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{{Infobox ET}}
{{Infobox ET}}
The 52 equal division of 3, the tritave, divides it into 52 equal parts of 36.576 cents each, corresponding to 32.808 edo. It is something of a curiosity as it really needs to be considered as a 29-limit no-twos system. While not super-accurate, it gets the entire no-twos 29-limit to within 18 cents. It is distinctly flat, in the sense that 5, 7, 11, 13, 17, 19, 23 and 29 are all flat, so using something other than pure-threes tuning might be advisable. It is contorted in the 11-limit, so that it tempers out the same commas as [[26edt|26edt]] in the 11-limit and [[13edt|13edt]] in the 7-limit. Other commas it tempers out includes 121/119, 209/207, 247/245, 275/273, 299/297, 325/323, 345/343, 363/361, 377/375, 437/435, 495/493, 627/625, 665/663, 667/665, 847/845, 1127/1125, 1311/1309 and 1617/1615. It is the eleventh [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos zeta peak edt]].
{{ED intro}}
 
It is something of a curiosity as it really needs to be considered as a 29-limit no-twos system. While not exceptionally accurate, it gets the entire no-twos 29-limit to within 18 cents. It is distinctly flat, in the sense that 5, 7, 11, 13, 17, 19, 23 and 29 are all flat, so using something other than pure-threes tuning might be advisable. It is contorted in the 7-limit, so that it tempers out the same commas as [[13edt]]. Other commas it tempers out includes 121/119, 209/207, 247/245, 275/273, 299/297, 325/323, 345/343, 363/361, 377/375, 437/435, 495/493, 627/625, 665/663, 667/665, 847/845, 1127/1125, 1311/1309 and 1617/1615. It is the eleventh [[the Riemann zeta function and tuning#Removing primes|no-twos zeta peak edt]].


== Harmonics ==
== Harmonics ==
While 52edt does not improve much on [[26edt]]'s mapping of the odd primes, it is notable for mapping 11, 13, 19, and 29 ''slightly'' better than 26edt, giving them (despite being nearly half a step off in the case of 11) a flat tendency in keeping with all the other odd primes, which allows 52edt to be the first edt that is [[consistent]] to the no-twos 29-[[throdd limit]], which is a record unbeaten until [[144edt]].
While 52edt does not improve much on [[26edt]]'s mapping of the odd primes, it still maps 11, 13, 19, and 29 ''slightly'' better than 26edt, which is notable for giving them (despite being nearly half a step off in the case of 11) a flat tendency in keeping with all the other odd primes, which allows 52edt to be the first edt that is [[consistent]] to the no-twos 29-[[throdd limit]], which is a record unbeaten until [[144edt]].
{{Harmonics in equal|52|3|1|intervals = prime|columns = 9}}
{{Harmonics in equal|52|3|1|intervals=prime|columns=9}}
{{Harmonics in equal|52|3|1|start = 12|collapsed = 1|intervals = odd}}
{{Harmonics in equal|52|3|1|start=12|collapsed=1|intervals=odd}}


== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Steps
! Steps
! | Cents
! Cents
!Hekts
! Hekts
! | BP nonatonic degree
! BP nonatonic degree
! | Diatonic degree
! Diatonic degree
! | Corresponding JI intervals
! Corresponding JI intervals
! | Comments
! Comments
! | Generator for...
! Generator for...
