39edt: Difference between revisions

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{{Harmonics in equal|39|3|1|intervals=prime}}
{{Harmonics in equal|39|3|1|intervals=prime}}


==Intervals==
== Intervals ==
{| class="wikitable"
{| class="wikitable center-all right-2 right-3"
|-
|-
! |Degree
! Steps
! | Cents
! [[Cent]]s
! |Cents<br>([[octave reduction|octave-reduced]])
! [[Hekt]]s
! |Hekts
! [[4L 5s (3/1-equivalent)|Enneatonic]] degree
! |Degree of [[Bohlen-Pierce|BP]]
! Corresponding
! |Comments
3.5.7.11.13 subgroup <br>
intervals
! [[Lambda ups and downs notation|Lambda]]  
(sLsLsLsLs, <br>
J = 1/1)
! Mintaka[7]
(E macro-Phrygian)
|-
|-
| | 0
| 0
| |0
| 0
| |
| 0
| |0
| P1
| | 0
| [[1/1]]
| |1/1
| J
| E
|-
|-
| |1
| 1
| | 48.768
| 48.8
| |
| 33.3
|33.333
| SP1
| |
| [[77/75]] (+3.2c); [[65/63]] (-5.3c)
| |39th root of 3
| ^J
| ^E, vF
|-
|-
| |2
| 2
| |97.536
| 97.5
| |
| 66.7
|66.667
| sA1/sm2
| |
| [[35/33]] (-4.3c); [[81/77]] (+9.9c)
| |
| vK
| F
|-
|-
| |3
| 3
| |146.304
| 146.3
| |
| 100
|100
| A1/m2
| |1
| [[49/45]] (-1.1c); [[27/25]] (+13.1c)
| |13th root of 3
| K
| ^F, vGb
|-
|-
| | 4
| 4
| |195.072
| 195.1
| |
| 133.3
|133.333
| SA1/Sm2
| |
| [[55/49]] (-4.9c); [[39/35]] (+7.7c)
| |
| ^K
| Gb, vE#
|-
|-
| | 5
| 5
| |243.840
| 243.8
| |
| 166.7
|166.667
| sM2/sd3
| |
| [[15/13]] (-3.9c); [[63/55]] (+8.7c)
| |
| vK#, vLb
| ^Gb, E#
|-
|-
| |6
| 6
| | 292.608
| 292.6
| |
| 200
|200
| M2/d3
| |2
| [[13/11]] (+3.4c); [[25/21]] (-9.2c)
| |
| K#, Lb
| vF#, ^E#
|-
|-
| |7
| 7
| | 341.377
| 341.4
| |
| 233.3
|233.333
| SM2/Sd3
| |
| [[11/9]] (-6.0c)
| |
| ^K#, ^Lb
| F#
|-
|-
| |8
| 8
| |390.145
| 390.1
| |
| 266.7
|266.667
| sA2/sP3/sd4
| |
| [[49/39]] (-5.0c); [[81/65]] (+9.2c)
| |
| vL
| ^F#, vG
|-
|-
| | 9
| 9
| |438.913
| 438.9
| |
| 300
| 300
| |3
| A2/P3/d4
| |
| [[9/7]] (+3.8c), [[35/27]] (-10.3c)
| L
| G
|-
|-
| |10
| 10
| | 487.681
| 487.7
| |
| 333.3
|333.333
| SA2/SP3/Sd4
| |
| [[65/49]] (-1.5c), [[33/25]] (+7.0c)
| |
| ^L
| ^G, vAb
|-
|-
| | 11
| 11
| |536.449
| 536.4
| |
| 366.7
| 366.667
| sA3/sm4/sd5
| |
| [[15/11]] (-0.5c)
| |
| vM
| Ab
|-
|-
| |12
| 12
| |585.217
| 585.2
| |
| 400
| 400
| |4
| A3/m4/d5
| |
| [[7/5]] (+2.7c)
| M
| ^Ab, Fx
|-
|-
| |13
| 13
| | 633.985
| 634.0
| |
| 433.3
|433.333
| SA3/Sm4/Sd5
| |
| [[13/9]] (-2.6c)
| |cube root of 3
| ^M
| vG#
|-
|-
| |14
| 14
| |682.753
| 682.7
| |
| 466.7
|466.667
| sM4/sm5
| |
| [[49/33]] (-1.6c), [[81/55]] (+12.6c)
| |
| vM#, vNb
| G#
|-
|-
| |15
| 15
| |731.521
| 731.5
| |
| 500
|500
| M4/m5
| |5
| [[75/49]] (-5.4c), [[117/77]] (+7.2c)
