent all the way up to the 21-limit.
Linear temperaments
List of edo-distinct 53et rank two temperaments
Just 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 pure 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.
Intervals
| degree | solfege | cents | approximate ratios | ups and downs notation | generator for | ||
|---|---|---|---|---|---|---|---|
| 0 | do | 0.00 | 1/1 | P1 | unison | D | |
| 1 | di | 22.64 | 81/80, 64/63, 50/49 | ^1 | up unison | D^ | |
| 2 | daw | 45.28 | 49/48, 36/35, 33/32, 128/125 | ^^1,
vvm2 |
double-up unison,
double-down minor 2nd |
D^^,
Ebvv |
Quartonic |
| 3 | ro | 67.92 | 27/26, 26/25, 25/24, 22/21 | vm2 | downminor 2nd | Ebv | |
| 4 | rih | 90.57 | 21/20, 256/243 | m2 | minor 2nd | Eb | |
| 5 | ra | 113.21 | 16/15, 15/14 | ^m2 | upminor 2nd | Eb^ | |
| 6 | ru | 135.85 | 14/13, 13/12, 27/25 | v~2 | downmid 2nd | Eb^^ | |
| 7 | ruh | 158.49 | 12/11, 11/10, 800/729 | ^~2 | upmid 2nd | Evv | Hemikleismic |
| 8 | reh | 181.13 | 10/9 | vM2 | downmajor 2nd | Ev | |
| 9 | re | 203.77 | 9/8 | M2 | major 2nd | E | |
| 10 | ri | 226.42 | 8/7, 256/225 | ^M2 | upmajor 2nd | E^ | |
| 11 | raw | 249.06 | 15/13, 144/125 | ^^M2,
vvm3 |
double-up major 2nd,
double-down minor 3rd |
E^^,
Fvv |
Hemischis |
| 12 | ma | 271.70 | 7/6, 75/64 | vm3 | downminor 3rd | Fv | Orwell |
| 13 | meh | 294.34 | 13/11, 32/27 | m3 | minor 3rd | F | |
| 14 | me | 316.98 | 6/5 | ^m3 | upminor 3rd | F^ | Hanson/Catakleismic |
| 15 | mu | 339.62 | 11/9, 243/200 | v~3 | downmid 3rd | F^^ | Amity/Hitchcock |
| 16 | muh | 362.26 | 16/13, 100/81 | ^~3 | upmid 3rd | F#vv | |
| 17 | mi | 384.91 | 5/4 | vM3 | downmajor 3rd | F#v | |
| 18 | maa | 407.55 | 81/64 | M3 | major 3rd | F# | |
| 19 | mo | 430.19 | 9/7, 14/11 | ^M3 | upmajor 3rd | F#^ | Hamity |
| 20 | maw | 452.83 | 13/10, 125/96 | ^^M3,
vv4 |
double-up major 3rd,
double-down 4th |
F#^^,
Gvv |
|
| 21 | fe | 475.47 | 21/16, 675/512, 320/243 | v4 | down 4th | Gv | Vulture/Buzzard |
| 22 | fa | 498.11 | 4/3 | P4 | perfect 4th | G | |
| 23 | fih | 520.75 | 27/20 | ^4 | up 4th | G^ | |
| 24 | fu | 543.40 | 11/8, 15/11 | ^^4 | double-up 4th | G^^ | |
| 25 | fuh | 566.04 | 18/13 | vvA4,
vd5 |
double-down aug 4th,
downdim 5th |
G#vv,
Abv |
Tricot |
| 26 | fi | 588.68 | 7/5, 45/32 | vA4,
d5 |
downaug 4th,
dim 5th |
G#v,
Ab |
|
| 27 | se | 611.32 | 10/7, 64/45 | A4,
^d5 |
aug 4th,
updim 5th |
G#,
Ab^ |
|
| 28 | suh | 633.96 | 13/9 | ^A4,
^^d5 |
upaug 4th,
double-up dim 5th |
G#^,
Ab^^ |
|
| 29 | su | 656.60 | 16/11, 22/15 | vv5 | double-down 5th | Avv | |
| 30 | sih | 679.25 | 40/27 | v5 | down 5th | Av | |