|-
|-
| | 1
| 1
| | 36.6
| 36.6
|25
| 25
| | Qa1/3d2
| Qa1/3d2
| | Sa1
| Sa1
| | 50/49~33/32~49/48
| 50/49~33/32~49/48
| |  
|  
| |  
|  
|-
|-
| | 2
| 2
| | 73.15
| 73.15
|50
| 50
| | Sa1/sd2
| Sa1/sd2
| | A1/dd2
| A1/dd2
| | 25/24~28/27~22/21~27/26~24/23~21/20~29/28
| 25/24~28/27~22/21~27/26~24/23~21/20~29/28
| |  
|  
| |  
|  
|-
|-
| | 3
| 3
| | 109.7
| 109.7
|75
| 75
| | 3A1/qd2
| 3A1/qd2
| | A+1/d-2
| A+1/d-2
| | 15/14~16/15~29/27~121/112
| 15/14~16/15~29/27~121/112
| |  
|  
| |  
|  
|-
|-
| | 4
| 4
| | 146.3
| 146.3
|100
| 100
| | A1/m2
| A1/m2
| | AA1/sm2
| AA1/sm2
| | 27/25~25/23~49/45~13/12~14/13~11/10~169/162
| 27/25~25/23~49/45~13/12~14/13~11/10~169/162
| |  
|  
| |  
|  
|-
|-
| | 5
| 5
| | 182.9
| 182.9
|125
| 125
| | Sm2
| Sm2
| | sm+2
| sm+2
| | 10/9
| 10/9
| |  
|  
| |  
|  
|-
|-
| | 6
| 6
| | 219.5
| 219.5
|150
| 150
| | N2
| N2
| | m2
| m2
| | 9/8~8/7~44/39
| 9/8~8/7~44/39
| |  
|  
| |  
|  
|-
|-
| | 7
| 7
| | 256.0
| 256.0
|175
| 175
| | sM2
| sM2
| | N2
| N2
| | 147/128~7/6
| 147/128~7/6
| |  
|  
| |  
|  
|-
|-
| | 8
| 8
| | 292.6
| 292.6
|200
| 200
| | M2/d3
| M2/d3
| | M2
| M2
| | 32/27~25/21~13/11~27/23
| 32/27~25/21~13/11~27/23
| |  
|  
| |  
|  
|-
|-
| | 9
| 9
| | 329.2
| 329.2
|225
| 225
| | Qa2/3d3
| Qa2/3d3
| | SM-2/d-2
| SM-2/d-2
| | 6/5
| 6/5
| | 11/9-
| 11/9-
| |  
|  
|-
|-
| | 10
| 10
| | 365.8
| 365.8
|250
| 250
| | Sa2/sd3
| Sa2/sd3
| | SM2/dd3
| SM2/dd3
| | 5/4~16/13
| 5/4~16/13
| | 11/9+
| 11/9+
| |  
|  
|-
|-
| | 11
| 11
| | 402.3
| 402.3
|275
| 275
| | 3A3/qd3
| 3A3/qd3
| | SM+2
| SM+2
| | 81/64~63/50~33/26~23/18
| 81/64~63/50~33/26~23/18
| |  
|  
| |  
|  
|-
|-
| | 12
| 12
| | 438.9
| 438.9
|300
| 300
| | A2/P3/d4
| A2/P3/d4
| | AA2/sm3
| AA2/sm3
| | 32/25~9/7~14/11~104/81~13/10
| 32/25~9/7~14/11~104/81~13/10
| |  
|  
| |  
|  
|-
|-
| | 13
| 13
| | 475.5
| 475.5
|325
| 325
| | Qa3/3d4
| Qa3/3d4
| | sm+3
| sm+3
| | 21/16~98/75
| 21/16~98/75
| |  
|  
| |  
|  
|-
|-
| | 14
| 14
| | 512.1
| 512.1
|350
| 350
| | Sa3/sd4
| Sa3/sd4
| | m3
| m3
| | 4/3~27/20~162/121
| 4/3~27/20~162/121
| |  
|  
| |  
|  
|-
|-
| | 15
| 15
| | 548.6
| 548.6
|375
| 375
| | 3A3/qd4
