Parakleismic: Difference between revisions
Jump to navigation
Jump to search
Cmloegcmluin (talk | contribs) →Tuning spectrum: improve and standardize tuning spectra tables |
m Text replacement - "(unchanged-interval)" to "(unchanged interval)" |
||
| (6 intermediate revisions by 4 users not shown) | |||
| Line 1: | Line 1: | ||
'''Parakleismic''' is the microtemperament tempering out the [[parakleisma]] in the 5-limit. This article also assumes the canonical mapping for 7, which means tempering out [[3136/3125]] and [[4375/4374]] in the 7-limit. | '''Parakleismic''' is the microtemperament tempering out the [[parakleisma]] in the 5-limit. This article also assumes the canonical mapping for 7, which means tempering out [[3136/3125]] and [[4375/4374]] in the 7-limit. | ||
Parakleismic is much | Parakleismic is much like [[catakleismic]] but a good tuning has the generator ([[6/5]]) flat, instead of sharp, than the just version. The sixth generator step is half a [[syntonic comma]] flat of the harmonic 3. Consequently, the 12th generator step is mapped to [[10/9]] instead of [[9/8]], and the 13th generator step is mapped to [[4/3]] instead of [[27/20]]. | ||
Extensions for harmonic 11 includes ''undecimal parakleismic'', mapping it to +36 steps, ''paralytic'', to -82 steps, ''parkleismic'', to -63 steps, and ''paradigmic'', to +17 steps. | Extensions for harmonic 11 includes ''undecimal parakleismic'', mapping it to +36 steps, ''paralytic'', to -82 steps, ''parkleismic'', to -63 steps, and ''paradigmic'', to +17 steps. | ||
| Line 112: | Line 112: | ||
== Tuning spectrum == | == Tuning spectrum == | ||
{| class="wikitable center-all" | {| class="wikitable center-all left-4" | ||
! | ! EDO<br>generator | ||
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]] | ! [[eigenmonzo|eigenmonzo<br>(unchanged interval)]] | ||
! generator<br>(¢) | ! generator<br>(¢) | ||
! comments | ! comments | ||
| Line 225: | Line 225: | ||
[[Category:Parakleismic| ]] <!-- main article --> | [[Category:Parakleismic| ]] <!-- main article --> | ||
[[Category:Rank-2 temperaments]] | |||
[[Category:Hemimean clan]] | [[Category:Hemimean clan]] | ||
[[Category:Ragismic microtemperaments]] | [[Category:Ragismic microtemperaments]] | ||
[[Category:Hemimage temperaments]] | [[Category:Hemimage temperaments]] | ||
[[Category:Canousmic temperaments]] | [[Category:Canousmic temperaments]] | ||
Latest revision as of 06:59, 21 June 2025
Parakleismic is the microtemperament tempering out the parakleisma in the 5-limit. This article also assumes the canonical mapping for 7, which means tempering out 3136/3125 and 4375/4374 in the 7-limit.
Parakleismic is much like catakleismic but a good tuning has the generator (6/5) flat, instead of sharp, than the just version. The sixth generator step is half a syntonic comma flat of the harmonic 3. Consequently, the 12th generator step is mapped to 10/9 instead of 9/8, and the 13th generator step is mapped to 4/3 instead of 27/20.
Extensions for harmonic 11 includes undecimal parakleismic, mapping it to +36 steps, paralytic, to -82 steps, parkleismic, to -63 steps, and paradigmic, to +17 steps.
See Ragismic microtemperaments #Parakleismic for technical data.
Interval chain
| # | Cents* | Approximate Ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 315.2 | 6/5 |
| 2 | 630.4 | 36/25 |
| 3 | 945.5 | 140/81 |
| 4 | 60.7 | 28/27 |
| 5 | 375.9 | 56/45 |
| 6 | 691.1 | 112/75 |
| 7 | 1006.3 | 25/14 |
| 8 | 121.4 | 15/14 |
| 9 | 436.6 | 9/7 |
| 10 | 751.8 | 54/35 |
| 11 | 1067.0 | 50/27 |
| 12 | 182.2 | 10/9 |
| 13 | 497.4 | 4/3 |
| 14 | 812.5 | 8/5 |
| 15 | 1127.7 | 48/25 |
| 16 | 242.9 | 144/125 |
| 17 | 558.1 | 112/81 |
| 18 | 873.7 | 224/135 |
| 19 | 1188.4 | 125/63, 448/225, 486/245 |
| 20 | 303.6 | 25/21 |
| 21 | 618.8 | 10/7 |
| 22 | 934.0 | 12/7 |
| 23 | 49.2 | 36/35 |
* in 7-limit POTE tuning
Tuning spectrum
| EDO generator |
eigenmonzo (unchanged interval) |
generator (¢) |
comments |
|---|---|---|---|
| 16\61 | 314.754 | Lower bound of 9-odd-limit diamond monotone | |
| 15/14 | 314.930 | ||
| 21\80 | 315.000 | ||
| 9/7 | 315.009 | ||
| 7/5 | 315.118 | ||
| 7/6 | 315.142 | ||
| 26\99 | 315.152 | ||
| 21/20 | 315.163 | ||
| 49/48 | 315.163 | ||
| 36/35 | 315.164 | ||
| 8/7 | 315.176 | 7-odd-limit minimax (error = 1.217¢) | |
| 80/63 | 315.183 | 9-odd-limit minimax (error = 1.345¢) | |
| 10/9 | 315.200 | ||
| 4/3 | 315.234 | ||
| 16/15 | 315.249 | 5-odd-limit minimax (error = 0.196¢) | |
| 31\118 | 315.254 | ||
| 5/4 | 315.263 | ||
| 25/24 | 315.289 | ||
| 6/5 | 315.641 | ||
| 28/27 | 315.740 | ||
| 5\19 | 315.789 | Upper bound of 9-odd-limit diamond monotone |