Olympic clan: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings
Line 12: Line 12:
: mapping generators: ~2, ~3, ~7
: mapping generators: ~2, ~3, ~7


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.0805{{c}}, ~7/4 = 969.0275{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9460{{c}}, ~3/2 = 702.0489{{c}}, ~7/4 = 968.9839{{c}}
: [[error map]]: {{val| -0.054 +0.040 +0.050 +0.038 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0853{{c}}, ~7/4 = 969.0242{{c}}
: error map: {{val| 0.000 +0.130 +0.198 +0.207 }}
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0805{{c}}, ~7/4 = 969.0275{{c}} -->


{{Optimal ET sequence|legend=1| 41, 87, 89, 94, 130, 135, 359, 400, 494, 535, 670, 805, 1164, 1299, 1834, 1969, 5102bde, 5237bde, 7206bddee, 10339bbdddeee }}
{{Optimal ET sequence|legend=1| 41, 87, 89, 94, 130, 135, 359, 400, 494, 535, 670, 805, 1164, 1299, 1834, 1969, 5102bde, 5237bde, 7206bddee, 10339bbdddeee }}


[[Badness]] (Smith): 0.0183 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.267


=== Overview to extensions ===
=== Overview to extensions ===
Line 38: Line 43:
{{Mapping|legend=1| 1 0 15 0 17 | 0 1 -8 0 -5 | 0 0 0 1 -2 }}
{{Mapping|legend=1| 1 0 15 0 17 | 0 1 -8 0 -5 | 0 0 0 1 -2 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.7405{{c}}, ~7/4 = 969.6950{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9335{{c}}, ~3/2 = 701.6817{{c}}, ~7/4 = 969.6199{{c}}
: [[error map]]: {{val| -0.067 -0.340 -0.233 +0.661 +0.502 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7248{{c}}, ~7/4 = 969.6728{{c}}
: error map: {{val| 0.000 -0.230 -0.112 +0.847 +0.713 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7405{{c}}, ~7/4 = 969.6950{{c}} -->


{{Optimal ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 537, 578, 761d, 985d, 1115de }}
{{Optimal ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 537, 578, 761d, 985d, 1115de }}


[[Badness]] (Smith): 1.18 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.41


=== 13-limit ===
=== 13-limit ===
Line 51: Line 61:
Mapping: {{mapping| 1 0 15 0 17 -3 | 0 1 -8 0 -5 6 | 0 0 0 1 -2 -1 }}
Mapping: {{mapping| 1 0 15 0 17 -3 | 0 1 -8 0 -5 6 | 0 0 0 1 -2 -1 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.7333{{c}}, ~7/4 = 969.7085{{c}}
Optimal tunings:
* WE: ~2 = 1199.9399{{c}}, ~3/2 = 701.6899{{c}}, ~7/4 = 969.6107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7264{{c}}, ~7/4 = 969.6672{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7333{{c}}, ~7/4 = 969.7085{{c}} -->


{{Optimal ET sequence|legend=0| 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d }}
{{Optimal ET sequence|legend=0| 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d }}


Badness (Smith): 0.833 × 10<sup>-3</sup>
Badness (Sintel): 0.779


== Baffin ==
== Baffin ==
Line 66: Line 79:
: mapping generators: ~2, ~8192/4725, ~5
: mapping generators: ~2, ~8192/4725, ~5


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~8192/4725 = 951.0868{{c}}, ~5/4 = 386.6183{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9033{{c}}, ~8192/4725 = 951.0102{{c}}, ~5/4 = 386.5872{{c}}
: [[error map]]: {{val| -0.097 +0.065 +0.080 +0.059 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8192/4725 = 951.0899{{c}}, ~5/4 = 386.6184{{c}}
: error map: {{val| 0.000 +0.225 +0.305 +0.308 }}
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~8192/4725 = 951.0868{{c}}, ~5/4 = 386.6183{{c}} -->


{{Optimal ET sequence|legend=1| 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd }}
{{Optimal ET sequence|legend=1| 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd }}


[[Badness]] (Smith): 0.865 × 10<sup>-3</sup>
[[Badness]] (Sintel): 3.82


=== 11-limit ===
=== 11-limit ===
Line 78: Line 96:


{{Mapping|legend=1| 1 0 0 13 -9 | 0 2 0 -7 4 | 0 0 1 -2 4 }}
{{Mapping|legend=1| 1 0 0 13 -9 | 0 2 0 -7 4 | 0 0 1 -2 4 }}
: mapping generators: ~2, ~400/231, ~5


