143ed11: Difference between revisions

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143 equal divisions of the 11th harmonic (abbreviated 143ed11), is the tuning system that divides the 11th harmonic into 143 equal parts of about 29.0302 ¢ each. Each step represents a frequency ratio of <math>11^{~~\frac~~{1}{~~143~~}}~~</math>, or the 143rd root of 11.~~

== Theory ==

==~~= Why~~ 143ed11 ~~matters ===~~

143ed11 is extremely strong in the 8.9.5.7.11.13.17.23 subgroup. All ratios in the 8.9.5.7.11.13.17.23 subgroup and 25-integer-limit are approximated in 143ed11 with less than 2.5 ¢ error. According to the finite Euler product with sigma = 1, the maxima in the 8.9.5.7.11.13.17.23 subgroup is about 41.33601-ED2 (or 29.03038 ¢). The Tenney–Euclidean regular temperement in the 8.9.5.7.11.13.17.23 subgroup mapped with [⟨124 131 96 116 143 153 169 187]] gives 41.33650-ED2 (or 29.03004 ¢). The finite Euler product with sigma = 1/2 gives about the same result. 143ed11, with its size of about 41.33627-ED2 (or 29.03020 ¢), gives a simple, very near compromise between the different methods approaching 8.9.5.7.11.13.17.23 in this area.

~~143ed11 is extremely strong in the 5.7.8.9.11.13.17.23 subgroup~~.

It serves as a slightly stretched version of [[96ed5]].

~~All ratios~~ in the 5.~~7.8.9.11.13.17.23 subgroup and 25-integer-limit are approximated in 143ed11 with less than 2.5~~ ¢ ~~error.~~

=== Proposed name for tunings / temperaments in the ~29.03 ¢ area: Noctrino ===

'''Noctrino'''

~~According to the finite Euler product with sigma = 1, the maxima in the 5.7.8.9.11.13.17.23 subgroup is about 41.33601~~-~~ED2 (or 29.03038 ¢).~~

Etymology: No-octave no-tritave -> No-oct no-tri -> Noct trino -> Noctrino

~~The Tenney–Euclidean regular temperement in the 5.7.8.9.11.13.17.23 subgroup mapped with [⟨96 116 124 131 143 153 169 187]] gives 41.33650~~-~~ED2 (or 29.03005 ¢). The finite Euler product with sigma = 1/2 gives about the same result.~~

~~143ed11, with its size of about 41.33627~~-~~ED2 (or 29.03020 ¢), gives a simple, very near compromise between the different methods approaching 5.7.8.9.11.13.17.23 in this area.~~

== Intervals and approximation to JI ==

{{Todo|inline=1|complete table|comment=Add the missing scale degrees.}}

{| class="wikitable sortable center-all right-2 left-3"

|-

! 143ed11 degree

! Cents

! Ratios in the 5.7~~.8.9~~.11.13.17.23 subgroup

! Ratios in the 8.9.5.7.11.13.17.23 subgroup

! Error (abs, [[Cent|¢]])

! Error (rel, [[Relative cent|%]])

Line 292:

Line 289:

== Harmonics ==

== Commas of the 5.7~~.8.9~~.11.13.17.23 subgroup tempered out in 143ed11 ==

== Commas of the 8.9.5.7.11.13.17.23 subgroup tempered out in 143ed11 ==

Commas with numerator < 100000 :

Commas with numerator < 100000 include:

