User:Eufalesio/12edo detemperament sequences: Difference between revisions

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== 13-limit ==

Finding 13-limit 12edo detemperaments is harder, because 12edo notably does a horrible job of approximating primes 11 and 13, which means we have another vals to consider; which is to say, we don't take the direct approximation of some primes, but the next best. Here are 3 vals that I found to be worth of mention.

[[File:12detempSequence.png|center|thumb|931x931px|Summary of the 12edo paths I took my time to analyze. For the unnamed ones, I won't be the one to name them.]]

=== 12e ===

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Maps 11/8 to a tritone, 13/8 to a major sixth. It is the best val temperament wise. All the corrections of primes 5 to 13 are in the same direction. There are two good paths to consider, through tridecimal compton and through diaschismic. Through tridecimal compton, the path ends at [[Marvel family#Helios|helios]], yet another marvel temperament, which either way is very similar to compton still... so just use 72edo.

Through diaschismic, things get brighter. I don't really like diaschismic, but it can be detempered into the rank-3 46 & 58 & 270, which is more accurate than Ultimate, altering primes 5-13 by -2, -3, -5, -6 half-[[176/175|valinorsmas]]. It can be detempered into the ultra-accurate rank-4 {9801/9800, 123201/123200} by ~~observing~~ [[~~Ibnsinma|ibnsinmas~~]], and ends at rank-5 [[chalmersic]].

Through diaschismic, things get brighter. I don't really like diaschismic, but it can be detempered into the rank-3 46 & 58 & 270, which is more accurate than Ultimate, altering primes 5-13 by -2, -3, -5, -6 half-[[176/175|valinorsmas]]. It can be detempered into the ultra-accurate rank-4 [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13&reduce=off&weights=unweighted&target=&edos=46+%26+58+%26+270+%26+2460&submit_edo=submit&commas= {9801/9800, 123201/123200}] by subtracting an [[ibnsinma]] to primes 7,11,13, and ends at rank-5 [[chalmersic]].

=== 12de ===

Maps 7/4 to a major sixth, 11/8 to a fourth, 13/8 to a minor sixth. It is an [[exotemperament]]; but... if you can stomach the weirdly mapped 7/4, you can go places. The diaschisma is also tempered out but you go through [[srutal]], and this can be dempered into the best 13-limit temperament: [[donar]]. It is much more accurate than Ultimate and 46 & 58 & 270. It alters primes 5-13 by -2, +8, +6, +5 half-valinorsmas. Donar then can be detempered into {9801/9800, 123201/123200} or {4375/4374, 4225/4224}, both to chalmersic.

Maps 7/4 to a major sixth, 11/8 to a fourth, 13/8 to a minor sixth. It is an [[exotemperament]]; but... if you can stomach the weirdly mapped 7/4, you can go places. The diaschisma is also tempered out but you go through [[srutal]], and this can be dempered into the best 13-limit temperament: [[donar]]. It is much more accurate than Ultimate and 46 & 58 & 270. It alters primes 5-13 by -2, +8, +6, +5 half-valinorsmas. Donar then can be detempered into {9801/9800, 123201/123200} or [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13&reduce=off&weights=unweighted&target=&edos=46+%26+58+%26+270+%26+2460&commas=4375%2F4374%0D%0AS65&submit_comma=submit {4375/4374, 4225/4224}], both to chalmersic.

Through the pontiac route you get [[Schismatic family#Quadrapont|Quadrapont]], pontiac with 4th octaves, You either way can detemper into donar, or whatever the hell 12de & 53 & 171 is. That thing can only detemper into {4375/4374, 4225/4224} by observing the schisma.

== 19-limit ==

Extending the paths into the 19-limit is tricky for some val sequences. Here is a brief summary of my finds:

=== 12e ===

Cassandra and Ultimate are both easy 19-limit temperaments. By tempering out 1225/1224 and 1216/1215, the temps can be strongly extended to include the two new primes. Ultimate is notable as arguably the best [[yazalathana]], altering the minor third of cassandra by +1 minicomma with great accuracy and negligible extra error.

17 is more complex and less accurately tuned, but still within reasonable use and accuracy... though, it needs the 12e'''g''' val. This can mitigated if you use [[Buzzardsma|buzzardmas]] as accidentals... but still, kinda sorta not the best.