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~400/231 = 951.0585{{c}}, ~5/4 = 386.7912{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9208{{c}}, ~400/231 = 950.9957{{c}}, ~5/4 = 386.7656{{c}}
: [[error map]]: {{val| -0.079 +0.036 +0.293 -0.040 -0.193 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 951.0635{{c}}, ~5/4 = 386.7769{{c}}
: error map: {{val| 0.000 +0.172 +0.463 +0.176 +0.044 }}
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~400/231 = 951.0585{{c}}, ~5/4 = 386.7912{{c}} -->


{{Optimal ET sequence|legend=0| 34, 43, 53, 87, 130, 183, 270, 670, 940, 1210, 2063c }}
{{Optimal ET sequence|legend=0| 34, 43, 53, 87, 130, 183, 270, 670, 940, 1210, 2063c }}


[[Badness]] (Smith): 0.976 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.17


=== 13-limit ===
=== 13-limit ===
Line 92: Line 116:
Mapping: {{mapping| 1 0 0 13 -9 1 | 0 2 0 -7 4 3 | 0 0 1 -2 4 1 }}
Mapping: {{mapping| 1 0 0 13 -9 1 | 0 2 0 -7 4 3 | 0 0 1 -2 4 1 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~26/15 = 951.0882{{c}}, ~5/4 = 386.7507{{c}}
Optimal tunings:
* WE: ~2 = 1199.9633{{c}}, ~400/231 = 951.0591{{c}}, ~5/4 = 386.7388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.0896{{c}}, ~5/4 = 386.7451{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0882{{c}}, ~5/4 = 386.7507{{c}} -->


{{Optimal ET sequence|legend=0| 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f }}
{{Optimal ET sequence|legend=0| 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f }}


Badness (Smith): 0.604 × 10<sup>-3</sup>
Badness (Sintel): 0.565


Complexity spectrum: 15/13, 16/15, 13/12, 4/3, 16/13, 5/4, 18/13, 13/10, 6/5, 9/8, 11/10, 8/7, 7/5, 15/11, 10/9, 13/11, 15/14, 11/8, 7/6, 14/13, 12/11, 9/7, 11/9, 14/11
Complexity spectrum: 15/13, 16/15, 13/12, 4/3, 16/13, 5/4, 18/13, 13/10, 6/5, 9/8, 11/10, 8/7, 7/5, 15/11, 10/9, 13/11, 15/14, 11/8, 7/6, 14/13, 12/11, 9/7, 11/9, 14/11
Line 106: Line 133:


{{Mapping|legend=1| 1 0 2 3 11 | 0 1 0 0 -5 | 0 0 5 -3 6 }}
{{Mapping|legend=1| 1 0 2 3 11 | 0 1 0 0 -5 | 0 0 5 -3 6 }}
: mapping generators: ~2, ~3, ~256/245


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.3024{{c}}, ~256/245 = 77.1952{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9947{{c}}, ~3/2 = 702.2994{{c}}, ~256/245 = 77.1949{{c}}
: [[error map]]: {{val| -0.005 +0.339 -0.350 -0.426 +0.323 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3025{{c}}, ~256/245 = 77.1949{{c}}
: error map: {{val| 0.000 +0.347 -0.339 -0.411 +0.339 }}
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3024{{c}}, ~256/245 = 77.1952{{c}} -->


{{Optimal ET sequence|legend=1| 46, 94, 140, 171, 217, 311, 979, 1290 }}
{{Optimal ET sequence|legend=1| 46, 94, 140, 171, 217, 311, 979, 1290 }}


[[Badness]] (Smith): 3.78 × 10<sup>-3</sup>
[[Badness]] (Sintel): 4.54


=== 13-limit ===
=== 13-limit ===
Line 120: Line 153:
Mapping: {{mapping| 1 0 2 3 11 7 | 0 1 0 0 -5 -2 | 0 0 5 -3 6 -2 }}
Mapping: {{mapping| 1 0 2 3 11 7 | 0 1 0 0 -5 -2 | 0 0 5 -3 6 -2 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.3319{{c}}, ~117/112 = 77.2152{{c}}
Optimal tunings:
* WE: ~2 = 1199.9732{{c}}, ~3/2 = 702.3162{{c}}, ~117/112 = 77.2135{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3335{{c}}, ~117/112 = 77.2145{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3319{{c}}, ~117/112 = 77.2152{{c}} -->


