39edt: Difference between revisions

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It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]; in fact it has a better no-twos 13-[[odd limit]] relative error than any other edt up to [[914edt]]. Like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being [[contorted]] in the no-twos 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]], [[1575/1573]], and [[847/845]]. An efficient traversal is therefore given by [[Mintra]] temperament, which in the 13-limit tempers out 275/273 and 1575/1573 alongside 245/243, and is generated by the interval of [[11/7]], which serves as a [[macrodiatonic]] "superpyth" fourth and splits the [[BPS]] generator of [[9/7]], up a tritave, in three.
It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]; in fact it has a better no-twos 13-[[odd limit]] relative error than any other edt up to [[914edt]]. Like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being [[contorted]] in the no-twos 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]], [[1575/1573]], and [[847/845]]. An efficient traversal is therefore given by [[Mintra]] temperament, which in the 13-limit tempers out 275/273 and 1575/1573 alongside 245/243, and is generated by the interval of [[11/7]], which serves as a [[macrodiatonic]] "superpyth" fourth and splits the [[BPS]] generator of [[9/7]], up a tritave, in three.


If octaves are inserted, 39edt is related to the {{nowrap|49f & 172f}} temperament in the full 13-limit, known as [[Sensamagic clan#Triboh|triboh]], tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [{{val|1 0 0 0 0 0}}, {{val|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 [[The Riemann zeta function and tuning#Removing primes|no-twos zeta peak edt]].
If octaves are inserted, 39edt is related to the {{nowrap|49f & 172f}} temperament in the full 13-limit, known as [[Sensamagic clan#Triboh|triboh]], tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [{{val|1 0 0 0 0 0}}, {{val|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 [[The Riemann zeta function and tuning#Removing primes|no-twos zeta peak edt]].
{{Harmonics in equal|39|3|1|intervals=prime|columns=12}}
{{Harmonics in equal|39|3|1|intervals=prime|columns=12}}