| 3A3/qd4
| | N3
| N3
| | 11/8~243/169
| 11/8~243/169
| | 18/13-
| 18/13-
| |  
|  
|-
|-
| | 16
| 16
| | 585.2
| 585.2
|400
| 400
| | A3/m4/d5
| A3/m4/d5
| | M3
| M3
| | 7/5~25/18~112/81~88/63~32/23~29/21
| 7/5~25/18~112/81~88/63~32/23~29/21
| | 18/13+
| 18/13+
| |  
|  
|-
|-
| | 17
| 17
| | 621.8
| 621.8
|425
| 425
| | Sm4/3d5
| Sm4/3d5
| | SM-3
| SM-3
| | 10/7~36/25~81/56~63/44~23/16~42/29
| 10/7~36/25~81/56~63/44~23/16~42/29
| | 13/9-
| 13/9-
| |  
|  
|-
|-
| | 18
| 18
| | 658.4
| 658.4
|450
| 450
| | N4/sd5
| N4/sd5
| | SM3/dd4
| SM3/dd4
| | 16/11~338/243
| 16/11~338/243
| | 13/9+
| 13/9+
| |  
|  
|-
|-
| | 19
| 19
| | 694.95
| 694.95
|475
| 475
| | sM4/qd5
| sM4/qd5
| | SM+3/d-4
| SM+3/d-4
| | 3/2~40/27~121/81
| 3/2~40/27~121/81
| |  
|  
| |  
|  
|-
|-
| | 20
| 20
| | 731.5
| 731.5
|500
| 500
| | M4/m5
| M4/m5
| | AA3/d4
| AA3/d4
| | 32/21~75/49
| 32/21~75/49
| |  
|  
| |  
|  
|-
|-
| | 21
| 21
| | 768.1
| 768.1
|525
| 525
| | Qa4/Sm5
| Qa4/Sm5
| | d+4
| d+4
| | 25/16~14/9~11/7~81/52
| 25/16~14/9~11/7~81/52
| |  
|  
| |  
|  
|-
|-
| | 22
| 22
| | 804.7
| 804.7
|550
| 550
| | Sa4/N5
| Sa4/N5
| | P4
| P4
| | 8/5~36/23
| 8/5~36/23
| |  
|  
| |  
|  
|-
|-
| | 23
| 23
| | 841.25
| 841.25
|575
| 575
| | 3A4/sM5
| 3A4/sM5
| | A-4
| A-4
| | 13/8
| 13/8
| |  
|  
| |  
|  
|-
|-
| | 24
| 24
| | 877.8
| 877.8
|600
| 600
| | A4/M5/d6
| A4/M5/d6
| | A4
| A4
| | 5/3
| 5/3
| |  
|  
| |  
|  
|-
|-
| | 25
| 25
| | 914.4
| 914.4
|625
| 625
| | Qa5/3d6
| Qa5/3d6
| | A+4
| A+4
| | 27/16~42/25~22/13~46/27
| 27/16~42/25~22/13~46/27
| |  
|  
| |  
|  
|-
|-
| | 26
| 26
| | 951
| 951
|650
| 650
| | Sa5/sd6
| Sa5/sd6
| | AA4/dd5
| AA4/dd5
| | 125/72
| 125/72
| |  
|  
| |  
|  
|-
|-
| | 27
| 27
| | 987.55
| 987.55
|675
| 675
| | 3A5/qd6
| 3A5/qd6
| | d-5
| d-5
| | 16/9~8/7~39/22~75/46
| 16/9~8/7~39/22~75/46
| |  
|  
| |  
|  
|-
|-
| | 28
| 28
| | 1024.1
| 1024.1
|700
| 700
| | A5/m6/d7
| A5/m6/d7
| | d5
| d5
| | 9/5
| 9/5
| |  
|  
| |  
|  
|-
|-
| | 29
| 29
| | 1060.7
| 1060.7
|725
| 725
| | Sm6/3d7
| Sm6/3d7
| | d+5
| d+5
| | 50/27~46/25~90/49~24/13~13/7~20/11
| 50/27~46/25~90/49~24/13~13/7~20/11
| |  
|  
| |  
|  
