Ploidacot/Omega-pentacot: Difference between revisions

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== Intervals and notation ==
== Intervals and notation ==
Due to dividing the fifth into so many steps, standard notation becomes almost useless for omega-pentacot. Regardless, notation has been provided for where [[Ploidacot/Monocot|monocot]] intervals appear in this system.
While there is no agreed-upon notation system for omega-pentacot, the following is based on interpreting the generator as a semitone (1/5 of a fourth), allowing for an ^ or v to stand for a 1/5 of an ''inversed'' diminished second (equivalent to the [[Pythagorean comma]]), so vvvC# and ^^Db are enharmonic.


{| class="wikitable"
{| class="wikitable"
Line 12: Line 12:
! Notation
! Notation
! Name
! Name
|-
| −30
| 611.730
| F#
| augmented fourth
|-
| −29
| 711.339
| ^^G
|
|-
| −28
| 810.948
| vG#
|
|-
| −27
| 910.557
| ^A
|
|-
| −26
| 1010.166
| vvA#
|
|-
| −25
| 1109.775
| B
| major seventh
|-
| −24
| 9.384
| ^^C
|
|-
| −23
| 108.993
| vC#
|
|-
| −22
| 208.602
| ^D
|
|-
| −21
| 308.211
| vvD#
|
|-
| −20
| 407.820
| E
| major third
|-
| −19
| 507.429
| ^^F
|
|-
| −18
| 607.038
| vF#
|
|-
| −17
| 706.647
| ^G
|
|-
|-
| −16
| −16
| 806.256
| 806.256
|  
| vvG#
|  
|  
|-
|-
Line 25: Line 95:
| −14
| −14
| 1005.474
| 1005.474
|  
| ^^Bb
|  
|  
|-
|-
| −13
| −13
| 1105.083
| 1105.083
|  
| vB
|  
|  
|-
|-
| −12
| −12
| 4.692
| 4.692
|  
| ^C
|  
|  
|-
|-
| −11
| −11
| 104.301
| 104.301
|  
| vvC#
|  
|  
|-
|-
Line 50: Line 120:
| −9
| −9
| 303.519
| 303.519
|  
| ^^Eb
|  
|  
|-
|-
| −8
| −8
| 403.128
| 403.128
|  
| vE
|  
|  
|-
|-
| −7
| −7
| 502.737
| 502.737
|  
| ^F
|  
|  
|-
|-
| −6
| −6
| 602.346
| 602.346
|  
| vvF#
|  
|  
|-
|-
Line 75: Line 145:
| −4
| −4
| 801.564
| 801.564
|  
| ^^Ab
|  
|  
|-
|-
| −3
| −3
| 901.173
| 901.173
|  
| vA
|  
|  
|-
|-
| −2
| −2
| 1000.782
| 1000.782
|  
| ^Bb
|  
|  
|-
|-
| −1
| −1
| 1100.391
| 1100.391
|  
| vvB
|  
|  
|-
|-
Line 100: Line 170:
| 1
| 1
| 99.609
| 99.609
|  
| ^^Db
|  
|  
|-
|-
| 2
| 2
| 199.218
| 199.218
|  
| vD
|  
|  
|-
|-
| 3
| 3
| 298.827
| 298.827
|  
| ^Eb
|  
|  
|-
|-
| 4
| 4
| 398.436
| 398.436
|  
| vvE
|  
|  
|-
|-
Line 125: Line 195:
| 6
| 6
| 597.654
| 597.654
|  
| ^^Gb
|  
|  
|-
|-
| 7
| 7
| 697.263
| 697.263
|  
| vG
|  
|  
|-
|-
| 8
| 8
| 796.872
| 796.872
|  
| ^Ab
|  
|  
|-
|-
| 9
| 9
| 896.481
| 896.481
|  
| vvA
|  
|  
|-
|-
Line 150: Line 220:
| 11
| 11
| 1095.699
| 1095.699
|  
| ^^Cb
|  
|  
|-
|-
| 12
| 12
| 1195.308
| 1195.308
|  
| vC
|  
|  
|-
|-
| 13
| 13
| 94.917
| 94.917
|  
| ^Db
|  
|  
|-
|-
| 14
| 14
| 194.526
| 194.526
|  
| vvD
|  
|  
|-
|-
Line 175: Line 245:
| 16
| 16
| 393.744
| 393.744
| ^^Fb
|  
|  
|-
| 17
| 493.353
| vF
|  
|  
|-
| 18
| 592.962
| ^Gb
|
|-
| 19
| 692.571
| vvG
|
|-
| 20
| 792.180
| Ab
| minor sixth
|-
| 21
| 891.789
| ^^Bbb
|
|-
| 22
| 991.398
| vBb
|
|-
| 23
| 1091.007
| ^Cb
|
|-
| 24
| 1190.616
| vvC
|
|-
| 25
| 90.225
| Db
| minor second
|-
| 26
| 189.834
| ^^Ebb
|
|-
| 27
| 289.443
| vEb
|
|-
| 28
| 389.052
| ^Fb
|
|-
| 29
| 488.661
| vvF
|
|-
| 30
| 588.270
| Gb
| diminished fifth
|}
|}
A notable feature of omega-pentacot is the small comma, encountered after 12 steps, which represents one-fifth of a Pythagorean comma (or its equivalence, ''inversed'' diminished second). This makes omega-pentacot scales cluster around [[12edo]].