Line 20: Line 20:
! [[4L 5s (3/1-equivalent)|Enneatonic]] degree
! [[4L 5s (3/1-equivalent)|Enneatonic]] degree
! Corresponding<br />3.5.7.11.13 subgroup<br />intervals
! Corresponding<br />3.5.7.11.13 subgroup<br />intervals
! [[Lambda ups and downs notation|Lambda]] <br />(sLsLsLsLs,<br />J = 1/1)
! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs,<br />{{nowrap|J {{=}} 1/1}})
! Mintaka[7]<br />(E macro-Phrygian)
! Mintaka[7]<br />(E macro-Phrygian)
|-
|-
Line 35: Line 35:
| 33.3
| 33.3
| SP1
| SP1
| [[77/75]] (+3.2c); [[65/63]] (-5.3c)
| [[77/75]] (+3.2¢); [[65/63]] (&minus;5.3¢)
| ^J
| ^J
| ^E, vF
| ^E, vF
Line 43: Line 43:
| 66.7
| 66.7
| sA1/sm2
| sA1/sm2
| [[35/33]] (-4.3c); [[81/77]] (+9.9c)
| [[35/33]] (&minus;4.3¢); [[81/77]] (+9.9¢)
| vK
| vK
| F
| F
Line 51: Line 51:
| 100
| 100
| A1/m2
| A1/m2
| [[99/91]] (+0.4c); [[49/45]] (-1.1c); [[27/25]] (+13.1c)
| [[99/91]] (+0.4¢); [[49/45]] (&minus;1.1¢); [[27/25]] (+13.1¢)
| K
| K
| ^F, vGb, Dx
| ^F, vGb, Dx
Line 59: Line 59:
| 133.3
| 133.3
| SA1/Sm2
| SA1/Sm2
| [[55/49]] (-4.9c); [[91/81]] (-6.5c); [[39/35]] (+7.7c)
| [[55/49]] (&minus;4.9¢); [[91/81]] (&minus;6.5¢); [[39/35]] (+7.7¢)
| ^K
| ^K
| Gb, vE#
| Gb, vE#
Line 67: Line 67:
| 166.7
| 166.7
| sM2/sd3
| sM2/sd3
| [[15/13]] (-3.9c); [[63/55]] (+8.7c)
| [[15/13]] (&minus;3.9¢); [[63/55]] (+8.7¢)
| vK#, vLb
| vK#, vLb
| ^Gb, E#
| ^Gb, E#
Line 75: Line 75:
| 200
| 200
| M2/d3
| M2/d3
| [[77/65]] (-0.7c); [[13/11]] (+3.4c); [[25/21]] (-9.2c)
| [[77/65]] (&minus;0.7¢); [[13/11]] (+3.4¢); [[25/21]] (&minus;9.2¢)
| K#, Lb
| K#, Lb
| vF#, ^E#
| vF#, ^E#
Line 83: Line 83:
| 233.3
| 233.3
| SM2/Sd3
| SM2/Sd3
| [[11/9]] (-6.0c); [[91/75]] (+6.6c)
| [[11/9]] (&minus;6.0¢); [[91/75]] (+6.6¢)
| ^K#, ^Lb
| ^K#, ^Lb
| F#
| F#
Line 91: Line 91:
| 266.7
| 266.7
| sA2/sP3/sd4
| sA2/sP3/sd4
| [[49/39]] (-5.0c); [[81/65]] (+9.2c)
| [[49/39]] (&minus;5.0¢); [[81/65]] (+9.2¢)
| vL
| vL
| vG, ^F#
| vG, ^F#
Line 99: Line 99:
| 300
| 300
| A2/P3/d4
| A2/P3/d4
| [[9/7]] (+3.8c); [[35/27]] (-10.3c)
| [[9/7]] (+3.8¢); [[35/27]] (&minus;10.3¢)
| L
| L
| G
| G
Line 107: Line 107:
| 333.3
| 333.3
| SA2/SP3/Sd4
| SA2/SP3/Sd4
| [[65/49]] (-1.5c); [[33/25]] (+7.0c)
| [[65/49]] (&minus;1.5¢); [[33/25]] (+7.0¢)
| ^L
| ^L
| ^G, vAb
| ^G, vAb
Line 115: Line 115:
| 366.7
| 366.7
| sA3/sm4/sd5
| sA3/sm4/sd5
| [[15/11]] (-0.5c)
| [[15/11]] (&minus;0.5¢)
| vM
| vM
| Ab
| Ab
Line 123: Line 123:
| 400
| 400
| A3/m4/d5
| A3/m4/d5
| [[7/5]] (+2.7c)
| [[7/5]] (+2.7¢)
| M
| M
| ^Ab, Fx
| ^Ab, Fx
Line 131: Line 131:
| 433.3
| 433.3
| SA3/Sm4/Sd5
| SA3/Sm4/Sd5
| [[13/9]] (-2.6c)
| [[13/9]] (&minus;2.6¢)
| ^M
| ^M
| vG#
| vG#
Line 139: Line 139:
| 466.7
| 466.7
| sM4/sm5
| sM4/sm5
| [[135/91]] (+0.07c); [[49/33]] (-1.6c); [[81/55]] (+12.6c)
| [[135/91]] (+0.07¢); [[49/33]] (&minus;1.6¢); [[81/55]] (+12.6¢)
| vM#, vNb
| vM#, vNb
| G#
| G#
Line 147: Line 147:
| 500
| 500
| M4/m5
| M4/m5
| [[75/49]] (-5.4c); [[117/77]] (+7.2c)
| [[75/49]] (&minus;5.4¢); [[117/77]] (+7.2¢)
| M#, Nb
| M#, Nb
| vA, ^G#
| vA, ^G#
Line 155: Line 155:
| 533.3
| 533.3
| SM4/Sm5
| SM4/Sm5
| [[11/7]] (-2.2c); [[39/25]] (+10.4c)
| [[11/7]] (&minus;2.2¢); [[39/25]] (+10.4¢)
| ^M#, ^Nb
| ^M#, ^Nb
| A
| A
Line 163: Line 163:
| 566.7
| 566.7
| sA4/sM5
| sA4/sM5
| [[21/13]] (-1.2c)
| [[21/13]] (&minus;1.2¢)
| vN
| vN
| ^A, vBb
| ^A, vBb
Line 171: Line 171:
| 600
| 600
| A4/M5
| A4/M5
| [[91/55]] (+6.1c); [[5/3]] (-6.5c); [[81/49]] (+7.7c)
