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

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-This is a generalisation of [[Gordon Wery]]'s [[343edo#Scales|frost23]] scale into a [[regular temperament]] in the [[17-limit]].
+These are generalisations of [[Gordon Wery]]'s [[343edo#Scales|lucite23]] scale into a [[regular temperament]].
-It could be called '''frost''' 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 ===
+<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
+	3	5	13	17
+[ ⟨	34	54	79	126	139	]
+⟨	91	144	211	337	372	] ⟩
+Reduced Mapping
+	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 ==
+=== Links ===
 * [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 (x31eq) ===
+==== [[EDO]]s including it ====
+''(not exhaustive)''
+* {{EDOs|23de, 34e, 11b, 57ddee, 12bbcde, 45bce, 46bcddee, 68dee, 22bbcf, 80bcdddeee}}
-=== Mappings ===
+==== Mappings ====
 Equal Temperament Mappings
@@ Line 29: / Line 148: @@
 ⟨	0	18	15	9	5	8	1	] ⟩
-=== TE tuning ===
+==== TE tuning ====
 TE Generator Tunings (cents)
@@ Line 49: / Line 168: @@
 ⟨2.156, 1.245, 1.842, -12.915, -16.184, 11.806, 9.400]
-=== POTE tuning ===
+==== POTE tuning ====
 POTE Generator Tunings (cents)
@@ Line 69: / Line 188: @@
 ⟨0.000, -2.167, -3.157, -18.932, -23.599, 3.822, 0.588]
-=== Complexity and error ===
+==== Complexity and error ====
 Complexity	3.761826
@@ Line 76: / Line 195: @@
 TE Error	3.033395 cents/octave
-=== Commas ===
+==== Commas ====
 Unison Vectors
@@ Line 88: / Line 207: @@
 [-1, -1, 1, -1, 1, 1, -1⟩ ([[715/714]])
-=== Lower rank temperaments including it ===
-''(not exhaustive)''
-* {{EDOs|23de, 34e, 11b, 57ddee, 12bbcde, 45bce, 46bcddee, 68dee, 22bbcf, 80bcdddeee}}