In Olympic, you '''must''' use the weak extension [[Catalog of rank-4 temperaments#Metaolympic|Metaolympic]], in which your garischisma is split in two heavily inflated [[10241/10240]] halves, and you use one of these halves to reach 17/16 and 19/16, -1 and +1 respectively. Through [[User:Eufalesio/Ultimate#Insatinismic AKA Batshittin|Batshittin]] (a weak extension of Batshit), you use the same 10241/10240 and [[Mina|minas]]; 4096/4095 ~ 4225/4224.

=== 12f ===

17 is very easy to reach in diaschismic, equated with 16/15. 19/16 is kinda shit... the best extension I could find is 46h & 58, which is the one that can extend beyond better. [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.19&reduce=off&weights=unweighted&target=&edos=46h+%26+58+%26+270&submit_edo=submit&commas=%0D%0AS65%0D%0A4375%2F4374%0D%0A14400%2F14399%0D%0AS2431 46h & 58 & 270] is very good in the yazalathana, but sucks for prime 17 [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.17.19&reduce=off&weights=unweighted&target=&edos=46h+%26+58+%26+270&submit_edo=submit&commas=%0D%0AS65%0D%0A4375%2F4374%0D%0A14400%2F14399%0D%0AS2431 (as seen here)], which is considerably less accurate.

However, things get better as you use the rank-4, as 10241/10240 and the [[scintillisma]] are added into the comma mix. This alone is a top-tier temperament, to an extent easily writable in a score (though 19/16 is extremely painful). The chalmersic extension adds the scintillisma and 104976/104975 or 228096/228095 into the mix, requiring kalismas to notate 17/16 and 19/16, not primes 2-13.

As an extra... the rank-4 apparently has a [https://sintel.pythonanywhere.com/result?subgroup=23&reduce=off&weights=unweighted&target=&edos=46h+%26+58+%26+270+%26+2460&submit_edo=submit&commas=%0D%0AS65%0D%0A4375%2F4374%0D%0A14400%2F14399%0D%0AS2431 strong extension] to the 23-limit! I don't know what that is, though.

=== 12de ===

This is one sequence I haven't explored too much beyond the 19-limit, as results were inconclusive. 17 is very easy to reach in srutal, equated with 16/15. 19-limit? Not that good, and not that extendable. Donar has an accurate but unusable extension to the 19-limit, requiring a jaw dropping +24 half-valinorsmas to reach qualities of 17/16 and 19/16. I don't deem it easily extendable beyond this.

