Rastmic rank-3 clan: Difference between revisions

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- CTE tunings
List & sort the temps by strength and badness
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* ''[[Hagrid]]'' (+9801/9800) → [[Cataharry family #Hagrid|Cataharry family]]
* ''[[Hagrid]]'' (+9801/9800) → [[Cataharry family #Hagrid|Cataharry family]]
* ''[[Rabic]]'' (+131769/131072) → [[Alphaxenic rank-3 clan #Betarabian|Alphaxenic rank-3 clan]]
* ''[[Rabic]]'' (+131769/131072) → [[Alphaxenic rank-3 clan #Betarabian|Alphaxenic rank-3 clan]]
Considered below are spectacle, mirwomo, mandos, cuckoo, parahemif, urania, and mirage.


== Spectacle ==
== Spectacle ==
Line 54: Line 56:


Badness (Sintel): 0.944
Badness (Sintel): 0.944
== Cuckoo ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 126/125, 243/242
{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 -4 5 | 0 0 1 3 0 }}
: mapping generators: ~2, ~11/9, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8222{{c}}, ~11/9 = 350.4356{{c}}, ~5/4 = 389.8478{{c}}
: [[error map]]: {{val| -0.178 -1.262 +3.178 -1.558 +0.504 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.4213{{c}}, ~5/4 = 389.7308{{c}}
: error map: {{val| 0.000 -1.112 +3.417 -1.318 +0.788 }}
{{Optimal ET sequence|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}
[[Badness]] (Sintel): 1.12
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 196/195, 243/242
Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 -4 5 -10 | 0 0 1 3 0 5 }}
Optimal tunings:
* WE: ~2 = 1199.7103{{c}}, ~11/9 = 350.5840{{c}}, ~5/4 = 389.8071{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.5682{{c}}, ~5/4 = 389.6104{{c}}
{{Optimal ET sequence|legend=0| 27e, 31, 58, 96d, 154 }}
Badness (Sintel): 1.23


== Mirwomo ==
== Mirwomo ==
Line 128: Line 96:
[[Badness]] (Sintel): 0.770
[[Badness]] (Sintel): 0.770


== Mirage ==
== Mandos ==
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 225/224, 243/242, 385/384
[[Comma list]]: 176/175, 243/242


{{Mapping|legend=1| 1 1 3 3 2 0 | 0 6 -7 -2 15 0 | 0 0 0 0 0 1 }}
{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 5 5 | 0 0 1 -2 0 }}


: mapping generators: ~2, ~15/14, ~13
: mapping generators: ~2, ~11/9, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7626{{c}}, ~15/4 = 116.7069{{c}}, ~13/8 = 838.2364{{c}}
* [[WE]]: ~2 = 1199.1949{{c}}, ~11/9 = 350.6135{{c}}, ~5/4 = 390.4090{{c}}
: [[error map]]: {{val| +0.763 -0.951 -0.974 +0.048 +0.810 -0.004 }}
: [[error map]]: {{val| -0.805 -1.533 +2.485 +1.814 +0.139 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 116.6469{{c}}, ~13/8 = 838.2123{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.5548{{c}}, ~5/4 = 390.2690{{c}}
: error map: {{val| 0.000 -2.074 -2.842 -2.120 -1.615 -2.315 }}
: error map: {{val| 0.000 -0.845 +3.955 +3.410 +1.456 }}


{{Optimal ET sequence|legend=1| 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}
{{Optimal ET sequence|legend=1| 24, 27e, 31, 58, 89, 154d, 181cde, 212cde, 301ccde }}


Badness (Sintel): 0.691
[[Badness]] (Sintel): 0.902


=== 17-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 243/242, 273/272, 385/384
Comma list: 144/143, 176/175, 243/242


Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 0 4 | 0 0 0 0 0 1 1 }}
Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 0 5 5 -1 | 0 0 1 -2 0 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.7628{{c}}, ~15/14 = 116.6995{{c}}, ~13/8 = 837.1672{{c}}
* WE: ~2 = 1198.5555{{c}}, ~11/9 = 351.0300{{c}}, ~5/4 = 391.0458{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.6395{{c}}, ~13/8 = 837.1424{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.1554{{c}}, ~5/4 = 391.1227{{c}}


{{Optimal ET sequence|legend=0| 31, 41, 62, 72, 103, 175f, 360bcdff, 463bccdeff }}
{{Optimal ET sequence|legend=0| 24, 27e, 31, 58, 123df, 181cdeff, 239ccddeefff }}


Badness (Sintel): 0.715
Badness (Sintel): 0.863


== Mandos ==
== Cuckoo ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 176/175, 243/242
[[Comma list]]: 126/125, 243/242