|-
|-
| | 30
| 30
| | 1097.3
| 1097.3
|750
| 750
| | N6/sd7
| N6/sd7
| | P5
| P5
| | 15/8
| 15/8
| |  
|  
| |  
|  
|-
|-
| | 31
| 31
| | 1133.9
| 1133.9
|775
| 775
| | sM6/qd7
| sM6/qd7
| | A-5
| A-5
| | 48/25~27/14~21/11~52/27~23/12~40/21~56/29
| 48/25~27/14~21/11~52/27~23/12~40/21~56/29
| |  
|  
| |  
|  
|-
|-
| | 32
| 32
| | 1170.4
| 1170.4
|800
| 800
| | M6/m7
| M6/m7
| | A5/dd6
| A5/dd6
| | 49/25~64/33~96/49
| 49/25~64/33~96/49
| |  
|  
| |  
|  
|-
|-
| | 33
| 33
| | 1207.0
| 1207.0
|825
| 825
| | Qa6/Sm7
| Qa6/Sm7
| | A+5
| A+5
| | 2/1
| 2/1
| |  
|  
| |  
|  
|-
|-
| | 34
| 34
| | 1243.6
| 1243.6
|850
| 850
| | Sa6/N7
| Sa6/N7
| | AA5/sm6
| AA5/sm6
| | 33/16~100/49~49/24~729/338
| 33/16~100/49~49/24~729/338
| |  
|  
| |  
|  
|-
|-
| | 35
| 35
| | 1280.2
| 1280.2
|875
| 875
| | 3A6/sM7
| 3A6/sM7
| | sm+6
| sm+6
| | 25/12~56/27~44/21~27/13
| 25/12~56/27~44/21~27/13
| |  
|  
| |  
|  
|-
|-
| | 36
| 36
| | 1316.7
| 1316.7
|900
| 900
| | A6/M7/d8
| A6/M7/d8
| | m6
| m6
| | 15/7~32/15~58/27
| 15/7~32/15~58/27
| |  
|  
| |  
|  
|-
|-
| | 37
| 37
| | 1353.3
| 1353.3
|925
| 925
| | Qa7/3d8
| Qa7/3d8
| | N6
| N6
| | 54/25~50/23~98/45~13/6~169/81
| 54/25~50/23~98/45~13/6~169/81
| |  
|  
| |  
|  
|-
|-
| | 38
| 38
| | 1389.9
| 1389.9
|950
| 950
| | Sa7/sd8
| Sa7/sd8
| | M6
| M6
| | 20/9~9/4-
| 20/9~9/4-
| |  
|  
| |  
|  
|-
|-
| | 39
| 39
| | 1426.5
| 1426.5
|975
| 975
| | 3A7/qd8
| 3A7/qd8
| | SM-6
| SM-6
| | 9/4+~16/7
| 9/4+~16/7
| |  
|  
| |  
|  
|-
|-
| | 40
| 40
| | 1463.0
| 1463.0
|1000
| 1000
| | A7/P8/d9
| A7/P8/d9
| | SM6/dd7
| SM6/dd7
| | 147/64~7/3
| 147/64~7/3
| |  
|  
| |  
|  
|-
|-
| | 41
| 41
| | 1499.6
| 1499.6
|1025
| 1025
| | Qa8/3d9
| Qa8/3d9
| | SM+6/sm-7
| SM+6/sm-7
| | 64/27~50/21~26/11~81/23
| 64/27~50/21~26/11~81/23
| |  
|  
| |  
|  
|-
|-
| | 42
| 42
| | 1536.2
| 1536.2
|1050
| 1050
| | Sa8/sd9
| Sa8/sd9
| | AA6/sm7
| AA6/sm7
| | 12/5
| 12/5
| | 22/9-
| 22/9-
| |  
|  
|-
|-
| | 43
| 43
| | 1572.7
| 1572.7
|1075
| 1075
| | 3A8/qd9
| 3A8/qd9
| | sm-7
| sm-7
| | 5/2~32/13
| 5/2~32/13
| | 22/9+
| 22/9+
| |  
|  
|-
|-
| | 44
| 44
| | 1609.3
| 1609.3
|1100
| 1100
| | A8/m9
| A8/m9