| |  
| M#, Nb
| ^G#, vA
|-
|-
| | 16
| 16
| |780.289
| 780.3
| |
| 533.3
| 533.333
| SM4/Sm5
| |
| [[11/7]] (-2.2c), [[39/25]] (+10.4c)
| |
| ^M#, ^Nb
| A
|-
|-
| | 17
| 17
| |829.057
| 829.0
| |
| 566.7
|566.667
| sA4/sM5
| |
| [[21/13]] (-1.2c)
| |
| vN
| ^A, vBb
|-
|-
| | 18
| 18
| | 877.825
| 877.8
| |
| 600
|600
| A4/M5
| |6
| [[5/3]] (-6.5c), [[81/49]] (+9.5c)
| |
| N
| Bb
|-
|-
| | 19
| 19
| |926.593
| 926.6
| |  
| 633.3
|633.333
| SA4/SM5
| |
| [[77/45]] (-3.3c)
| |
| ^N
| ^Bb, vCb, Gx
|-
|-
| | 20
| 20
| |975.362
| 975.3
| |  
| 666.7
|666.667
| sA5/sm6/sd7
| |
| [[135/77]] (+3.3c)
| |
| vO
| vA#, Cb
|-
|-
| |21
| 21
| |1024.13
| 1024.1
| |
| 700
|700
| A5/m6/d7
| |7
| [[9/5]] (+6.5c), [[49/27]] (-9.5c)
| |
| O
| A#, ^Cb
|-
|-
| | 22
| 22
| |1072.898
| 1072.9
| |
| 733.3
|733.333
| SA5/Sm6/Sd7
| |
| [[13/7]] (+1.2c)
| |
| ^O
| ^A#, vB
|-
|-
| | 23
| 23
| | 1121.666
| 1121.6
| |
| 766.7
|766.667
| sM6/sm7
| |
| [[21/11]] (+2.2c), [[25/13]] (-10.4c)
| |
| vO#, vPb
| B
|-
|-
| |24
| 24
| |1170.434
| 1170.4
| |
| 800
|800
| M6/m7
| |8
| [[49/25]] (+5.4c), [[77/39]] (-7.2c)
| |
| O#, Pb
| ^B, vC
|-
|-
| | 25
| 25
| | 1219.202
| 1219.2
| |19.202
| 833.3
|833.333
| SM6/Sm7
| |
| [[99/49]] (+1.6c), [[55/27]] (-12.6c)
| |
| ^O#, ^Pb
| C
|-
|-
| |26
| 26
| | 1267.97
| 1267.9
| |67.97
| 866.7
| 866.667
| sA6/sM7/sd8
| |
| [[27/13]] (+2.6c)
| |
| vP
| ^C, vDb
|-
|-
| |27
| 27
| |1316.738
| 1316.7
| |116.738
| 900
|900
| A6/M7/d8
| |9
| [[15/7]] (-2.7c)
| |
| P
| Db, vB#
|-
|-
| |28
| 28
| |1365.506
| 1365.5
| |165.506
| 933.3
|933.333
| SA6/SM7/Sd8
| |
| [[11/5]] (+0.5c)
| |
| ^P
| ^Db, B#
|-
|-
| |29
| 29
| |1414.274
| 1414.2
| | 214.274
| 966.7
|966.667
| sP8/sd9
| |
| [[147/65]] (+1.5c), [[25/11]] (-7.0c)
| |
| vQ
| vC#, ^B#
|-
|-
| |30
| 30
| |1463.042
| 1463.0
| |263.042
| 1000
|1000
| P8/d9
| |10
| [[7/3]] (-3.8c), [[81/35]] (+10.3c)
| |
| Q
| C#
|-
|-
| | 31
| 31
| |1511.81
| 1511.8
| |311.81
| 1033.3
| 1033.333
| SP8/Sd9
| |
| [[117/49]] (+5.0c); [[65/27]] (-9.2c)
| |
| ^Q
| ^C#, vD
|-
|-
| | 32
| 32
| |1560.578
| 1560.5
| |360.578
| 1066.7
|1066.667
| sA8/sm9
| |
| [[27/11]] (+6.0c)
| |
| vQ#, vRb
| D
|-
|-
| |33
| 33
| |1609.347
| 1609.3
| |409.347
| 1100
|1100
| A8/m9
| | 11
| [[33/13]] (-3.4c), [[63/25]] (+9.2c)
| |
| Q#, Rb
| ^D, vEb
|-
|-
| |34
| 34
| |1658.115
| 1658.1
| | 458.115
| 1133.3
|1133.333
| SA8/Sm9
| |
| [[13/5]] (+3.9c); [[55/21]] (-8.7c)
| |
| ^Q#, ^Rb
| Eb
|-
|-
| |35
| 35
| |1706.883
| 1706.9
| | 506.883
| 1166.7
|1166.667
| sM9/sd10
| |