| 31 | sol | 701.89 | 3/2 | P5 | perfect 5th | A | Helmholtz/Garibaldi |
| 32 | si | 724.53 | 32/21, 243/160, 1024/675 | ^5 | up 5th | A^ | |
| 33 | saw | 747.17 | 20/13, 192/125 | ^^5,
vvm6 |
double-up 5th,
double-down minor 6th |
A^^,
Bbvv |
|
| 34 | lo | 769.81 | 14/9, 25/16, 11/7 | vm6 | downminor 6th | Bbv | |
| 35 | leh | 792.45 | 128/81 | m6 | minor 6th | Bb | |
| 36 | le | 815.09 | 8/5 | ^m6 | upminor 6th | Bb^ | |
| 37 | lu | 837.74 | 13/8, 81/50 | v~6 | downmid 6th | Bb^^ | |
| 38 | luh | 860.38 | 18/11, 400/243 | ^~6 | upmid 6th | Bvv | |
| 39 | la | 883.02 | 5/3 | vM6 | downmajor 6th | Bv | |
| 40 | laa | 905.66 | 22/13, 27/16 | M6 | major 6th | B | |
| 41 | lo | 928.30 | 12/7 | ^M6 | upmajor 6th | B^ | |
| 42 | law | 950.94 | 26/15, 125/72 | ^^M6
vvm7 |
double-up major 6th,
double-down minor 7th |
B^^,
Cvv |
|
| 43 | ta | 973.58 | 7/4 | vm7 | downminor 7th | Cv | |
| 44 | teh | 996.23 | 16/9 | m7 | minor 7th | C | |
| 45 | te | 1018.87 | 9/5 | ^m7 | upminor 7th | C^ | |
| 46 | tu | 1041.51 | 11/6, 20/11, 729/400 | v~7 | downmid 7th | C^^ | |
| 47 | tuh | 1064.15 | 13/7, 24/13, 50/27 | ^~7 | upmid 7th | C#vv | |
| 48 | ti | 1086.79 | 15/8 | vM7 | downmajor 7th | C#v | |
| 49 | tih | 1109.43 | 40/21, 243/128 | M7 | major 7th | C# | |
| 50 | to | 1132.08 | 48/25, 27/14 | ^M7 | upmajor 7th | C#^ | |
| 51 | taw | 1154.72 | 125/64 | ^^M7,
vv8 |
double-up major 7th,
double-down 8ve |
C#^^,
Dvv |
|
| 52 | da | 1177.36 | 160/81 | v8 | down 8ve | Dv | |
| 53 | do | 1200 | 2/1 | P8 | perfect 8ve | D | |
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}, b < -1 | 32/27, 16/9 |
| upminor | gu | {a, b, -1} | 6/5, 9/5 |
| downmid | lova | {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}, 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. Here are the zo, gu, lova, 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 Ebv G | C.vm | C downminor |
| gu | 10:12:15 | 0-14-31 | C Eb^ G | C.^m | C upminor |
| lova | 18:22:27 | 0-15-31 | C Eb^^ G | C.v~ | C downmid |
| yo | 4:5:6 | 0-17-31 | C Ev G | C.v | C downmajor or C dot down |
| ru | 14:18:27 | 0-19-31 | C E^ G | C.^ | C upmajor or C dot up |
For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.
Compositions
Bach WTC1 Prelude 1 in 53 by Bach and Mykhaylo Khramov
Bach WTC1 Fugue 1 in 53 by Bach and Mykhaylo Khramov
Whisper Song in 53EDO play by Prent Rodgers
Trio in Orwell play by Gene Ward Smith
Desert Prayer by Aaron Krister Johnson
Whisper Song in 53 EDO by Prent Rodgers
Elf Dine on Ho Ho play and Spun play by Andrew Heathwaite
The Fallen of Kleismic15play by Chris Vaisvil
mothers by Cam Taylor