{{Optimal ET sequence|legend=0| 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290 }}
{{Optimal ET sequence|legend=0| 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290 }}


Badness (Smith): 1.67 × 10<sup>-3</sup>
Badness (Sintel): 1.56


=== 17-limit ===
=== 17-limit ===
Line 133: Line 169:
Mapping: {{mapping| 1 0 2 3 11 7 7 | 0 1 0 0 -5 -2 -2 | 0 0 5 -3 6 -2 4 }}
Mapping: {{mapping| 1 0 2 3 11 7 7 | 0 1 0 0 -5 -2 -2 | 0 0 5 -3 6 -2 4 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 702.3205{{c}}, ~68/65 = 77.2255{{c}}
Optimal tunings:
* WE: ~2 = 1199.9956{{c}}, ~3/2 = 702.3179{{c}}, ~68/65 = 77.2252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3207{{c}}, ~68/65 = 77.2254{{c}}
<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3205{{c}}, ~68/65 = 77.2255{{c}} -->


{{Optimal ET sequence|legend=0| 46, 77e, 94, 140, 171, 217, 311, 668, 839e, 979g }}
{{Optimal ET sequence|legend=0| 46, 77e, 94, 140, 171, 217, 311, 668, 839e, 979g }}


Badness (Smith): 0.989 × 10<sup>-3</sup>
Badness (Sintel): 0.940


[[Category:Temperament clans]]
[[Category:Temperament clans]]

Revision as of 12:36, 6 February 2026

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 olympic clan of rank-3 temperaments tempers out the olympia (ratio: 131072/130977, monzo: [17 -5 0 -2 -1⟩). This has the effect of equating the undecimal quartertone (33/32) with a stack of two septimal commas (64/63).

For the rank-4 olympic temperament, see Rank-4 temperament #Olympic (131072/130977).

Olympian

Subgroup: 2.3.7.11

Comma list: 131072/130977

Subgroup-val mapping: [⟨1 0 0 17], ⟨0 1 0 -5], ⟨0 0 1 -2]]

mapping generators: ~2, ~3, ~7

Optimal tunings:

  • WE: ~2 = 1199.9460 ¢, ~3/2 = 702.0489 ¢, ~7/4 = 968.9839 ¢
error map: ⟨-0.054 +0.040 +0.050 +0.038]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0853 ¢, ~7/4 = 969.0242 ¢
error map: ⟨0.000 +0.130 +0.198 +0.207]

Optimal ET sequence: 41, 87, 89, 94, 130, 135, 359, 400, 494, 535, 670, 805, 1164, 1299, 1834, 1969, 5102bde, 5237bde, 7206bddee, 10339bbdddeee

Badness (Sintel): 0.267

Overview to extensions

The second comma in the comma list determines how we extend olympian to include the harmonic 5. Akea adds 385/384, and finds the harmonic 5 by equating the syntonic comma (81/80) with the septimal comma. Orthoschismic adds 32805/32768, and finds the harmonic 5 on the chain of fifths. Cassaschismic adds 19712/19683 with an independent generator for harmonic 5. Pessoal adds 9801/9800, splitting the octave into two. Lif adds 2401/2400, splitting the perfect fifth into two. Baffin adds 5632/5625, splitting the perfect twelfth into two. Lux adds 3025/3024, splitting the ~21/16 into two. Hera adds 6144/6125 or 8019/8000, splitting the ~21/16 into three. Finally, sophia adds 42875/42768, splitting the ~8/7 into three. These all have neat extensions to the 13-limit via tempering out both 2080/2079 and 4096/4095.

Temperaments discussed elsewhere are:

Considered below are orthoschismic, baffin and sophia.

Orthoschismic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 32805/32768

Mapping: [⟨1 0 15 0 17], ⟨0 1 -8 0 -5], ⟨0 0 0 1 -2]]

Optimal tunings:

  • WE: ~2 = 1199.9335 ¢, ~3/2 = 701.6817 ¢, ~7/4 = 969.6199 ¢
error map: ⟨-0.067 -0.340 -0.233 +0.661 +0.502]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7248 ¢, ~7/4 = 969.6728 ¢
error map: ⟨0.000 -0.230 -0.112 +0.847 +0.713]

Optimal ET sequence: 41, 53, 89, 94, 130, 183, 224, 354, 537, 578, 761d, 985d, 1115de

