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{{Infobox ET}}
{{Infobox ET}}
'''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 [[cent]]s 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.
{{ED intro}} 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 24.606[[edo]]. 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.
39edt can be described as approximately 24.606[[edo]]. 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 ==
== Theory ==
It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]. 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-[[throdd 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 1331/1323 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 [[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]].
39edt also supports the temperaments: [[suhail]] (generators ~634.1c, ~49.7c), [[erigone]] (3/1, ~682.4c), [[electra]] (3/1, ~536.1c), [[bohlenic]] (1\13edt, ~11/1) and [[deneb]] (3/1, ~892.6c).


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


When treated as an octave-repeating tuning with the sharp octave of 25 steps (about 1219 cents), and the other primes chosen by their best octave-reduced mappings, it functions as a tuning of [[mavila]] temperament, analogous to [[25edo]]'s mavila.
Mavila is one of the few places where octave-stretching makes sense, due to how flat the fifth and often the major third are; this fifth of 683 cents is much more recognizable as a perfect fifth of 3/2 than the 672-cent tuning with just octaves.
{{Harmonics in equal|39|3|1|intervals=prime|columns=12}}


== Intervals ==
== Intervals ==
All intervals shown are within the 77-[[odd limit#Nonoctave equaves|throdd limit]] and are consistently represented.
All intervals shown are within the 91-[[odd limit#Nonoctave equaves|throdd limit]] and are consistently represented.