@@ Line 78: / Line 78: @@
 == 13-limit ==
 Finding 13-limit 12edo detemperaments is harder, because 12edo notably does a horrible job of approximating primes 11 and 13, which means we have another vals to consider; which is to say, we don't take the direct approximation of some primes, but the next best. Here are 3 vals that I found to be worth of mention.
+[[File:12detempSequence.png|center|thumb|931x931px|Summary of the 12edo paths I took my time to analyze. For the unnamed ones, I won't be the one to name them.]]
 === 12e ===
@@ Line 89: / Line 90: @@
 Maps 11/8 to a tritone, 13/8 to a major sixth. It is the best val temperament wise. All the corrections of primes 5 to 13 are in the same direction. There are two good paths to consider, through tridecimal compton and through diaschismic. Through tridecimal compton, the path ends at [[Marvel family#Helios|helios]], yet another marvel temperament, which either way is very similar to compton still... so just use 72edo.
-Through diaschismic, things get brighter. I don't really like diaschismic, but it can be detempered into the rank-3 46 & 58 & 270, which is more accurate than Ultimate, altering primes 5-13 by -2, -3, -5, -6 half-[[176/175|valinorsmas]]. It can be detempered into the ultra-accurate rank-4 {9801/9800, 123201/123200} by observing [[Ibnsinma|ibnsinmas]], and ends at rank-5 [[chalmersic]].
+Through diaschismic, things get brighter. I don't really like diaschismic, but it can be detempered into the rank-3 46 & 58 & 270, which is more accurate than Ultimate, altering primes 5-13 by -2, -3, -5, -6 half-[[176/175|valinorsmas]]. It can be detempered into the ultra-accurate rank-4 [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13&reduce=off&weights=unweighted&target=&edos=46+%26+58+%26+270+%26+2460&submit_edo=submit&commas= {9801/9800, 123201/123200}] by subtracting an [[ibnsinma]] to primes 7,11,13, and ends at rank-5 [[chalmersic]].
 === 12de ===
-Maps 7/4 to a major sixth, 11/8 to a fourth, 13/8 to a minor sixth. It is an [[exotemperament]]; but... if you can stomach the weirdly mapped 7/4, you can go places. The diaschisma is also tempered out but you go through [[srutal]], and this can be dempered into the best 13-limit temperament: [[donar]]. It is much more accurate than Ultimate and 46 & 58 & 270. It alters primes 5-13 by -2, +8, +6, +5 half-valinorsmas. Donar then can be detempered into {9801/9800, 123201/123200} or {4375/4374, 4225/4224}, both to chalmersic.
+Maps 7/4 to a major sixth, 11/8 to a fourth, 13/8 to a minor sixth. It is an [[exotemperament]]; but... if you can stomach the weirdly mapped 7/4, you can go places. The diaschisma is also tempered out but you go through [[srutal]], and this can be dempered into the best 13-limit temperament: [[donar]]. It is much more accurate than Ultimate and 46 & 58 & 270. It alters primes 5-13 by -2, +8, +6, +5 half-valinorsmas. Donar then can be detempered into {9801/9800, 123201/123200} or [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13&reduce=off&weights=unweighted&target=&edos=46+%26+58+%26+270+%26+2460&commas=4375%2F4374%0D%0AS65&submit_comma=submit {4375/4374, 4225/4224}], both to chalmersic.
 Through the pontiac route you get [[Schismatic family#Quadrapont|Quadrapont]], pontiac with 4th octaves, You either way can detemper into donar, or whatever the hell 12de & 53 & 171 is. That thing can only detemper into {4375/4374, 4225/4224} by observing the schisma.
+== 19-limit ==
+Extending the paths into the 19-limit is tricky for some val sequences. Here is a brief summary of my finds:
+=== 12e ===
+Cassandra and Ultimate are both easy 19-limit temperaments. By tempering out 1225/1224 and 1216/1215, the temps can be strongly extended to include the two new primes. Ultimate is notable as arguably the best [[yazalathana]], altering the minor third of cassandra by +1 minicomma with great accuracy and negligible extra error.
+is more complex and less accurately tuned, but still within reasonable use and accuracy... though, it needs the 12e'''g''' val. This can mitigated if you use [[Buzzardsma|buzzardmas]] as accidentals... but still, kinda sorta not the best.
+In Olympic, you '''must''' use the weak extension [[Catalog of rank-4 temperaments#Metaolympic|Metaolympic]], in which your garischisma is split in two heavily inflated [[10241/10240]] halves, and you use one of these halves to reach 17/16 and 19/16, -1 and +1 respectively. Through [[User:Eufalesio/Ultimate#Insatinismic AKA Batshittin|Batshittin]] (a weak extension of Batshit), you use the same 10241/10240 and [[Mina|minas]]; 4096/4095 ~ 4225/4224.
+=== 12f ===
+is very easy to reach in diaschismic, equated with 16/15. 19/16 is kinda shit... the best extension I could find is 46h & 58, which is the one that can extend beyond better. [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.19&reduce=off&weights=unweighted&target=&edos=46h+%26+58+%26+270&submit_edo=submit&commas=%0D%0AS65%0D%0A4375%2F4374%0D%0A14400%2F14399%0D%0AS2431 46h & 58 & 270] is very good in the yazalathana, but sucks for prime 17 [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.17.19&reduce=off&weights=unweighted&target=&edos=46h+%26+58+%26+270&submit_edo=submit&commas=%0D%0AS65%0D%0A4375%2F4374%0D%0A14400%2F14399%0D%0AS2431 (as seen here)], which is considerably less accurate.
+However, things get better as you use the rank-4, as 10241/10240 and the [[scintillisma]] are added into the comma mix. This alone is a top-tier temperament, to an extent easily writable in a score (though 19/16 is extremely painful). The chalmersic extension adds the scintillisma and 104976/104975 or 228096/228095 into the mix, requiring kalismas to notate 17/16 and 19/16, not primes 2-13.
+As an extra... the rank-4 apparently has a [https://sintel.pythonanywhere.com/result?subgroup=23&reduce=off&weights=unweighted&target=&edos=46h+%26+58+%26+270+%26+2460&submit_edo=submit&commas=%0D%0AS65%0D%0A4375%2F4374%0D%0A14400%2F14399%0D%0AS2431 strong extension] to the 23-limit! I don't know what that is, though.
+=== 12de ===
+This is one sequence I haven't explored too much beyond the 19-limit, as results were inconclusive. 17 is very easy to reach in srutal, equated with 16/15. 19-limit? Not that good, and not that extendable. Donar has an accurate but unusable extension to the 19-limit, requiring a jaw dropping +24 half-valinorsmas to reach qualities of 17/16 and 19/16. I don't deem it easily extendable beyond this.