Mabilic and trismegistus: Difference between revisions

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Created page with "{{Infobox regtemp|Optimization method=POTE|Generator tuning=527.236|Mapping=1; -3 5|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}}'''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, b..."
 
Cleanup on infobox
 
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{{Infobox regtemp|Optimization method=POTE|Generator tuning=527.236|Mapping=1; -3 5|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}}'''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
| Title = Mabilic; Trismegistus
| Subgroups = 2.5.7; 2.3.5.7
| Comma basis = 1071875/1048576 (2.5.7); <br>1029/1024, 3125/3072 (2.3.5.7)
| Edo join 1 = 16 | Edo join 2 = 25
| Mapping = 1; -15 -3 5
| Generators = 175/128
| Generators tuning = 526.7
| Optimization method = CWE
| Ploidacot = alpha-triseph
| Odd limit 1 = 2.5.7 35 | Mistuning 1 = ? | Complexity 1 = 16
| Odd limit 2 = add-25 add-35 7 | Mistuning 2 = 5.9 | Complexity 2 = 25
}}
'''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. 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.  
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:Magic family]]
[[Category:Gamelismic clan]]
[[Category:Gariboh clan]]

Latest revision as of 10:15, 9 February 2026

Mabilic; Trismegistus
Subgroups 2.5.7; 2.3.5.7
Comma basis 1071875/1048576 (2.5.7);
1029/1024, 3125/3072 (2.3.5.7)
Reduced mapping ⟨1; -15 -3 5]
ET join 16 & 25
Generators (CWE) ~175/128 = 526.7 ¢
MOS scales n/a
Ploidacot alpha-triseph
Minimax error 2.5.7 35-odd-limit: ? ¢;
add-25 add-35 7-odd-limit: 5.9 ¢
Target scale size 2.5.7 35-odd-limit: 16 notes;
add-25 add-35 7-odd-limit: 25 notes

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 armodue temperament. As such, mabilic can be associated with mos scales like antidiatonic and armotonic. Thus, while the generating interval is not 3/2, 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.

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.

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 antidiatonic generator.

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

Intervals

In the following tables, odd harmonics and subharmonics 1–15 are labeled in bold.

Generators up Generators down
# Cents Approximate ratios Cents Approximate ratios
0 0.000 1/1 1200.000 2/1
1 527.236 672.764
2 1054.472 64/35 145.528 35/32
3 381.708 5/4 818.292 8/5
4 908.944 42/25 291.056 25/21
5 236.18 8/7 963.82 7/4
6 763.416 25/16 436.584 32/25
7 90.652 1109.348
8 617.888 10/7 582.112 7/5
9 1145.124 54.876
10 472.36 21/16 727.64 32/21
11 999.596 25/14 200.404 28/25
12 326.832 6/5 873.168 5/3
13 854.068 345.932
14 181.304 1018.696
15 708.54 3/2 491.46 4/3
16 35.776 1164.224
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