39edt: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{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. | ||
It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]] | == 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 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 {{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]]. | |||
{| class="wikitable" | 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 == | |||
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" | |||
|- | |- | ||
! | ! Steps | ||
! | ! [[Cent]]s | ||
! | ! [[Hekt]]s | ||
! | ! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree | ||
! | ! Corresponding 3.5.7.11.13 subgroup<br />intervals | ||
! | ! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs,<br />{{nowrap|J {{=}} 1/1}}) | ||
! Mintaka[7]<br />(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 | | 300 | ||
| |3 | | 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 | | 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 | |||
|1300 | | A9/P10 | ||
| | | [[3/1]] | ||
| | | J | ||
| | | E | ||
| | |} | ||
| | |||
== Approximation to JI == | |||
=== No-2 zeta peak === | |||
{| class="wikitable" | |||
|+ | |||
| | !Steps | ||
| | per octave | ||
!Steps | |||
per tritave | |||
!Step size | |||
(cents) | |||
!Height | |||
!Tritave size | |||
(cents) | |||
!Tritave stretch | |||
(cents) | |||
|- | |- | ||
| | |24.573831630 | ||
| | |38.948601633 | ||
| | |48.832433543 | ||
| | |4.665720 | ||
| | |1904.464908194 | ||
| | |2.509907328 | ||
|} | |} | ||
Every 7 steps of the [[172edo|172f]] val is an excellent approximation of the ninth no-2 zeta peak in the 15-limit. | |||
== 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) |