User:Lucius Chiaraviglio/Musical Mad Science: Difference between revisions
→Various Lumatone mappings: Insert Shaping Consonance with Harmonic Timbre after this, starting with ''The Physics of Dissonance'' by minutephysics (2025) |
→Musical Mad Science Musings on Diatonicized Third-Tone Sub-Chromaticism: No neat one size fits all mapping for third-tones |
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## For those EDOs having a less flat fifth (but also including 19), the extension is actually much simpler, needing only 11 bright generators to get ~[[7/4]] — optimal ET sequence: 17c, 19, 36, 55 of these, only 36 qualifies for Mothra. The comma for this is a spectrum of less complicated (but still very complicated) commas from |47 0 0 1 0 0 0 0 -11⟩ ~ |-74 -11 11 1 0 0 0 0 0 0 0 0 0 0 0 11⟩, of which |3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩ has the closest 53-limit just intonation value to 0 (2.35850949135{{c}}, made using 7 instances of 23/16 and 4 instances of 384/265). | ## For those EDOs having a less flat fifth (but also including 19), the extension is actually much simpler, needing only 11 bright generators to get ~[[7/4]] — optimal ET sequence: 17c, 19, 36, 55 of these, only 36 qualifies for Mothra. The comma for this is a spectrum of less complicated (but still very complicated) commas from |47 0 0 1 0 0 0 0 -11⟩ ~ |-74 -11 11 1 0 0 0 0 0 0 0 0 0 0 0 11⟩, of which |3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩ has the closest 53-limit just intonation value to 0 (2.35850949135{{c}}, made using 7 instances of 23/16 and 4 instances of 384/265). | ||
## This leaves out 129edo, which we don't want to miss because it has a very accurate 7th harmonic; for 129edo, if we want a strong extension, we need -44 bright generators (which is +44 dark generators) — optimal ET sequence: 55, 74, 129; however, this is overly complex for all 3 members, since 55 and 74 also belong to much simpler strong extensions, while 129 qualifies for Mothra. For 129edo, this means that we can proceed by -3 bright generators (+3 dark generators), octave-reduce, and divide by 3, which simplifies to +1 dark generator and +1/3 octave; furthermore, this also works for the other EDO sizes divisible by 3, which suggests the name Alpha-Mothra (since these are both tricot and alpha-tricot, which simplifies to triploid alpha-dark_generator); optimal ET sequence: 36, 93, 129. The comma for the simplified form is a spectrum of merely highly complicated commas from [[4173281/4194304]] (|3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩) ~ [[18966528/18609625]] (|11 3 -3 3 0 0 0 0 0 0 0 0 0 0 0 -3⟩), of which [[544341/542720]] (|-11 1 -1 3 0 0 0 0 2 0 0 0 0 0 0 -1⟩) has the closest 53-limit just intonation value to 0 (5.1631554689{{c}}, smaller than the [[1029/1024|Gamelisma]] for which it substitutes). | ## This leaves out 129edo, which we don't want to miss because it has a very accurate 7th harmonic; for 129edo, if we want a strong extension, we need -44 bright generators (which is +44 dark generators) — optimal ET sequence: 55, 74, 129; however, this is overly complex for all 3 members, since 55 and 74 also belong to much simpler strong extensions, while 129 qualifies for Mothra. For 129edo, this means that we can proceed by -3 bright generators (+3 dark generators), octave-reduce, and divide by 3, which simplifies to +1 dark generator and +1/3 octave; furthermore, this also works for the other EDO sizes divisible by 3, which suggests the name Alpha-Mothra (since these are both tricot and alpha-tricot, which simplifies to triploid alpha-dark_generator); optimal ET sequence: 36, 93, 129. The comma for the simplified form is a spectrum of merely highly complicated commas from [[4173281/4194304]] (|3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩) ~ [[18966528/18609625]] (|11 3 -3 3 0 0 0 0 0 0 0 0 0 0 0 -3⟩), of which [[544341/542720]] (|-11 1 -1 3 0 0 0 0 2 0 0 0 0 0 0 -1⟩) has the closest 53-limit just intonation value to 0 (5.1631554689{{c}}, smaller than the [[1029/1024|Gamelisma]] for which it substitutes). | ||
# Embedded within 17L 2s is the standard diatonic scale [[5L 2s]], of which each large step consists of 3 large steps of 17L 2s, and each small step (the standard diatonic semitone) consists of 1 large step + 1 small step of 17L 2s. In the parent of 17L 2s, which is [[2L 15s]], the large step of 2L 15s; is the standard diatonic semitone of 5L 2s. This points the way to a rank-3 elevation of this temperament. | |||
# The large step of 17L 2s maps to the septimal third-tone ~[[28/27]], at least in 19edo, 55edo, 36edo, 53edo, and 17edo; however, this mapping stops working fairly quickly, for instance failing for 74edo and 93edo, for which it is necessary to use the large tridecimal third-tone ~[[26/25]] instead (which, however, fails for 36edo and 17edo as patent val and 17c), which means that no universal or near-universal third-tone mapping will suffice other than that obtained by stacking and octave-reducing pairs of bright generators. | |||
# The dark and bright generators of 17L 2s (and 2L 15s) consist of stacks of 7 and 8 (respectively) third-tones + 1 standard diatonic semitone; this maps correctly at least for 19edo, 55edo, 36edo, 53edo, and 17edo. Need to choose which mapping of diatonic semitone to use to get the widest range of EDOs for which this maps correctly. These potential mappings for the bright generator have not yet been added to the table (actually group of tables) below of odd harmonics for various EDO values supporting 17L 2s. | |||
Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 08:20, 4 April 2025 (UTC)<br> | Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 08:20, 4 April 2025 (UTC)<br> | ||
Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) | Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:55, 6 September 2025 (UTC) | ||
=== Table of odd harmonics for various EDO values supporting 17L 2s === | === Table of odd harmonics for various EDO values supporting 17L 2s === | ||