| [[147/55]] (+4.9c); [[35/13]] (-7.7c)
| |
| vR
| ^Eb, vFb, Cx
|-
|-
| |36
| 36
| |1755.651
| 1755.7
| |555.651
| 1200
|1200
| M9/d10
| | 12
| [[135/49]] (+1.1c); [[25/9]] (-13.1c)
| |
| R
| vD#, Fb
|-
|-
| |37
| 37
| |1804.419
| 1804.5
| | 604.419
| 1233.3
|1233.333
| SM9/Sd10
| |
| [[99/35]] (+4.3c); [[77/27]] (-9.9c)
| |
| ^R
| D#, ^Fb, Dx
|-
|-
| |38
| 38
| |1853.187
| 1853.2
| | 653.187
| 1266.7
|1266.667
| sA9/sP10
| |
| [[225/77]] (-3.2c); [[189/65]] (+5.3c)
| |
| vJ
| ^D#
|-
|-
| |39
| 39
| |1901.955
| 1902.0
| |701.955
| 1300
|1300
| A9/P10
| | 13
| [[3/1]]
| | 3/1 (tritave)
| J
|-
| E
| | 40
| |1950.723
| |750.723
| 1333.333
| |
| |
|-
| |41
| |1999.491
| |799.491
|1366.667
| |
| |
|-
| | 42
| |2048.259
| | 848.259
| 1400
| |14
| |
|-
| |43
| |2097.027
| |897.027
| 1433.333
| |
| |
|-
| |44
| |2145.795
| | 945.795
|1466.667
| |
| |
|-
| |45
| |2194.563
| |994.563
|1500
| |15
| |
|-
| |46
| |2243.332
| |1043.332
|1533.333
| |
| |
|-
| |47
| |2292.100
| |1092.100
| 1566.667
| |
| |
|-
| |48
| |2340.868
| |1140.868
|1600
| |16
| |
|-
| |49
| |2389.636
| |1189.636
| 1633.333
| |
| |
|-
| | 50
| |2438.404
| | 38.404
|1666.667
| |
| |
|-
| |51
| | 2487.172
| |87.172
|1700
| |17
| |
|-
| | 52
| |2535.940
| | 135.940
|1733.333
| |
| |
|-
| | 53
| |2584.708
| | 184.708
|1766.667
| |
| |
|-
| |54
| |2633.476
| |233.476
|1800
| |18
| |
|-
| | 55
| |2682.244
| | 282.244
|1833.333
| |
| |
|-
| | 56
| |2731.012
| | 331.012
|1866.667
| |
| |
|-
| |57
| |2779.78
| |379.78
|1900
| |19
| |
|-
| | 58
| |2828.548
| |428.548
|1933.333
| |
| |
|-
| | 59
| |2877.317
| |477.317
|1966.667
| |
| |
|-
| |60
| |2926.085
| |526.085
|2000
| |20
| |
|-
| | 61
| |2974.853
| |574.853
|2033.333
| |
| |
|-
| | 62
| |3023.621
| |623.621
|2066.667
| |
| |
|-
| |63
| |3072.389
| |672.389
|2100
| |21
| |
|-
| | 64
| |3121.157
| |721.157
| 2133.333
| |
| |
|-
| |65
| |3169.925
| |769.925
|2166.667
| |
| |
|-
| | 66
| |3218.693
| | 818.693
| 2200
| |22
| |
|-
| |67
| |3267.461
| |867.461
|2233.333
| |
| |
|-
| |68
| |3316.229
| |916.229
|2266.667
| |
| |
|-
| |69
| | 3364.997
| |964.997
|2300
| |23
| |
|-
| |70
| |3413.765
| |1013.765
| 2333.333
| |
| |
|-
| | 71
| |3462.533
| | 1062.533
|2366.667
| |
| |
|-
| |72
| |3511.302
| |1111.302
|2400
| |24
| |
|-
| |73
| |3560.07
| |1160.07
|2433.333
| |
| |
|-
| |74
| |3608.838
| |8.838
|2466.667
| |
| |
|-
| |75
| |3657.606
| |57.606
|2500
| |25
| |
|-
| |76
| |3706.374
| |106.374
|2533.333
| |
| |
|-
| |77
| |3755.142
| |155.142
|2566.667
| |
| |
|-
| |78
| |3803.91
| |203.91
|2600
| |26
| |9/1
|}
|}