| [[91/55]] (+6.1¢); [[5/3]] (&minus;6.5¢); [[81/49]] (+7.7¢)
| N
| N
| Bb
| Bb
Line 179: Line 179:
| 633.3
| 633.3
| SA4/SM5
| SA4/SM5
| [[77/45]] (-3.3c)
| [[77/45]] (&minus;3.3¢)
| ^N
| ^N
| ^Bb, vCb, Gx
| ^Bb, vCb, Gx
Line 187: Line 187:
| 666.7
| 666.7
| sA5/sm6/sd7
| sA5/sm6/sd7
| [[135/77]] (+3.3c)
| [[135/77]] (+3.3¢)
| vO
| vO
| vA#, Cb
| vA#, Cb
Line 195: Line 195:
| 700
| 700
| A5/m6/d7
| A5/m6/d7
| [[165/91]] (-6.1c); [[9/5]] (+6.5c); [[49/27]] (-7.7c)
| [[165/91]] (&minus;6.1¢); [[9/5]] (+6.5¢); [[49/27]] (&minus;7.7¢)
| O
| O
| A#, ^Cb
| A#, ^Cb
Line 203: Line 203:
| 733.3
| 733.3
| SA5/Sm6/Sd7
| SA5/Sm6/Sd7
| [[13/7]] (+1.2c)
| [[13/7]] (+1.2¢)
| ^O
| ^O
| vB, ^A#
| vB, ^A#
Line 211: Line 211:
| 766.7
| 766.7
| sM6/sm7
| sM6/sm7
| [[21/11]] (+2.2c); [[25/13]] (-10.4c)
| [[21/11]] (+2.2¢); [[25/13]] (&minus;10.4¢)
| vO#, vPb
| vO#, vPb
| B
| B
Line 219: Line 219:
| 800
| 800
| M6/m7
| M6/m7
| [[49/25]] (+5.4c); [[77/39]] (-7.2c)
| [[49/25]] (+5.4¢); [[77/39]] (&minus;7.2¢)
| O#, Pb
| O#, Pb
| ^B, vC
| ^B, vC
Line 227: Line 227:
| 833.3
| 833.3
| SM6/Sm7
| SM6/Sm7
| [[91/45]] (+0.07c); [[99/49]] (+1.6c); [[55/27]] (-12.6c)
| [[91/45]] (+0.07¢); [[99/49]] (+1.6¢); [[55/27]] (&minus;12.6¢)
| ^O#, ^Pb
| ^O#, ^Pb
| C
| C
Line 235: Line 235:
| 866.7
| 866.7
| sA6/sM7/sd8
| sA6/sM7/sd8
| [[27/13]] (+2.6c)
| [[27/13]] (+2.6¢)
| vP
| vP
| ^C, vDb
| ^C, vDb
Line 243: Line 243:
| 900
| 900
| A6/M7/d8
| A6/M7/d8
| [[15/7]] (-2.7c)
| [[15/7]] (&minus;2.7¢)
| P
| P
| Db, vB#
| Db, vB#
Line 251: Line 251:
| 933.3
| 933.3
| SA6/SM7/Sd8
| SA6/SM7/Sd8
| [[11/5]] (+0.5c)
| [[11/5]] (+0.5¢)
| ^P
| ^P
| ^Db, B#
| ^Db, B#
Line 259: Line 259:
| 966.7
| 966.7
| sP8/sd9
| sP8/sd9
| [[147/65]] (+1.5c); [[25/11]] (-7.0c)
| [[147/65]] (+1.5¢); [[25/11]] (&minus;7.0¢)
| vQ
| vQ
| vC#, ^B#
| vC#, ^B#
Line 267: Line 267:
| 1000
| 1000
| P8/d9
| P8/d9
| [[7/3]] (-3.8c); [[81/35]] (+10.3c)
| [[7/3]] (&minus;3.8¢); [[81/35]] (+10.3¢)
| Q
| Q
| C#
| C#
Line 275: Line 275:
| 1033.3
| 1033.3
| SP8/Sd9
| SP8/Sd9
| [[117/49]] (+5.0c); [[65/27]] (-9.2c)
| [[117/49]] (+5.0¢); [[65/27]] (&minus;9.2¢)
| ^Q
| ^Q
| vD, ^C#
| vD, ^C#
Line 283: Line 283:
| 1066.7
| 1066.7
| sA8/sm9
| sA8/sm9
| [[27/11]] (+6.0c); [[225/91]] (+6.6c)
| [[27/11]] (+6.0¢); [[225/91]] (+6.6¢)
| vQ#, vRb
| vQ#, vRb
| D
| D
Line 291: Line 291:
| 1100
| 1100
| A8/m9
| A8/m9
| [[195/77]] (-0.7c); [[33/13]] (-3.4c); [[63/25]] (+9.2c)
| [[195/77]] (&minus;0.7¢); [[33/13]] (&minus;3.4¢); [[63/25]] (+9.2¢)
| Q#, Rb
| Q#, Rb
| ^D, vEb
| ^D, vEb
Line 299: Line 299:
| 1133.3
| 1133.3
| SA8/Sm9
| SA8/Sm9
| [[13/5]] (+3.9c); [[55/21]] (-8.7c)
| [[13/5]] (+3.9¢); [[55/21]] (&minus;8.7¢)
| ^Q#, ^Rb
| ^Q#, ^Rb
| Eb
| Eb
Line 307: Line 307:
| 1166.7
| 1166.7
| sM9/sd10
| sM9/sd10
| [[147/55]] (+4.9c); [[243/91]] (+6.5c); [[35/13]] (-7.7c)
| [[147/55]] (+4.9¢); [[243/91]] (+6.5¢); [[35/13]] (&minus;7.7¢)
| vR
| vR
| ^Eb, vFb, Cx
| ^Eb, vFb, Cx
Line 315: Line 315:
| 1200
| 1200
| M9/d10
| M9/d10
| [[91/33]] (+0.4c); [[135/49]] (+1.1c); [[25/9]] (-13.1c)
| [[91/33]] (+0.4¢); [[135/49]] (+1.1¢); [[25/9]] (&minus;13.1¢)
| R
| R
| vD#, Fb
| vD#, Fb
Line 323: Line 323:
| 1233.3
| 1233.3
| SM9/Sd10
| SM9/Sd10
| [[99/35]] (+4.3c); [[77/27]] (-9.9c)
| [[99/35]] (+4.3¢); [[77/27]] (&minus;9.9¢)
| ^R
| ^R
| D#, ^Fb
| D#, ^Fb
Line 331: Line 331:
| 1266.7
| 1266.7
| sA9/sP10
| sA9/sP10
| [[225/77]] (-3.2c); [[189/65]] (+5.3c)
| [[225/77]] (&minus;3.2¢); [[189/65]] (+5.3¢)
| vJ
| vJ
| vE, ^D#
| vE, ^D#