[[392/391]], [[441/440]], [[576/575]], [[729/728]], [[833/832]], [[847/845]], [[936/935]], [[1001/1000]], [[1089/1088]], [[1127/1125]], [[1225/1224]], [[1288/1287]], [[1377/1375]], [[1449/1445]], [[1496/1495]], [[1575/1573]], [[1771/1768]], [[1863/1859]], [[2024/2023]], [[2025/2023]], [[2025/2024]], [[2200/2197]], [[2205/2197]], [[2277/2275]], [[2401/2392]], [[2601/2600]], [[2695/2691]], [[2875/2873]], [[2880/2873]], [[3128/3125]], [[3136/3125]], [[3185/3179]], [[3520/3519]], [[3584/3575]], [[3773/3757]], [[4096/4095]], [[4232/4225]], [[4235/4232]], [[4459/4455]], [[4761/4760]], [[4928/4913]], [[5103/5083]], [[5544/5525]], [[5632/5625]], [[5635/5632]], [[5824/5819]], [[5831/5819]], [[5832/5819]], [[5832/5831]], [[6561/6545]], [[6655/6647]], [[6656/6647]], [[6656/6655]], [[6664/6655]], [[6877/6875]], [[6885/6877]], [[7371/7360]], [[7497/7475]], [[7616/7605]], [[7744/7735]], [[7889/7865]], [[8019/8000]], [[8281/8280]], [[9000/8993]], [[9009/8993]], [[9317/9315]], [[9801/9775]], [[9801/9800]], [[10304/10285]], [[10647/10625]], [[10648/10625]], [[10648/10647]], [[11016/10985]], [[11016/11011]], [[11583/11560]], [[11781/11776]], [[11979/11960]], [[12005/11968]], [[12005/11979]], [[12167/12155]], [[12168/12167]], [[12376/12375]], [[12397/12393]], [[13041/13000]], [[13013/13005]], [[13689/13685]], [[13720/13689]], [[13923/13915]], [[14175/14144]], [[14161/14157]], [[14399/14375]], [[14400/14399]], [[14641/14625]], [[14651/14641]], [[14875/14872]], [[14904/14875]], [[16807/16731]], [[16807/16767]], [[17595/17576]], [[17920/17901]], [[18515/18496]], [[18515/18513]], [[19136/19125]], [[19208/19125]], [[20480/20449]], [[20493/20480]], [[21609/21505]], [[21879/21875]], [[21952/21879]], [[22295/22264]], [[22680/22627]], [[23040/23023]], [[24219/24200]], [[24696/24565]], [[25047/25000]], [[25025/25024]], [[25088/25047]], [[25515/25432]], [[25921/25857]], [[25921/25920]], [[26411/26325]], [[27209/27200]], [[27225/27209]], [[27783/27625]], [[28611/28561]], [[28672/28561]], [[28672/28611]], [[30613/30600]], [[30625/30613]], [[31185/31096]], [[31752/31603]], [[31752/31625]], [[32768/32725]], [[32805/32768]], [[33327/33275]], [[33327/33280]], [[33957/33800]], [[33856/33813]], [[33880/33813]], [[33957/33856]], [[34391/34375]], [[34391/34385]], [[34496/34385]], [[34496/34425]], [[34983/34969]], [[35000/34969]], [[35000/34983]], [[35640/35581]], [[36179/36125]], [[36288/36125]], [[36288/36179]], [[39325/39304]], [[39375/39304]], [[39445/39304]], [[40131/40000]], [[40733/40625]], [[40817/40625]], [[40824/40625]], [[41400/41327]], [[41472/41327]], [[41503/41400]], [[41472/41405]], [[41503/41472]], [[42840/42757]], [[42875/42757]], [[42875/42849]], [[43659/43520]], [[44275/44217]], [[45056/44965]], [[45927/45760]], [[46529/46475]], [[46648/46475]], [[46656/46475]], [[46656/46529]], [[46592/46575]], [[46648/46575]], [[46656/46585]], [[47320/47311]], [[47385/47311]], [[47432/47385]], [[49000/48841]], [[49005/48841]], [[49049/48875]], [[49049/48960]], [[50575/50531]], [[50625/50531]], [[50688/50531]], [[50688/50575]], [[51597/51425]], [[52488/52325]], [[53248/53125]], [[53361/53125]], [[53235/53176]], [[53361/53176]], [[53361/53248]], [[54080/54043]], [[55223/54925]], [[55223/55000]], [[55125/55016]], [[55223/55080]], [[56000/55913]], [[57024/56875]], [[57967/57800]], [[59049/58880]], [[60025/59823]], [[60025/59904]], [[60928/60835]], [[60984/60835]], [[61952/61893]], [[61965/61952]], [[62951/62920]], [[64000/63869]], [[64009/63869]], [[64152/63869]], [[64009/64000]], [[64141/64000]], [[65219/65000]], [[65219/65025]], [[66248/66125]], [[66339/66125]], [[67392/67375]], [[69632/69575]], [[71001/70720]], [[72128/71825]], [[72171/71825]], [[72072/71875]], [[72171/71875]], [[72171/71944]], [[72171/72128]], [[73125/73117]], [[73304/73125]], [[75712/75625]], [[75803/75625]], [[75816/75625]], [[75735/75712]], [[78336/78125]], [[78408/78125]], [[83655/83521]], [[83720/83521]], [[83835/83521]], [[83853/83521]], [[84035/83521]], [[83835/83776]], [[85169/85000]], [[85293/85000]], [[85184/85169]], [[85293/85169]], [[85293/85184]], [[86625/86411]], [[86751/86515]], [[86625/86528]], [[86751/86528]], [[86779/86528]], [[88209/87880]], [[88088/87975]], [[89600/89401]], [[91125/91091]], [[92664/92575]], [[94208/93925]], [[94325/94208]], [[95823/95744]], [[95832/95795]], [[97461/97336]], [[97405/97344]], [[98923/98865]], [[99127/98865]], [[99176/98865]], [[99225/98923]], [[99127/99000]]