Revision as of 23:50, 5 October 2024

← 38edt 39edt 40edt →
Prime factorization 3 × 13
Step size 48.7681 ¢ 
Octave 25\39edt (1219.2 ¢)
Consistency limit 3
Distinct consistency limit 3

39 equal divisions of the tritave (39edt) is the nonoctave tuning system derived by dividing the tritave (3/1) into 39 equal steps of approximately 48.7 cents each, or the 39th root of 3. It is also known as the Triple Bohlen-Pierce scale (Triple BP), since it divides each step of the equal-tempered Bohlen-Pierce scale (13edt) into three equal parts.

39edt can be described as approximately 24.606edo. This implies that each step of 39edt can be approximated by 5 steps of 123edo. 39edt contains within it a close approximation of 4ed11/5: every seventh step of 39edt equates to a step of 4ed11/5.

It is a strong no-twos 13-limit system, a fact first noted by Paul Erlich, and like 26edt and 52edt, it is a multiple of 13edt and so contains the Bohlen-Pierce scale. It is contorted in the 7-limit, tempering out the same BP commas 245/243 and 3125/3087 as 13edt. In the 11-limit it tempers out 1331/1323 and in the 13-limit 275/273, 847/845 and 1575/1573. It is related to the 49f&172f temperament tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [⟨1 0 0 0 0 0], ⟨0 39 57 69 85 91]]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth no-twos zeta peak edt.

Harmonics

Approximation of prime harmonics in 39edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +19.2 +0.0 -6.5 -3.8 -6.0 -2.6 +20.6 +23.1 -15.0 +22.6 +4.7
Relative (%) +39.4 +0.0 -13.4 -7.9 -12.4 -5.4 +42.3 +47.4 -30.8 +46.3 +9.6
Steps
(reduced) 		25
(25) 		39
(0) 		57
(18) 		69
(30) 		85
(7) 		91
(13) 		101
(23) 		105
(27) 		111
(33) 		120
(3) 		122
(5)

Intervals

Steps Cents Hekts Enneatonic degree Corresponding

3.5.7.11.13 subgroup
intervals

Lambda

(sLsLsLsLs,
J = 1/1)

Mintaka[7]

(E macro-Phrygian)