== Temperament interpretations ==
== Temperament interpretations ==

Revision as of 23:37, 2 January 2026

< Ploidacot
Omega-pentacot
Pergen [P8, P4/5]
Numeral form 4-sheared 5-cot
Pure generator size 99.61 ¢
Pure period size 1200 ¢
Forms 12, 13, 25, 37
Characteristic multival entry 5

Omega-pentacot is a temperament archetype where the generator is a semitone, five of which stack to form a perfect fourth of 4/3, and the period is a 2/1 octave. Omega-pentacot temperaments usually generate the 1L 11s and 12L 1s MOS structures. Regular temperaments of omega-pentacot are cluster temperaments with 12 clusters of notes in an octave.

Intervals and notation

While there is no agreed-upon notation system for omega-pentacot, the following is based on interpreting the generator as a semitone (1/5 of a fourth), allowing for an ^ or v to stand for a 1/5 of an inversed diminished second (equivalent to the Pythagorean comma), so vvvC# and ^^Db are enharmonic.

Omega-pentacot intervals (assuming pure fifth and octave)
# Cents Notation Name
−30 611.730 F# augmented fourth
−29 711.339 ^^G
−28 810.948 vG#
−27 910.557 ^A
−26 1010.166 vvA#
−25 1109.775 B major seventh
−24 9.384 ^^C
−23 108.993 vC#
−22 208.602 ^D
−21 308.211 vvD#
−20 407.820 E major third
−19 507.429 ^^F
−18 607.038 vF#
−17 706.647 ^G
−16 806.256 vvG#
−15 905.865 A major sixth
−14 1005.474 ^^Bb
−13 1105.083 vB
−12 4.692 ^C
−11 104.301 vvC#
−10 203.910 D major second
−9 303.519 ^^Eb
−8 403.128 vE
−7 502.737 ^F
−6 602.346 vvF#
−5 701.955 G perfect fifth
−4 801.564 ^^Ab
−3 901.173 vA
−2 1000.782 ^Bb
−1 1100.391 vvB
0 0.000 C perfect unison
1 99.609 ^^Db
2 199.218 vD
3 298.827 ^Eb
4 398.436 vvE
5 498.045 F perfect fourth
6 597.654 ^^Gb
7 697.263 vG
8 796.872 ^Ab
9 896.481 vvA
10 996.090 Bb minor seventh
11 1095.699 ^^Cb
12 1195.308 vC
13 94.917 ^Db
14 194.526 vvD
15 294.135 Eb minor third
16 393.744 ^^Fb
17 493.353 vF
18 592.962 ^Gb
19 692.571 vvG
20 792.180 Ab minor sixth
21 891.789 ^^Bbb
22 991.398 vBb
23 1091.007 ^Cb
24 1190.616 vvC
25 90.225 Db minor second
26 189.834 ^^Ebb
27 289.443 vEb
28 389.052 ^Fb
29 488.661 vvF
30 588.270 Gb diminished fifth

A notable feature of omega-pentacot is the small comma, encountered after 12 steps, which represents one-fifth of a Pythagorean comma (or its equivalence, inversed diminished second). This makes omega-pentacot scales cluster around 12edo.

Temperament interpretations

Quinticular

See also: Quinticular comma

Omega-pentacot temperaments are generally interpretated as quinticular temperaments; the generator is 18/17, five of them gives 4/3, so the quinticular comma (1419857/1417176) is tempered out.

Quindromeda

Main article: Quindromeda family

In quindromeda, the generator is 18/17, three generators make 19/16, five make 4/3, and 28 make 5th harmonic in the 2.3.5.17.19 subgroup, so 1216/1215, 1445/1444, and 6144/6137 are tempered out. This temperament is supported by 12, 169, 181, 193, 205, 217, 229, and 241 edos.

Equating 225/224 with 256/255 leads to quintakwai (12 & 193), which tempers out 400/399 (also equating 20/19 and 21/20) in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads to quintagar (12 & 217), which tempers out 476/475 (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.

Quintaleap

Main article: Quintaleap family

In quintaleap, the generator is 18/17, three generators make 19/16, five make 4/3, and 16 make 5/2 in the 2.3.5.17.19 subgroup, so 256/255, 361/360, and 4624/4617 are tempered out. This temperament is supported by 12, 109, 121, 133, 145, and 157 edos.

In the 2.3.5.7.17.19 subgroup, tempering out 400/399 (equating 20/19 and 21/20) leads to quintupole (12 & 121), and tempering out 476/475 (equating 19/17 with 28/25) leads to quinticosiennic (12 & 145).

Passion

Main article: Passion family

In passion, the generator is 16/15, four generators make 5/4, and five make 4/3. It is best tuned with a slightly flat generator of about 98.7 ¢, and follows that both 3 and 5 should be tuned sharp. The canonical mapping of 7 places 7/4 at 10 generators, and follows that the generator should be tuned flatter (about 98.1 ¢).

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