{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 5 5 | 0 0 1 -2 0 }}
{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 -4 5 | 0 0 1 3 0 }}


: mapping generators: ~2, ~11/9, ~5
: mapping generators: ~2, ~11/9, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.1949{{c}}, ~11/9 = 350.6135{{c}}, ~5/4 = 390.4090{{c}}
* [[WE]]: ~2 = 1199.8222{{c}}, ~11/9 = 350.4356{{c}}, ~5/4 = 389.8478{{c}}
: [[error map]]: {{val| -0.805 -1.533 +2.485 +1.814 +0.139 }}
: [[error map]]: {{val| -0.178 -1.262 +3.178 -1.558 +0.504 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.5548{{c}}, ~5/4 = 390.2690{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.4213{{c}}, ~5/4 = 389.7308{{c}}
: error map: {{val| 0.000 -0.845 +3.955 +3.410 +1.456 }}
: error map: {{val| 0.000 -1.112 +3.417 -1.318 +0.788 }}


{{Optimal ET sequence|legend=1| 24, 27e, 31, 58, 89, 154d, 181cde, 212cde, 301ccde }}
{{Optimal ET sequence|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}


[[Badness]] (Sintel): 0.902
[[Badness]] (Sintel): 1.12


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 144/143, 176/175, 243/242
Comma list: 126/125, 196/195, 243/242


Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 0 5 5 -1 | 0 0 1 -2 0 0 }}
Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 -4 5 -10 | 0 0 1 3 0 5 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1198.5555{{c}}, ~11/9 = 351.0300{{c}}, ~5/4 = 391.0458{{c}}
* WE: ~2 = 1199.7103{{c}}, ~11/9 = 350.5840{{c}}, ~5/4 = 389.8071{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.1554{{c}}, ~5/4 = 391.1227{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.5682{{c}}, ~5/4 = 389.6104{{c}}


{{Optimal ET sequence|legend=0| 24, 27e, 31, 58, 123df, 181cdeff, 239ccddeefff }}
{{Optimal ET sequence|legend=0| 27e, 31, 58, 96d, 154 }}


Badness (Sintel): 0.863
Badness (Sintel): 1.23


== Parahemif ==
== Parahemif ==
Line 278: Line 246:


Scales: [[urania24]]
Scales: [[urania24]]
== Mirage ==
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: 225/224, 243/242, 385/384
{{Mapping|legend=1| 1 1 3 3 2 0 | 0 6 -7 -2 15 0 | 0 0 0 0 0 1 }}
: mapping generators: ~2, ~15/14, ~13
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7626{{c}}, ~15/4 = 116.7069{{c}}, ~13/8 = 838.2364{{c}}
: [[error map]]: {{val| +0.763 -0.951 -0.974 +0.048 +0.810 -0.004 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 116.6469{{c}}, ~13/8 = 838.2123{{c}}
: error map: {{val| 0.000 -2.074 -2.842 -2.120 -1.615 -2.315 }}
{{Optimal ET sequence|legend=1| 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}
Badness (Sintel): 0.691
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 243/242, 273/272, 385/384
Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 0 4 | 0 0 0 0 0 1 1 }}
Optimal tunings:
* WE: ~2 = 1200.7628{{c}}, ~15/14 = 116.6995{{c}}, ~13/8 = 837.1672{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.6395{{c}}, ~13/8 = 837.1424{{c}}
{{Optimal ET sequence|legend=0| 31, 41, 62, 72, 103, 175f, 360bcdff, 463bccdeff }}
Badness (Sintel): 0.715


== References ==
== References ==

Revision as of 17:17, 8 September 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The rastmic rank-3 clan of temperaments tempers out the rastma, 243/242. Both no-5 rastmic and no-7 rastmic can be the head of this clan. These temperaments divide the fifth in half and use it as an 11/9 neutral third.

Temperaments discussed elsewhere include:

Considered below are spectacle, mirwomo, mandos, cuckoo, parahemif, urania, and mirage.

Spectacle

Spectacle, named by Gene Ward Smith in 2010[1], can be described as the 31 & 34d & 41 temperament. It tempers out 225/224, making it a sort of marvel infested with neutral thirds. It is therefore generated by octaves, major thirds, and neutral thirds. 3/2 is equated with a stack of two 11/9's as a corollary of 243/242 being tempered out, 7/4 is equated with a stack of four 11/9's and two 5/4's, 11/8 is equated with a stack of five 11/9's, 13/8 is equated with a stack of two 18/11's and four 5/4's, and 17/16 is equated with three 18/11's and three 5/4's. Every harmonic is reached with help of other intervals at most with three 5/4's.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242

Mapping: [⟨1 1 0 -3 2], ⟨0 2 0 4 5], ⟨0 0 1 2 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