Revision as of 16:16, 11 December 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, perfect twelfth, or 3rd harmonic (abbreviated 39edt or 39ed3), is a nonoctave tuning system that divides the interval of 3/1 into 39 equal parts of about 48.8 ¢ each. Each step represents a frequency ratio of 31/39, 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.

Theory

It is a strong no-twos 13-limit system, a fact first noted by Paul Erlich; in fact it has a better no-twos 13-odd limit relative error than any other edt up to 914edt. Like 26edt and 52edt, it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being contorted in the no-twos 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, 1575/1573, and 847/845. An efficient traversal is therefore given by Mintra temperament, which in the 13-limit tempers out 275/273 and 1575/1573 alongside 245/243, and is generated by the interval of 11/7, which serves as a macrodiatonic "superpyth" fourth and splits the BPS generator of 9/7, up a tritave, in three.

If octaves are inserted, 39edt is related to the 49f & 172f temperament in the full 13-limit, known as triboh, 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.

Approximation of prime harmonics in 39edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +19.2 +0.0 -6.5 -3.8 -6.0 -2.6 +20.6 +23.1 -15.0 +22.6 +4.7 -9.0
Relative (%) +39.4 +0.0 -13.4 -7.9 -12.4 -5.4 +42.3 +47.4 -30.8 +46.3 +9.6 -18.5
Steps
(reduced) 		25
(25) 		39
(0) 		57
(18) 		69
(30) 		85
(7) 		91
(13) 		101
(23) 		105
(27) 		111
(33) 		120
(3) 		122
(5) 		128
(11)

Intervals

All intervals shown are within the 91-throdd limit and are consistently represented.