@@ Line 1: / Line 1: @@
-equal divisions of the 11th harmonic (abbreviated 143ed11), is the tuning system that divides the 11th harmonic into 143 equal parts of about 29.0302 ¢ each. Each step represents a frequency ratio of <math>11^{\frac{1}{143}}</math>, or the 143rd root of 11.
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
-=== Why 143ed11 matters ===
+ed11 is extremely strong in the 8.9.5.7.11.13.17.23 subgroup. All ratios in the 8.9.5.7.11.13.17.23 subgroup and 25-integer-limit are approximated in 143ed11 with less than 2.5 ¢ error. According to the finite Euler product with sigma = 1, the maxima in the 8.9.5.7.11.13.17.23 subgroup is about 41.33601-ED2 (or 29.03038 ¢). The Tenney–Euclidean regular temperement in the 8.9.5.7.11.13.17.23 subgroup mapped with [⟨124 131 96 116 143 153 169 187]] gives 41.33650-ED2 (or 29.03004 ¢). The finite Euler product with sigma = 1/2 gives about the same result. 143ed11, with its size of about 41.33627-ED2 (or 29.03020 ¢), gives a simple, very near compromise between the different methods approaching 8.9.5.7.11.13.17.23 in this area.
-ed11 is extremely strong in the 5.7.8.9.11.13.17.23 subgroup.
+It serves as a slightly stretched version of [[96ed5]].
-All ratios in the 5.7.8.9.11.13.17.23 subgroup and 25-integer-limit are approximated in 143ed11 with less than 2.5 ¢ error.
+=== Proposed name for tunings / temperaments in the ~29.03 ¢ area: Noctrino ===
+'''Noctrino'''
-According to the finite Euler product with sigma = 1, the maxima in the 5.7.8.9.11.13.17.23 subgroup is about 41.33601-ED2 (or 29.03038 ¢).
+Etymology: No-octave no-tritave -> No-oct no-tri -> Noct trino -> Noctrino
-The Tenney–Euclidean regular temperement in the 5.7.8.9.11.13.17.23 subgroup mapped with [⟨96 116 124 131 143 153 169 187]] gives 41.33650-ED2 (or 29.03005 ¢). The finite Euler product with sigma = 1/2 gives about the same result.
-ed11, with its size of about 41.33627-ED2 (or 29.03020 ¢), gives a simple, very near compromise between the different methods approaching 5.7.8.9.11.13.17.23 in this area.
 == Intervals and approximation to JI ==
+{{Todo|inline=1|complete table|comment=Add the missing scale degrees.}}
 {| class="wikitable sortable center-all right-2 left-3"
 |-
 ! 143ed11<br>degree
 ! Cents
-! Ratios in the<br>5.7.8.9.11.13.17.23<br>subgroup
+! Ratios in the<br>8.9.5.7.11.13.17.23<br>subgroup
 ! Error (abs, [[Cent|¢]])
 ! Error (rel, [[Relative cent|%]])
@@ Line 292: / Line 289: @@
 == Harmonics ==
-{{Harmonics in equal|143|11|1|prec=1|columns=15}}
+{{Harmonics in equal|143|11|1|prec=1|columns=12}}
-{{Harmonics in equal|143|11|1|prec=1|columns=9|start=16}}
+{{Harmonics in equal|143|11|1|prec=1|columns=12|start=13}}
-== Commas of the 5.7.8.9.11.13.17.23 subgroup tempered out in 143ed11 ==
+== Commas of the 8.9.5.7.11.13.17.23 subgroup tempered out in 143ed11 ==
-Commas with numerator < 100000 :
+Commas with numerator < 100000 include:
 [[392/391]], [[441/440]], [[576/575]], [[729/728]], [[833/832]], [[847/845]], [[936/935]], [[1001/1000]], [[1089/1088]], [[1127/1125]], [[1225/1224]], [[1288/1287]], [[1377/1375]], [[1449/1445]], [[1496/1495]], [[1575/1573]], [[1771/1768]], [[1863/1859]], [[2024/2023]], [[2025/2023]], [[2025/2024]], [[2200/2197]], [[2205/2197]], [[2277/2275]], [[2401/2392]], [[2601/2600]], [[2695/2691]], [[2875/2873]], [[2880/2873]], [[3128/3125]], [[3136/3125]], [[3185/3179]], [[3520/3519]], [[3584/3575]], [[3773/3757]], [[4096/4095]], [[4232/4225]], [[4235/4232]], [[4459/4455]], [[4761/4760]], [[4928/4913]], [[5103/5083]], [[5544/5525]], [[5632/5625]], [[5635/5632]], [[5824/5819]], [[5831/5819]], [[5832/5819]], [[5832/5831]], [[6561/6545]], [[6655/6647]