| | m7
| m7
| | 81/32~63/25~33/13~23/9
| 81/32~63/25~33/13~23/9
| |  
|  
| |  
|  
|-
|-
| | 45
| 45
| | 1645.9
| 1645.9
|1125
| 1125
| | Sm9
| Sm9
| | N7
| N7
| | 64/45~18/7~28/11~208/81~13/5
| 64/45~18/7~28/11~208/81~13/5
| |  
|  
| |  
|  
|-
|-
| | 46
| 46
| | 1682.5
| 1682.5
|1150
| 1150
| | N9
| N9
| | M7
| M7
| | 21/8~196/75
| 21/8~196/75
| |  
|  
| |  
|  
|-
|-
| | 47
| 47
| | 1719.1
| 1719.1
|1175
| 1175
| | sM9
| sM9
| | SM-7
| SM-7
| | 8/3~27/10
| 8/3~27/10
| |  
|  
| |  
|  
|-
|-
| | 48
| 48
| | 1755.65
| 1755.65
|1200
| 1200
| | M9/d10
| M9/d10
| | SM7/dd8
| SM7/dd8
| | 69/25~135/49
| 69/25~135/49
| | 36/13-
| 36/13-
| |  
|  
|-
|-
| | 49
| 49
| | 1792.2
| 1792.2
|1225
| 1225
| | Qa9/3d10
| Qa9/3d10
| | SM+7/d-8
| SM+7/d-8
| | 14/5~25/9~224/81~176/63~64/23~58/21
| 14/5~25/9~224/81~176/63~64/23~58/21
| | 36/13+
| 36/13+
| |  
|  
|-
|-
| | 50
| 50
| | 1828.8
| 1828.8
|1250
| 1250
| | Sa9/sd10
| Sa9/sd10
| | A7/d8
| A7/d8
| | 20/7~72/25~81/28~63/22~23/8~84/29
| 20/7~72/25~81/28~63/22~23/8~84/29
| | 26/9-
| 26/9-
| |  
|  
|-
|-
| | 51
| 51
| | 1865.4
| 1865.4
|1275
| 1275
| | 3A9/qd10
| 3A9/qd10
| | P-8
| P-8
| | 147/50~32/11~338/81~144/49
| 147/50~32/11~338/81~144/49
| | 26/9+
| 26/9+
| |  
|  
|-
|-
| | 52
| 52
| | 1902.0
| 1902.0
|1300
| 1300
| | A9/P10
| A9/P10
| | P8
| P8
| | 3/1
| 3/1
| | Tritave
| Tritave
| |  
|  
|}
|}


Line 493: Line 494:
{| class="wikitable"
{| class="wikitable"
|-
|-
! | 52edt
! 52edt
! | 52edo
! 52edo
! | Discrepancy
! Discrepancy
|-
|-
| | 365.761
| 365.761
| | 369.231
| 369.231
| | -3.47
| −3.47
|-
|-
| | 512.065
| 512.065
| | 507.692
| 507.692
| | +4.373
| +4.373
|-
|-
| | 877.825
| 877.825
| | 876.923
| 876.923
| | +0.902
| +0.902
|-
|-
| | 1243.586
| 1243.586
| | 1246.154
| 1246.154
| | -2.168
| −2.168
|-
|-
| | 1389.89
| 1389.89
| | 1384.615
| 1384.615
| | +5.275
| +5.275
|-
|-
| | 1755.651
| 1755.651
| | 1753.846
| 1753.846
| | +1.805
| +1.805
|-
|-
| | 2121.411
| 2121.411
| | 2123.077
| 2123.077
| | -1.666
| −1.666
|-
|-
| | 2633.476
| 2633.476
| | 2630.769
| 2630.769
| | +2.647
| +2.647
|}
|}
…and so on
…and so on
Retrieved from "https://en.xen.wiki/w/52edt"