  • WE: ~2 = 1200.5486 ¢, ~11/9 = 350.2171 ¢, ~5/4 = 384.1078 ¢
error map: ⟨+0.549 -0.972 -1.109 +0.806 +0.864]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.1758 ¢, ~5/4 = 384.0951 ¢
error map: ⟨0.000 -1.603 -2.219 +0.068 -0.439]

Minimax tuning:

  • 11-odd-limit: ~2 = [1 0 0 0 0⟩, ~11/9 = [-2/5 0 0 0 1/5⟩, ~5 = [2/5 -2 1 0 4/5⟩
unchanged-interval (eigenmonzo) basis: 2.9/5.11

Optimal ET sequence: 31, 41, 65d, 72, 247c, 281, 353c, 425bc, 497bc

Badness (Sintel): 0.599

Projection pairs: 3 242/81, 7 366025/52488, 11 644204/59049 to 2.5.11/9

Scales: spectacle31

Associated temperament: marvo

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 351/350

Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 4 5 -2], ⟨0 0 1 2 0 4]]

Optimal tunings:

  • WE: ~2 = 1200.6024 ¢, ~11/9 = 350.1004 ¢, ~5/4 = 384.5435 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.0393 ¢, ~5/4 = 384.5866 ¢

Optimal ET sequence: 31, 65d, 72, 103, 175f, 312bf, 384bcf, 487bceff *

* optimal patent val: 240

Badness (Sintel): 0.944

Mirwomo

The mirwomo temperament tunes ~36/35 to exactly half of an apotome ~2187/2048, and tunes ~128/105 to exactly half of a perfect fifth ~3/2.

7-limit

Subgroup: 2.3.5.7

Comma list: 33075/32768

Mapping: [⟨1 1 0 6], ⟨0 2 0 -3], ⟨0 0 1 -1]]

mapping generators: ~2, ~128/105, ~5

Optimal tunings:

  • WE: ~2 = 1200.8046 ¢, ~11/9 = 350.3723 ¢, ~5/4 = 384.1239 ¢
error map: ⟨+0.805 -0.406 -0.581 -0.848]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.1448 ¢, ~5/4 = 383.8961 ¢
error map: ⟨0.000 -1.665 -2.418 -3.157]

Optimal ET sequence: 17, 21, 24, 31, 41, 72, 281d, 322cd, 353cd, 425bcdd, 497bcdd

Badness (Sintel): 3.40

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384

Mapping: [⟨1 1 0 6 2], ⟨0 2 0 -3 5], ⟨0 0 1 -1 0]]

Optimal tunings:

  • WE: ~2 = 1200.7360 ¢, ~11/9 = 350.1700 ¢, ~5/4 = 384.3403 ¢
error map: ⟨+0.736 -0.879 -0.501 -0.733 +1.004]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.0035 ¢, ~5/4 = 384.0785 ¢
error map: ⟨0.000 -1.948 -2.235 -2.915 -1.301]

Optimal ET sequence: 17, 21e, 24, 31, 41, 72, 247c, 312bd, 384bcdd, 456bcdde, 528bcdde, 631bbccdde

Badness (Sintel): 0.770

Mandos

Subgroup: 2.3.5.7.11

Comma list: 176/175, 243/242

Mapping: [⟨1 1 0 6 2], ⟨0 2 0 5 5], ⟨0 0 1 -2 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

  • WE: ~2 = 1199.1949 ¢, ~11/9 = 350.6135 ¢, ~5/4 = 390.4090 ¢
error map: ⟨-0.805 -1.533 +2.485 +1.814 +0.139]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.5548 ¢, ~5/4 = 390.2690 ¢
error map: ⟨0.000 -0.845 +3.955 +3.410 +1.456]

Optimal ET sequence: 24, 27e, 31, 58, 89, 154d, 181cde, 212cde, 301ccde

Badness (Sintel): 0.902

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 243/242

Mapping: [⟨1 1 0 6 2 4], ⟨0 2 0 5 5 -1], ⟨0 0 1 -2 0 0]]

Optimal tunings:

  • WE: ~2 = 1198.5555 ¢, ~11/9 = 351.0300 ¢, ~5/4 = 391.0458 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.1554 ¢, ~5/4 = 391.1227 ¢

Optimal ET sequence: 24, 27e, 31, 58, 123df, 181cdeff, 239ccddeefff

Badness (Sintel): 0.863

Cuckoo

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242

Mapping: [⟨1 1 0 -3 2], ⟨0 2 0 -4 5], ⟨0 0 1 3 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