{| class="wikitable center-all right-2 right-3"
{| class="wikitable center-all right-2 right-3"
Line 20: Line 24:
! [[Cent]]s
! [[Cent]]s
! [[Hekt]]s
! [[Hekt]]s
! [[4L 5s (3/1-equivalent)|Enneatonic]] degree
! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree
! Corresponding<br />3.5.7.11.13 subgroup<br />intervals
! Corresponding 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 37: Line 41:
| 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 45: Line 49:
| 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 53: Line 57:
| 100
| 100
| A1/m2
| A1/m2
| [[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
| ^F, vGb, Dx
|-
|-
| 4
| 4
Line 61: Line 65:
| 133.3
| 133.3
| SA1/Sm2
| SA1/Sm2
| [[55/49]] (-4.9c); [[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 69: Line 73:
| 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 77: Line 81:
| 200
| 200
| M2/d3
| M2/d3
| [[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 85: Line 89:
| 233.3
| 233.3
| SM2/Sd3
| SM2/Sd3
| [[11/9]] (-6.0c)
| [[11/9]] (&minus;6.0¢); [[91/75]] (+6.6¢)
| ^K#, ^Lb
| ^K#, ^Lb
| F#
| F#
Line 93: Line 97:
| 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
| ^F#, vG
| vG, ^F#
|-
|-
| 9
| 9
Line 101: Line 105:
| 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 109: Line 113:
| 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 117: Line 121:
| 366.7
| 366.7
| sA3/sm4/sd5
| sA3/sm4/sd5
| [[15/11]] (-0.5c)
| [[15/11]] (&minus;0.5¢)
| vM
| vM
| Ab
| Ab
Line 125: Line 129:
| 400
| 400
| A3/m4/d5
| A3/m4/d5
| [[7/5]] (+2.7c)
| [[7/5]] (+2.7¢)
| M
| M
| ^Ab, Fx
| ^Ab, Fx
Line 133: Line 137:
| 433.3
| 433.3
| SA3/Sm4/Sd5
| SA3/Sm4/Sd5
| [[13/9]] (-2.6c)
| [[13/9]] (&minus;2.6¢)
| ^M
| ^M
| vG#
| vG#
Line 141: Line 145:
| 466.7
| 466.7
| sM4/sm5
| sM4/sm5
| [[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 149: Line 153:
| 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
| ^G#, vA
| vA, ^G#
|-
|-
| 16
| 16
Line 157: Line 161:
| 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 165: Line 169:
| 566.7
| 566.7
| sA4/sM5
| sA4/sM5
| [[21/13]] (-1.2c)
| [[21/13]] (&minus;1.2¢)
| vN
| vN
| ^A, vBb
| ^A, vBb
Line 173: Line 177:
| 600
| 600
| A4/M5
| A4/M5
| [[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 181: Line 185:
| 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 189: Line 193:
| 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 197: Line 201:
| 700
| 700
| A5/m6/d7
| A5/m6/d7
| [[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 205: Line 209:
| 733.3
| 733.3
| SA5/Sm6/Sd7
| SA5/Sm6/Sd7
| [[13/7]] (+1.2c)
| [[13/7]] (+1.2¢)
| ^O
| ^O
| ^A#, vB
| vB, ^A#
|-
|-
| 23
| 23
Line 213: Line 217:
| 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 221: Line 225:
| 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 229: Line 233:
| 833.3
| 833.3
| SM6/Sm7
| SM6/Sm7
| [[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 237: Line 241:
| 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 245: Line 249:
| 900
| 900
| A6/M7/d8
| A6/M7/d8
| [[15/7]] (-2.7c)
| [[15/7]] (&minus;2.7¢)
| P
| P
| Db, vB#
| Db, vB#
Line 253: Line 257:
| 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 261: Line 265:
| 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 269: Line 273:
| 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 277: Line 281:
| 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
| ^C#, vD
| vD, ^C#
|-
|-
| 32
| 32
Line 285: Line 289:
| 1066.7
| 1066.7
| sA8/sm9
| sA8/sm9
| [[27/11]] (+6.0c)
| [[27/11]] (+6.0¢); [[225/91]] (+6.6¢)
| vQ#, vRb
| vQ#, vRb
| D
| D
Line 293: Line 297:
| 1100
| 1100
| A8/m9
| A8/m9
| [[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 301: Line 305:
| 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 309: Line 313:
| 1166.7
| 1166.7
| sM9/sd10
| sM9/sd10
| [[147/55]] (+4.9c); [[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 317: Line 321:
| 1200
| 1200
| M9/d10
| M9/d10
| [[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 325: Line 329:
| 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, Dx
| D#, ^Fb
|-
|-
| 38
| 38
Line 333: Line 337:
| 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
| ^D#
| vE, ^D#
|-
|-
| 39
| 39
Line 345: Line 349:
| E
| E
|}
|}
== Approximation to JI ==
According to the finite Euler product with sigma = 1, the 3.5.7.11.13 subgroup gets its maxima at 48.82085 ¢. With sigma = 1/2, the maxima is 48.82100 ¢.
The Tenney–Euclidean regular temperement in the 3.5.7.11.13 subgroup mapped with [⟨39 57 69 85 91]] gives 48.82201 ¢.
[[69ed7]], with a step size of 48.82356 ¢, is an equal division that approximates this area better than 39edt.
== Music ==
; [[Francium]]
* [https://www.youtube.com/watch?v=jstg4_B0jfY ''Strange Juice''] (2025)
;[https://www.youtube.com/@PhanomiumMusic Phanomium]
* ''[https://www.youtube.com/watch?v=GX79ZX1Z8C8 Polygonal]'' (2025)

Latest revision as of 04:24, 23 March 2026

← 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-throdd 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 1331/1323 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.

39edt also supports the temperaments: suhail (generators ~634.1c, ~49.7c), erigone (3/1, ~682.4c), electra (3/1, ~536.1c), bohlenic (1\13edt, ~11/1) and deneb (3/1, ~892.6c).

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.

When treated as an octave-repeating tuning with the sharp octave of 25 steps (about 1219 cents), and the other primes chosen by their best octave-reduced mappings, it functions as a tuning of mavila temperament, analogous to 25edo's mavila.

Mavila is one of the few places where octave-stretching makes sense, due to how flat the fifth and often the major third are; this fifth of 683 cents is much more recognizable as a perfect fifth of 3/2 than the 672-cent tuning with just octaves.

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

Approximation to JI

According to the finite Euler product with sigma = 1, the 3.5.7.11.13 subgroup gets its maxima at 48.82085 ¢. With sigma = 1/2, the maxima is 48.82100 ¢.

The Tenney–Euclidean regular temperement in the 3.5.7.11.13 subgroup mapped with [⟨39 57 69 85 91]] gives 48.82201 ¢.

69ed7, with a step size of 48.82356 ¢, is an equal division that approximates this area better than 39edt.

Music

Francium
Phanomium
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