User:BudjarnLambeth/Regular temperament interpretation of lucite23: Difference between revisions

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Created page with "This is a generalisation of Gordon Wery's frost23 scale into a regular temperament. It could be called '''frost''' temperament. == x31eq link == * [https://x31eq.com/pyscript/rt.html?ets=23de_11b&limit=17 23de&11b (17-limit) - x31eq.com] == Temperament data == === 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 ] ⟨..."
 
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This is a generalisation of [[Gordon Wery]]'s [[343edo#Scales|frost23]] scale into a regular temperament. It could be called '''frost''' temperament.
These are generalisations of [[Gordon Wery]]'s [[343edo#Scales|lucite23]] scale into a [[regular temperament]].


== x31eq link ==
== 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 ===
 
<big><big>"</big></big>
 
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.
 
<big><big>"</big></big>
 
=== Temperament data (x31eq) ===
 
==== [[EDO]]s including it ====
''(not exhaustive)''
* {{EDOs|34, 68, 102f, 91p, 125p, 57p, 159f, 23p, 136f, 193f}}
 
==== 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 ===
* [https://x31eq.com/pyscript/rt.html?ets=23de_11b&limit=17 23de&11b (17-limit) - x31eq.com]
* [https://x31eq.com/pyscript/rt.html?ets=23de_11b&limit=17 23de&11b (17-limit) - x31eq.com]
* [https://sintel.pythonanywhere.com/result?subgroup=17&reduce=on&weights=weil&target=&edos=&commas=49%2F48%2C+66%2F65%2C+275%2F273%2C+561%2F560%2C+715%2F714&submit_comma=submit 23de&11b (17-limit) - sintel.pythonanywhere.com]
=== Temperament data (x31eq) ===


== Temperament data ==
==== [[EDO]]s including it ====
''(not exhaustive)''
* {{EDOs|23de, 34e, 11b, 57ddee, 12bbcde, 45bce, 46bcddee, 68dee, 22bbcf, 80bcdddeee}}


=== Mappings ===
==== Mappings ====
Equal Temperament Mappings
Equal Temperament Mappings


Line 24: Line 148:
⟨ 0 18 15 9 5 8 1 ] ⟩
⟨ 0 18 15 9 5 8 1 ] ⟩
   
   
=== TE tuning ===
==== TE tuning ====
TE Generator Tunings (cents)
TE Generator Tunings (cents)


Line 44: Line 168:
⟨2.156, 1.245, 1.842, -12.915, -16.184, 11.806, 9.400]
⟨2.156, 1.245, 1.842, -12.915, -16.184, 11.806, 9.400]


=== POTE tuning ===
==== POTE tuning ====
POTE Generator Tunings (cents)
POTE Generator Tunings (cents)


Line 64: Line 188:
⟨0.000, -2.167, -3.157, -18.932, -23.599, 3.822, 0.588]
⟨0.000, -2.167, -3.157, -18.932, -23.599, 3.822, 0.588]


=== Complexity and error ===
==== Complexity and error ====
Complexity 3.761826
Complexity 3.761826


Line 71: Line 195:
TE Error 3.033395 cents/octave
TE Error 3.033395 cents/octave


=== Commas ===
==== Commas ====
Unison Vectors
Unison Vectors


Latest revision as of 02:48, 21 October 2025

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

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):

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)

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