User:BudjarnLambeth/Regular temperament interpretation of lucite23
These are generalisations of Gordon Wery's lucite23 scale into a regular temperament.
Triencata temperament
Described by Osmium in October 2025. Relatively high accuracy; tunes the 2.3.5.13.17 subgroup (so it's a type of no-7s temperament).
Using Sintel Temperament Finder's badness, it has a badness of: 1.179.
Osmium's post
"
Looks like the 7 and 11 in [lucite] temperament are suspicious; I recommend removing them to achieve a 2.3.5.13.17-subgroup temperament of considerably more accuracy. in order of decreasing sharpness of the fifth, here are the
- [CE/CEE](<https://sintel.pythonanywhere.com/result?subgroup=17&reduce=on&weights=unweighted&target=&edos=&commas=49%2F48+66%2F65+275%2F273+561%2F560+715%2F714&submit_comma=submit>),
- [CTE](<https://sintel.pythonanywhere.com/result?subgroup=17&reduce=on&weights=tenney&target=&edos=&commas=49%2F48+66%2F65+275%2F273+561%2F560+715%2F714&submit_comma=submit>)
- and [CWE](<https://sintel.pythonanywhere.com/result?subgroup=17&reduce=on&weights=weil&target=&edos=&commas=49%2F48+66%2F65+275%2F273+561%2F560+715%2F714&submit_comma=submit>)
tunings of [lucite] temp where we can see the issue with the 7 and 11 (which becomes more pronounced cuz theyre the only ones with an unambiguous flat tendency) persists in all of these.
By contrast, for the 2.3.5.13.17 subgroup that I suggest, here are the results in order of increasing sharpness of the fifth (to keep the order as CE/CEE, CTE, CWE):
- [CE/CEE](<https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=unweighted&target=&edos=11b+34&submit_edo=submit&commas=>),
- [CTE](<https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=tenney&target=&edos=11b+34&submit_edo=submit&commas=>) and
- [CWE](<https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=weil&target=&edos=11b+34&submit_edo=submit&commas=>)
for which we also notice many much larger viable EDOs of considerable accuracy and interest.
It also makes clearer that this is related to kleismic, specifically a version of kleismic splitting the ~6/5 gen into three equal parts of 17/16, so I'd suggest a working name of "triencata", as we extend the 2.3.5.13 subgroup of cata to prime 17 by splitting 6/5 into three 17/16's. Then we know a more accurate tuning range so we can look for extensions.
"
Temperament data (x31eq)
EDOs including it
(not exhaustive)
Mappings
Equal Temperament Mappings
2 3 5 13 17
[ ⟨ 34 54 79 126 139 ]
⟨ 91 144 211 337 372 ] ⟩
Reduced Mapping
2 3 5 13 17
[ ⟨ 1 0 1 3 4 ]
⟨ 0 18 15 8 1 ] ⟩
TE tuning
TE Generator Tunings (cents)
⟨1199.6283, 105.6795]
TE Step Tunings (cents)
⟨19.81091, 5.78085]
TE Tuning Map (cents)
⟨1199.628, 1902.232, 2784.821, 4444.321, 4904.193]
TE Mistunings (cents)
⟨-0.372, 0.277, -1.492, 3.794, -0.763]
POTE tuning
POTE Generator Tunings (cents)
⟨1200.0000, 105.7123]
POTE Step Tunings (cents)
⟨19.81705, 5.78264]
POTE Tuning Map (cents)
⟨1200.000, 1902.821, 2785.684, 4445.698, 4905.712]
POTE Mistunings (cents)
⟨-0.000, 0.866, -0.630, 5.171, 0.757]
Complexity and error
Complexity 4.329045
Adjusted Error 2.360479 cents
TE Error 0.577493 cents/octave
Commas
Unison Vectors
[1, 0, 1, -2, 1⟩ (170/169)
[4, -2, 3, -1, -1⟩ (2000/1989)
[-3, 2, -2, -1, 2⟩ (2601/2600)
[-6, -5, 6, 0, 0⟩ (15625/15552)
[5, -2, 4, -3, 0⟩ (20000/19773)
[13, 1, -1, 0, -3⟩ (24576/24565)
Lucite temperament
Described by Budjarn Lambeth in October 2025. Relatively low accuracy but includes the full 17-limit.
Using Sintel Temperament Finder's badness, it has a badness of: 3.052.
Links
Temperament data (x31eq)
EDOs including it
(not exhaustive)
Mappings
Equal Temperament Mappings
2 3 5 7 11 13 17
[ ⟨ 23 36 53 64 79 85 94 ]
⟨ 11 18 26 31 38 41 45 ] ⟩
Reduced Mapping
2 3 5 7 11 13 17
[ ⟨ 1 0 1 2 3 3 4 ]
⟨ 0 18 15 9 5 8 1 ] ⟩
TE tuning
TE Generator Tunings (cents)
⟨1202.1556, 105.7334]
TE Step Tunings (cents)
⟨39.08862, 27.55611]
TE Tuning Map (cents)
⟨1202.156, 1903.200, 2788.156, 3355.911, 4135.133, 4452.334, 4914.356]
TE Mistunings (cents)
⟨2.156, 1.245, 1.842, -12.915, -16.184, 11.806, 9.400]
POTE tuning
POTE Generator Tunings (cents)
⟨1200.0000, 105.5438]
POTE Step Tunings (cents)
⟨39.01853, 27.50670]
POTE Tuning Map (cents)
⟨1200.000, 1899.788, 2783.157, 3349.894, 4127.719, 4444.350, 4905.544]
POTE Mistunings (cents)
⟨0.000, -2.167, -3.157, -18.932, -23.599, 3.822, 0.588]
Complexity and error
Complexity 3.761826
Adjusted Error 12.398890 cents
TE Error 3.033395 cents/octave
Commas
Unison Vectors
[-4, -1, 0, 2, 0, 0, 0⟩ (49/48)
[1, 1, -1, 0, 1, -1, 0⟩ (66/65)
[0, -1, 2, -1, 1, -1, 0⟩ (275/273)
[-4, 1, -1, -1, 1, 0, 1⟩ (561/560)
[-1, -1, 1, -1, 1, 1, -1⟩ (715/714)