  • WE: ~2 = 1199.8222 ¢, ~11/9 = 350.4356 ¢, ~5/4 = 389.8478 ¢
error map: ⟨-0.178 -1.262 +3.178 -1.558 +0.504]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.4213 ¢, ~5/4 = 389.7308 ¢
error map: ⟨0.000 -1.112 +3.417 -1.318 +0.788]

Optimal ET sequence: 24d, 27e, 31, 58, 89, 154, 185

Badness (Sintel): 1.12

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 243/242

Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 -4 5 -10], ⟨0 0 1 3 0 5]]

Optimal tunings:

  • WE: ~2 = 1199.7103 ¢, ~11/9 = 350.5840 ¢, ~5/4 = 389.8071 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.5682 ¢, ~5/4 = 389.6104 ¢

Optimal ET sequence: 27e, 31, 58, 96d, 154

Badness (Sintel): 1.23

Parahemif

7-limit

Subgroup: 2.3.5.7

Comma list: 1605632/1594323

Mapping: [⟨1 1 0 -1], ⟨0 2 0 13], ⟨0 0 1 0]]

mapping generators: ~2, ~896/729, ~5

Optimal tunings:

  • WE: ~2 = 1199.7303 ¢, ~896/729 = 351.4056 ¢, ~5/4 = 386.8527 ¢
error map: ⟨-0.270 +0.586 -0.000 -0.284]
  • CWE: ~2 = 1200.0000 ¢, ~896/729 = 351.4569 ¢, ~5/4 = 386.6884 ¢
error map: ⟨0.000 +0.959 +0.375 +0.114]

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99, 239, 338

Badness (Sintel): 8.77

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 896/891

Mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5], ⟨0 0 1 0 0]]

Optimal tunings:

  • WE: ~2 = 1199.2633 ¢, ~11/9 = 351.3189 ¢, ~5/4 = 387.7835 ¢
error map: ⟨-0.737 -0.054 -0.004 -0.944 +3.803]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.4593 ¢, ~5/4 = 387.4226 ¢
error map: ⟨0.000 +0.964 +1.109 +0.145 +5.979]

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99e *

* optimal patent val: 123

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 243/242, 364/363

Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1], ⟨0 0 1 0 0 0]]

Optimal tunings:

  • WE: ~2 = 1198.7603 ¢, ~11/9 = 351.3275 ¢, ~5/4 = 388.7872 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.6042 ¢, ~5/4 = 388.4720 ¢

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99ef, 157eff, 290cdeeefff

Badness (Sintel): 1.12

Urania

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120

Mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5], ⟨0 0 0 1 0]]

mapping generators: ~2, ~11/9, ~7

Mapping to lattice: [⟨0 2 8 0 5], ⟨0 0 0 -1 0]]

Lattice basis:

11/9 length = 0.2536, 8/7 length = 2.807
Angle (11/9, 8/7) = 90 degrees

Optimal tunings:

  • WE: ~2 = 1201.8548 ¢, ~11/9 = 348.6318 ¢, ~5/4 = 965.0936 ¢
error map: ⟨+1.855 -2.836 +2.741 -0.023 -4.449]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.3793 ¢, ~7/4 = 965.6304 ¢
error map: ⟨0.000 -5.196 +0.721 -3.196 -9.421]

Optimal ET sequence: 7d, 14c, 17c, 24, 31, 100de, 131bdee, 162bdee

Badness (Sintel): 1.01

Complexity spectrum: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5

Scales: urania24

Mirage

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384

Mapping: [⟨1 1 3 3 2 0], ⟨0 6 -7 -2 15 0], ⟨0 0 0 0 0 1]]

mapping generators: ~2, ~15/14, ~13

Optimal tunings:

  • WE: ~2 = 1200.7626 ¢, ~15/4 = 116.7069 ¢, ~13/8 = 838.2364 ¢
error map: ⟨+0.763 -0.951 -0.974 +0.048 +0.810 -0.004]
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.6469 ¢, ~13/8 = 838.2123 ¢
error map: ⟨0.000 -2.074 -2.842 -2.120 -1.615 -2.315]

Optimal ET sequence: 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef

Badness (Sintel): 0.691

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 385/384

Mapping: [⟨1 1 3 3 2 0 0], ⟨0 6 -7 -2 15 0 4], ⟨0 0 0 0 0 1 1]]

Optimal tunings:

  • WE: ~2 = 1200.7628 ¢, ~15/14 = 116.6995 ¢, ~13/8 = 837.1672 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.6395 ¢, ~13/8 = 837.1424 ¢

Optimal ET sequence: 31, 41, 62, 72, 103, 175f, 360bcdff, 463bccdeff

Badness (Sintel): 0.715

References

  1. ↑ Yahoo! Tuning Group | Spectacle temperament?
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