0 0 0 P1 1/1 J E
1 48.8 33.3 SP1 77/75 (+3.2c); 65/63 (-5.3c) ^J ^E, vF
2 97.5 66.7 sA1/sm2 35/33 (-4.3c); 81/77 (+9.9c) vK F
3 146.3 100 A1/m2 49/45 (-1.1c); 27/25 (+13.1c) K ^F, vGb
4 195.1 133.3 SA1/Sm2 55/49 (-4.9c); 39/35 (+7.7c) ^K Gb, vE#
5 243.8 166.7 sM2/sd3 15/13 (-3.9c); 63/55 (+8.7c) vK#, vLb ^Gb, E#
6 292.6 200 M2/d3 13/11 (+3.4c); 25/21 (-9.2c) K#, Lb vF#, ^E#
7 341.4 233.3 SM2/Sd3 11/9 (-6.0c) ^K#, ^Lb F#
8 390.1 266.7 sA2/sP3/sd4 49/39 (-5.0c); 81/65 (+9.2c) vL ^F#, vG
9 438.9 300 A2/P3/d4 9/7 (+3.8c), 35/27 (-10.3c) L G
10 487.7 333.3 SA2/SP3/Sd4 65/49 (-1.5c), 33/25 (+7.0c) ^L ^G, vAb
11 536.4 366.7 sA3/sm4/sd5 15/11 (-0.5c) vM Ab
12 585.2 400 A3/m4/d5 7/5 (+2.7c) M ^Ab, Fx
13 634.0 433.3 SA3/Sm4/Sd5 13/9 (-2.6c) ^M vG#
14 682.7 466.7 sM4/sm5 49/33 (-1.6c), 81/55 (+12.6c) vM#, vNb G#
15 731.5 500 M4/m5 75/49 (-5.4c), 117/77 (+7.2c) M#, Nb ^G#, vA
16 780.3 533.3 SM4/Sm5 11/7 (-2.2c), 39/25 (+10.4c) ^M#, ^Nb A
17 829.0 566.7 sA4/sM5 21/13 (-1.2c) vN ^A, vBb
18 877.8 600 A4/M5 5/3 (-6.5c), 81/49 (+9.5c) N Bb
19 926.6 633.3 SA4/SM5 77/45 (-3.3c) ^N ^Bb, vCb, Gx
20 975.3 666.7 sA5/sm6/sd7 135/77 (+3.3c) vO vA#, Cb
21 1024.1 700 A5/m6/d7 9/5 (+6.5c), 49/27 (-9.5c) O A#, ^Cb
22 1072.9 733.3 SA5/Sm6/Sd7 13/7 (+1.2c) ^O ^A#, vB
23 1121.6 766.7 sM6/sm7 21/11 (+2.2c), 25/13 (-10.4c) vO#, vPb B
24 1170.4 800 M6/m7 49/25 (+5.4c), 77/39 (-7.2c) O#, Pb ^B, vC
25 1219.2 833.3 SM6/Sm7 99/49 (+1.6c), 55/27 (-12.6c) ^O#, ^Pb C
26 1267.9 866.7 sA6/sM7/sd8 27/13 (+2.6c) vP ^C, vDb
27 1316.7 900 A6/M7/d8 15/7 (-2.7c) P Db, vB#
28 1365.5 933.3 SA6/SM7/Sd8 11/5 (+0.5c) ^P ^Db, B#
29 1414.2 966.7 sP8/sd9 147/65 (+1.5c), 25/11 (-7.0c) vQ vC#, ^B#
30 1463.0 1000 P8/d9 7/3 (-3.8c), 81/35 (+10.3c) Q C#
31 1511.8 1033.3 SP8/Sd9 117/49 (+5.0c); 65/27 (-9.2c) ^Q ^C#, vD
32 1560.5 1066.7 sA8/sm9 27/11 (+6.0c) vQ#, vRb D
33 1609.3 1100 A8/m9 33/13 (-3.4c), 63/25 (+9.2c) Q#, Rb ^D, vEb
34 1658.1 1133.3 SA8/Sm9 13/5 (+3.9c); 55/21 (-8.7c) ^Q#, ^Rb Eb
35 1706.9 1166.7 sM9/sd10 147/55 (+4.9c); 35/13 (-7.7c) vR ^Eb, vFb, Cx
36 1755.7 1200 M9/d10 135/49 (+1.1c); 25/9 (-13.1c) R vD#, Fb
37 1804.5 1233.3 SM9/Sd10 99/35 (+4.3c); 77/27 (-9.9c) ^R D#, ^Fb, Dx
38 1853.2 1266.7 sA9/sP10 225/77 (-3.2c); 189/65 (+5.3c) vJ ^D#
39 1902.0 1300 A9/P10 3/1 J E
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