], [[6656/6647]], [[6656/6655]], [[6664/6655]], [[6877/6875]], [[6885/6877]], [[7371/7360]], [[7497/7475]], [[7616/7605]], [[7744/7735]], [[7889/7865]], [[8019/8000]], [[8281/8280]], [[9000/8993]], [[9009/8993]], [[9317/9315]], [[9801/9775]], [[9801/9800]], [[10304/10285]], [[10647/10625]], [[10648/10625]], [[10648/10647]], [[11016/10985]], [[11016/11011]], [[11583/11560]], [[11781/11776]], [[11979/11960]], [[12005/11968]], [[12005/11979]], [[12167/12155]], [[12168/12167]], [[12376/12375]], [[12397/12393]], [[13041/13000]], [[13013/13005]], [[13689/13685]], [[13720/13689]], [[13923/13915]], [[14175/14144]], [[14161/14157]], [[14399/14375]], [[14400/14399]], [[14641/14625]], [[14651/14641]], [[14875/14872]], [[14904/14875]], [[16807/16731]], [[16807/16767]], [[17595/17576]], [[17920/17901]], [[18515/18496]], [[18515/18513]], [[19136/19125]], [[19208/19125]], [[20480/20449]], [[20493/20480]], [[21609/21505]], [[21879/21875]], [[21952/21879]], [[22295/22264]], [[22680/22627]], [[23040/23023]], [[24219/24200]], [[24696/24565]], [[25047/25000]], [[25025/25024]], [[25088/25047]], [[25515/25432]], [[25921/25857]], [[25921/25920]], [[26411/26325]], [[27209/27200]], [[27225/27209]], [[27783/27625]], [[28611/28561]], [[28672/28561]], [[28672/28611]], [[30613/30600]], [[30625/30613]], [[31185/31096]], [[31752/31603]], [[31752/31625]], [[32768/32725]], [[32805/32768]], [[33327/33275]], [[33327/33280]], [[33957/33800]], [[33856/33813]], [[33880/33813]], [[33957/33856]], [[34391/34375]], [[34391/34385]], [[34496/34385]], [[34496/34425]], [[34983/34969]], [[35000/34969]], [[35000/34983]], [[35640/35581]], [[36179/36125]], [[36288/36125]], [[36288/36179]], [[39325/39304]], [[39375/39304]], [[39445/39304]], [[40131/40000]], [[40733/40625]], [[40817/40625]], [[40824/40625]], [[41400/41327]], [[41472/41327]], [[41503/41400]], [[41472/41405]], [[41503/41472]], [[42840/42757]], [[42875/42757]], [[42875/42849]], [[43659/43520]], [[44275/44217]], [[45056/44965]], [[45927/45760]], [[46529/46475]], [[46648/46475]], [[46656/46475]], [[46656/46529]], [[46592/46575]], [[46648/46575]], [[46656/46585]], [[47320/47311]], [[47385/47311]], [[47432/47385]], [[49000/48841]], [[49005/48841]], [[49049/48875]], [[49049/48960]], [[50575/50531]], [[50625/50531]], [[50688/50531]], [[50688/50575]], [[51597/51425]], [[52488/52325]], [[53248/53125]], [[53361/53125]], [[53235/53176]], [[53361/53176]], [[53361/53248]], [[54080/54043]], [[55223/54925]], [[55223/55000]], [[55125/55016]], [[55223/55080]], [[56000/55913]], [[57024/56875]], [[57967/57800]], [[59049/58880]], [[60025/59823]], [[60025/59904]], [[60928/60835]], [[60984/60835]], [[61952/61893]], [[61965/61952]], [[62951/62920]], [[64000/63869]], [[64009/63869]], [[64152/63869]], [[64009/64000]], [[64141/64000]], [[65219/65000]], [[65219/65025]], [[66248/66125]], [[66339/66125]], [[67392/67375]], [[69632/69575]], [[71001/70720]], [[72128/71825]], [[72171/71825]], [[72072/71875]], [[72171/71875]], [[72171/71944]], [[72171/72128]], [[73125/73117]], [[73304/73125]], [[75712/75625]], [[75803/75625]], [[75816/75625]], [[75735/75712]], [[78336/78125]], [[78408/78125]], [[83655/83521]], [[83720/83521]], [[83835/83521]], [[83853/83521]], [[84035/83521]], [[83835/83776]], [[85169/85000]], [[85293/85000]], [[85184/85169]], [[85293/85169]], [[85293/85184]], [[86625/86411]], [[86751/86515]], [[86625/86528]], [[86751/86528]], [[86779/86528]], [[88209/87880]], [[88088/87975]], [[89600/89401]], [[91125/91091]], [[92664/92575]], [[94208/93925]], [[94325/94208]], [[95823/95744]], [[95832/95795]], [[97461/97336]], [[97405/97344]], [[98923/98865]], [[99127/98865]], [[99176/98865]], [[99225/98923]], [[99127/99000]]