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Mabilic and trismegistus: Difference between revisions

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{{Infobox regtemp|Optimization method=POTE|Generator tuning=527.236|Mapping=1; -15 -3 5|Ploidacot = alpha-triseph|Subgroups=2.5.7; 2.3.5.7|Title=Mabilic; trismegistus|Comma basis=1071875/1048576 (2.5.7); <br> 1029/1024, 3125/3072 (2.3.5.7)|Generator=175/128|Edo join 1=16|Edo join 2=25|Odd limit 1 = (2.5.7) 35|Odd limit 2=(7-limit) 35|Complexity 1=25|Complexity 2=107}}'''Mabilic''' is a temperament in the 2.5.7 subgroup where 5/2 is split into three generators, five of which octave-reduced reach 8/7. The generator is a sharpened fourth (or, conversely, a flattened fifth) in size, best tuned around 528 cents. Mabilic, as a result, tempers out the '''mabilisma''' 1071875/1048576. Mabilic is arrived at by removing the inaccurate 3/2 mapping from [[Mavila family#Armodue|armodue]] temperament. As such, mabilic can be associated with MOS scales like [[antidiatonic]] and [[armotonic]]. Thus, while the generating interval is not 3/2, [[2L 5s#Notation|melodic]] chain-of-fifths notation makes some amount of sense to use here.
{{Infobox regtemp
| Optimization method = POTE
| Generator tuning = 527.236
| Mapping = 1; -15 -3 5
| Ploidacot = alpha-triseph
| Subgroups = 2.5.7; 2.3.5.7
| Title = Mabilic; trismegistus
| Comma basis = 1071875/1048576 (2.5.7); <br> 1029/1024, 3125/3072 (2.3.5.7)
| Generator = 175/128
| Edo join 1 = 16 | Edo join 2 = 25
| Odd limit 1 = (2.5.7) 35 | Odd limit 2 = (7-limit) 35
| Complexity 1 = 25 | Complexity 2 = 107
}}
'''Mabilic''' is a [[regular temperament|temperament]] in the [[2.5.7 subgroup|2.5.7]] [[subgroup]] where [[5/2]] is split into three [[generator]]s, five of which octave-reduced reach [[8/7]]. The generator is a sharpened fourth (or, conversely, a flattened fifth) in size, best tuned around 528 [[cent]]s. Mabilic, as a result, tempers out the [[mabilisma]] 1071875/1048576. Mabilic is arrived at by removing the inaccurate 3/2 mapping from [[armodue (temperament)|armodue]] temperament. As such, mabilic can be associated with [[mos scale]]s like [[antidiatonic]] and [[armotonic]]. Thus, while the generating interval is not [[3/2]], [[2L 5s #Notation|melodic]] [[chain-of-fifths notation]] makes some amount of sense to use here.


Mabilic can be extended into a full 7-limit temperament called '''trismegistus''' in which 3 is found at 15 steps. Since 5 is at 3 steps, that makes trismegistus a magic temperament. Similarly, since 8/7 is at 5 steps, trismegistus is a slendric temperament.Trismegistus is associated with the MOS scale [[9L 7s]], where 3/2 is found as the "augmented" version of the generator, and 25edo, 41edo, and 66edo make for good tunings.
Mabilic can be extended into a full [[7-limit]] temperament called '''trismegistus''' in which [[3/1|3]] is found at 15 steps. Since [[5/1|5]] is at 3 steps, that makes trismegistus a [[magic family|magic temperament]]. Similarly, since 8/7 is at 5 steps, trismegistus is a [[gamelismic clan|slendric temperament]]. Trismegistus is associated with the mos scale [[9L 7s]], where 3/2 is found as the "augmented" version of the generator, and [[25edo]], [[41edo]], and [[66edo]] make for good tunings.


There is an alternative extension, '''semabila''' ([[Mabila family#Semabila]]), which is a semaphore temperament (hence its name) and thus finds 4/3 at 10 generators. It is best tuned sharper than trismegistus.  
There is an alternative extension, '''semabila''' ([[Mabila family #Semabila]]), which is a [[semaphoresmic clan|semaphore temperament]] (hence its name) and thus finds [[4/3]] at 10 generators. It is best tuned sharper than trismegistus.  