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4096/4095

Mapping: [⟨1 0 15 0 17 -3], ⟨0 1 -8 0 -5 6], ⟨0 0 0 1 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1199.9399 ¢, ~3/2 = 701.6899 ¢, ~7/4 = 969.6107 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7264 ¢, ~7/4 = 969.6672 ¢

Optimal ET sequence: 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d

Badness (Sintel): 0.779

Baffin

7-limit (decovulture)

Subgroup: 2.3.5.7

Comma list: 67108864/66976875

Mapping: [⟨1 0 0 13], ⟨0 2 0 -7], ⟨0 0 1 -2]]

mapping generators: ~2, ~8192/4725, ~5

Optimal tunings:

  • WE: ~2 = 1199.9033 ¢, ~8192/4725 = 951.0102 ¢, ~5/4 = 386.5872 ¢
error map: ⟨-0.097 +0.065 +0.080 +0.059]
  • CWE: ~2 = 1200.0000 ¢, ~8192/4725 = 951.0899 ¢, ~5/4 = 386.6184 ¢
error map: ⟨0.000 +0.225 +0.305 +0.308]

Optimal ET sequence: 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd

Badness (Sintel): 3.82

11-limit

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 131072/130977

Mapping: [⟨1 0 0 13 -9], ⟨0 2 0 -7 4], ⟨0 0 1 -2 4]]

mapping generators: ~2, ~400/231, ~5

Optimal tunings:

  • WE: ~2 = 1199.9208 ¢, ~400/231 = 950.9957 ¢, ~5/4 = 386.7656 ¢
error map: ⟨-0.079 +0.036 +0.293 -0.040 -0.193]
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 951.0635 ¢, ~5/4 = 386.7769 ¢
error map: ⟨0.000 +0.172 +0.463 +0.176 +0.044]

Optimal ET sequence: 34, 43, 53, 87, 130, 183, 270, 670, 940, 1210, 2063c

Badness (Sintel): 1.17

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4096/4095

Mapping: [⟨1 0 0 13 -9 1], ⟨0 2 0 -7 4 3], ⟨0 0 1 -2 4 1]]

Optimal tunings:

  • WE: ~2 = 1199.9633 ¢, ~400/231 = 951.0591 ¢, ~5/4 = 386.7388 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 951.0896 ¢, ~5/4 = 386.7451 ¢

Optimal ET sequence: 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f

Badness (Sintel): 0.565

Complexity spectrum: 15/13, 16/15, 13/12, 4/3, 16/13, 5/4, 18/13, 13/10, 6/5, 9/8, 11/10, 8/7, 7/5, 15/11, 10/9, 13/11, 15/14, 11/8, 7/6, 14/13, 12/11, 9/7, 11/9, 14/11

Sophia

Subgroup: 2.3.5.7.11

Comma list: 42875/42768, 131072/130977

Mapping: [⟨1 0 2 3 11], ⟨0 1 0 0 -5], ⟨0 0 5 -3 6]]

mapping generators: ~2, ~3, ~256/245

Optimal tunings:

  • WE: ~2 = 1199.9947 ¢, ~3/2 = 702.2994 ¢, ~256/245 = 77.1949 ¢
error map: ⟨-0.005 +0.339 -0.350 -0.426 +0.323]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3025 ¢, ~256/245 = 77.1949 ¢
error map: ⟨0.000 +0.347 -0.339 -0.411 +0.339]

Optimal ET sequence: 46, 94, 140, 171, 217, 311, 979, 1290

Badness (Sintel): 4.54

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095, 13720/13689

Mapping: [⟨1 0 2 3 11 7], ⟨0 1 0 0 -5 -2], ⟨0 0 5 -3 6 -2]]

Optimal tunings:

  • WE: ~2 = 1199.9732 ¢, ~3/2 = 702.3162 ¢, ~117/112 = 77.2135 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3335 ¢, ~117/112 = 77.2145 ¢

Optimal ET sequence: 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290

Badness (Sintel): 1.56

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 833/832, 1156/1155, 4096/4095

Mapping: [⟨1 0 2 3 11 7 7], ⟨0 1 0 0 -5 -2 -2], ⟨0 0 5 -3 6 -2 4]]

Optimal tunings:

  • WE: ~2 = 1199.9956 ¢, ~3/2 = 702.3179 ¢, ~68/65 = 77.2252 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3207 ¢, ~68/65 = 77.2254 ¢

Optimal ET sequence: 46, 77e, 94, 140, 171, 217, 311, 668, 839e, 979g

Badness (Sintel): 0.940

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