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 (+3.2¢); 65/63 (−5.3¢) ^J ^E, vF
2 97.5 66.7 sA1/sm2 35/33 (−4.3¢); 81/77 (+9.9¢) vK F
3 146.3 100 A1/m2 99/91 (+0.4¢); 49/45 (−1.1¢); 27/25 (+13.1¢) K ^F, vGb, Dx
4 195.1 133.3 SA1/Sm2 55/49 (−4.9¢); 91/81 (−6.5¢); 39/35 (+7.7¢) ^K Gb, vE#
5 243.8 166.7 sM2/sd3 15/13 (−3.9¢); 63/55 (+8.7¢) vK#, vLb ^Gb, E#
6 292.6 200 M2/d3 77/65 (−0.7¢); 13/11 (+3.4¢); 25/21 (−9.2¢) K#, Lb vF#, ^E#
7 341.4 233.3 SM2/Sd3 11/9 (−6.0¢); 91/75 (+6.6¢) ^K#, ^Lb F#
8 390.1 266.7 sA2/sP3/sd4 49/39 (−5.0¢); 81/65 (+9.2¢) vL vG, ^F#
9 438.9 300 A2/P3/d4 9/7 (+3.8¢); 35/27 (−10.3¢) L G
10 487.7 333.3 SA2/SP3/Sd4 65/49 (−1.5¢); 33/25 (+7.0¢) ^L ^G, vAb
11 536.4 366.7 sA3/sm4/sd5 15/11 (−0.5¢) vM Ab
12 585.2 400 A3/m4/d5 7/5 (+2.7¢) M ^Ab, Fx
13 634.0 433.3 SA3/Sm4/Sd5 13/9 (−2.6¢) ^M vG#
14 682.7 466.7 sM4/sm5 135/91 (+0.07¢); 49/33 (−1.6¢); 81/55 (+12.6¢) vM#, vNb G#
15 731.5 500 M4/m5 75/49 (−5.4¢); 117/77 (+7.2¢) M#, Nb vA, ^G#
16 780.3 533.3 SM4/Sm5 11/7 (−2.2¢); 39/25 (+10.4¢) ^M#, ^Nb A
17 829.0 566.7 sA4/sM5 21/13 (−1.2¢) vN ^A, vBb
18 877.8 600 A4/M5 91/55 (+6.1¢); 5/3 (−6.5¢); 81/49 (+7.7¢) N Bb
19 926.6 633.3 SA4/SM5 77/45 (−3.3¢) ^N ^Bb, vCb, Gx
20 975.3 666.7 sA5/sm6/sd7 135/77 (+3.3¢) vO vA#, Cb
21 1024.1 700 A5/m6/d7 165/91 (−6.1¢); 9/5 (+6.5¢); 49/27 (−7.7¢) O A#, ^Cb
22 1072.9 733.3 SA5/Sm6/Sd7 13/7 (+1.2¢) ^O vB, ^A#
23 1121.6 766.7 sM6/sm7 21/11 (+2.2¢); 25/13 (−10.4¢) vO#, vPb B
24 1170.4 800 M6/m7 49/25 (+5.4¢); 77/39 (−7.2¢) O#, Pb ^B, vC
25 1219.2 833.3 SM6/Sm7 91/45 (+0.07¢); 99/49 (+1.6¢); 55/27 (−12.6¢) ^O#, ^Pb C
26 1267.9 866.7 sA6/sM7/sd8 27/13 (+2.6¢) vP ^C, vDb
27 1316.7 900 A6/M7/d8 15/7 (−2.7¢) P Db, vB#
28 1365.5 933.3 SA6/SM7/Sd8 11/5 (+0.5¢) ^P ^Db, B#
29 1414.2 966.7 sP8/sd9 147/65 (+1.5¢); 25/11 (−7.0¢) vQ vC#, ^B#
30 1463.0 1000 P8/d9 7/3 (−3.8¢); 81/35 (+10.3¢) Q C#
31 1511.8 1033.3 SP8/Sd9 117/49 (+5.0¢); 65/27 (−9.2¢) ^Q vD, ^C#
32 1560.5 1066.7 sA8/sm9 27/11 (+6.0¢); 225/91 (+6.6¢) vQ#, vRb D
33 1609.3 1100 A8/m9 195/77 (−0.7¢); 33/13 (−3.4¢); 63/25 (+9.2¢) Q#, Rb ^D, vEb
34 1658.1 1133.3 SA8/Sm9 13/5 (+3.9¢); 55/21 (−8.7¢) ^Q#, ^Rb Eb
35 1706.9 1166.7 sM9/sd10 147/55 (+4.9¢); 243/91 (+6.5¢); 35/13 (−7.7¢) vR ^Eb, vFb, Cx
36 1755.7 1200 M9/d10 91/33 (+0.4¢); 135/49 (+1.1¢); 25/9 (−13.1¢) R vD#, Fb
37 1804.5 1233.3 SM9/Sd10 99/35 (+4.3¢); 77/27 (−9.9¢) ^R D#, ^Fb
38 1853.2 1266.7 sA9/sP10 225/77 (−3.2¢); 189/65 (+5.3¢) vJ vE, ^D#
39 1902.0 1300 A9/P10 3/1 J E
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