Making the generator itself 4/3 leads to the exotemperament [[mavila]], after which mabilic is named.
Making the generator itself 4/3 leads to the [[exotemperament]] [[mavila]], after which mabilic is named.


The tuning optimum of mabilic is 527.2 cents, which is almost exactly the [[Golden sequences and tuning|golden]] antidiatonic generator.
The tuning optimum of mabilic is 527.2 cents, which is almost exactly the [[Golden sequences and tuning|golden]] antidiatonic generator.


For technical data, see [[No-threes subgroup temperaments#Mabilic]] and [[Magic family#Trismegistus]].
For technical data, see [[No-threes subgroup temperaments #Mabilic]] and [[Magic family #Trismegistus]].


== Intervals ==
== Intervals ==
In the following tables, odd harmonics and subharmonics 1–15 are labeled in '''bold'''.
In the following tables, odd harmonics and subharmonics 1–15 are labeled in '''bold'''.
{| class="wikitable"
{| class="wikitable center-1 right-2 right-4"
|+
|+
!
!
! colspan="2" |Generators up
! colspan="2" | Generators up
! colspan="2" |Generators down
! colspan="2" | Generators down
|-
|-
!#
! #
!Cents
! Cents
!Approximate ratios
! Approximate ratios
!Cents
! Cents
!Approximate ratios
! Approximate ratios
|-
|-
|0
| 0
|0
| 0.000
|'''1/1'''
| '''1/1'''
|1200
| 1200.000
|'''2/1'''
| '''2/1'''
|-
|-
|1
| 1
|527.236
| 527.236
|
|
|672.764
| 672.764
|
|
|-
|-
|2
| 2
|1054.472
| 1054.472
|64/35
| 64/35
|145.528
| 145.528
|35/32
| 35/32
|-
|-
|3
| 3
|381.708
| 381.708
|'''5/4'''
| '''5/4'''
|818.292
| 818.292
|'''8/5'''
| '''8/5'''
|-
|-
|4
| 4
|908.944
| 908.944
|42/25
| 42/25
|291.056
| 291.056
|25/21
| 25/21
|-
|-
|5
| 5
|236.18
| 236.18
|'''8/7'''
| '''8/7'''
|963.82
| 963.82
|'''7/4'''
| '''7/4'''
|-
|-
|6
| 6
|763.416
| 763.416
|25/16
| 25/16
|436.584
| 436.584
|32/25
| 32/25
|-
|-
|7
| 7
|90.652
| 90.652
|
|  
|1109.348
| 1109.348
|
|  
|-
|-
|8
| 8
|617.888
| 617.888
|10/7
| 10/7
|582.112
| 582.112
|7/5
| 7/5
|-
|-
|9
| 9
|1145.124
| 1145.124
|
|  
|54.876
| 54.876
|
|  
|-
|-
|10
| 10
|472.36
| 472.36
|21/16
| 21/16
|727.64
| 727.64
|32/21
| 32/21
|-
|-
|11
| 11
|999.596
| 999.596
|25/14
| 25/14
|200.404
| 200.404
|28/25
| 28/25
|-
|-
|12
| 12
|326.832
| 326.832
|6/5
| 6/5
|873.168
| 873.168
|5/3
| 5/3
|-
|-
|13
| 13
|854.068
| 854.068
|
|  
|345.932
| 345.932
|
|
|-
|-
|14
| 14
|181.304
| 181.304
|
|  
|1018.696
| 1018.696
|
|
|-
|-
|15
| 15
|708.54
| 708.54
|'''3/2'''
| '''3/2'''
|491.46
| 491.46
|'''4/3'''
| '''4/3'''
|-
|-
|16
| 16
|35.776
| 35.776
|
|  
|1164.224
| 1164.224
|
|  
|}
|}


{{Todo| unify precision }}
[[Category:Rank-2 temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Magic family]]
[[Category:Magic family]]
[[Category:Gamelismic clan]]
[[Category:Gariboh clan]]
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