Octoid: Difference between revisions

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Line 101: Line 101:
| [[9/5]]
| [[9/5]]
| 1167.895
| 1167.895
| [[49/25]]
| [[49/25]], [[55/28]]
|-
|-
| 3
| 3
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| [[15/11]]
| [[15/11]]
| 685.791
| 685.791
|  
| [[49/33]]
| 835.791
| 835.791
|  
|  
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| [[45/44]], [[49/48]]
| [[45/44]], [[49/48]]
| 187.634
| 187.634
|  
| [[49/44]]
| 337.634
| 337.634
|  
|  
Line 249: Line 249:
| POTE: ~7/5 = 583.9622{{c}}
| POTE: ~7/5 = 583.9622{{c}}
|}
|}
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo <br>generator
! [[Eigenmonzo|Unchanged interval <br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| 1\8
|
| 150.000
| 8d val, lower bound of 7-odd-limit diamond monotone
|-
| 12\88
|
| 163.636
| 88bcde val, lower bound of 9- and 11-odd-limit diamond monotone
|-
|
| 9/7
| 164.916
|
|-
| 11\80
|
| 165.000
|
|-
|
| 11/10
| 165.004
|
|-
| 32\232
|
| 165.517
| 232d val
|-
|
| 5/3
| 165.641
|
|-
| 21\152
|
| 165.789
|
|-
|
| 11/9
| 165.803
|
|-
|
| 5/4
| 165.922
| 5-odd-limit minimax
|-
| 52\376
|
| 165.957
|
|-
|
| 3/2
| 166.015
| 11-limit 15-odd-limit minimax
|-
| 31\224
|
| 166.071
|
|-
|
| 9/5
| 166.202
| 9- and 11-odd-limit minimax
|-
| 41\296
|
| 166.216
|
|-
|
| 11/8
| 166.227
|
|-
|
| 7/4
| 166.235
| 7-odd-limit minimax
|-
|
| 11/7
| 166.246
|
|-
|
| 7/6
| 166.565
|
|-
| 10\72
|
| 166.667
|
|-
|
| 7/5
| 167.488
|
|-
| 9\64
|
| 168.750
| 64cd val, upper bound of 9- and 11-odd-limit diamond monotone
|-
| 8\56
|
| 171.429
| 56bccdde val, upper bound of 7-odd-limit diamond monotone
|}
<nowiki/>* Besides the octave


== Music ==
== Music ==
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[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Mirkwai clan]]
[[Category:Canopic clan]]
[[Category:Stearnsmic clan]]
[[Category:Stearnsmic clan]]

Latest revision as of 10:30, 6 June 2026

Octoid
Subgroups 2.3.5.7, 2.3.5.7.11
Comma basis 4375/4374, 16875/16807 (7-limit);
540/539, 1375/1372, 4000/3993 (11-limit)
Reduced mapping ⟨8; 3 4 5 3]
ET join 72 & 80
Generators (CWE) ~7/5 = 583.948 ¢
MOS scales 8L 64s, 72L 8s
Ploidacot octaploid alpha-tricot
Pergen (P8/8, P4/3)
Minimax error 11-odd-limit: 1.286 ¢;
11-limit 15-odd-limit: 1.473 ¢
Target scale size 11-odd-limit: 56 notes;
11-limit 15-odd-limit: 64 notes

Octoid is a regular temperament which takes a period of 1/8 octave, which represents 12/11, and adds a single generator which represents 6/5, 7/5, 9/7 or 11/10. It tempers out 4375/4374 and 16875/16807 in the 7-limit, and 540/539, 1375/1372, and 4000/3993 in the 11-limit.

There are some extensions for the 13-limit including tridecimal octoid (72 & 224) and octopus (72 & 80).

See Ragismic microtemperaments #Octoid for technical details.

Interval chain

Generator Period 1 Period 2 Period 3 Period 4 Period 5 Period 6 Period 7 Period 8
Cents Approx. ratios Cents Approx. ratios Cents Approx. ratios Cents Approx. ratios Cents Approx. ratios Cents Approx. ratios Cents Approx. ratios Cents Approx. ratios
0 150.000 12/11 300.000 25/21 450.000 35/27 600.000 99/70, 140/99 750.000 54/35 900.000 42/25 1050.000 11/6 1200.000 2/1
1 133.948 27/25 283.948 33/28 433.948 9/7 583.948 7/5 733.948 883.948 5/3 1033.948 20/11 1183.948
2 117.895 15/14 267.895 7/6 417.895 14/11 567.895 25/18 717.895 50/33 867.895 33/20 1017.895 9/5 1167.895 49/25, 55/28
3 101.843 35/33 251.843 401.843 551.843 11/8 701.843 3/2 851.843 18/11 1001.843 25/14 1151.843 35/18
4 85.791 21/20 235.791 385.791 5/4 535.791 15/11 685.791 49/33 835.791 985.791 1135.791 27/14
5 69.739 25/24 219.739 25/22 369.739 519.739 27/20 669.739 819.739 45/28 969.739 7/4 1119.739 21/11
6 53.686 33/32 203.686 9/8 353.686 27/22 503.686 653.686 35/24 803.686 35/22 953.686 1103.686
7 37.634 45/44, 49/48 187.634 49/44 337.634 487.634 637.634 787.634 937.634 1087.634 15/8
8 21.582 81/80 171.582 321.582 471.582 21/16 621.582 771.582 921.582 1071.582

* in 11-limit CWE tuning

Scales

Tunings

Norm-based tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~7/5 = 583.9418 ¢ CWE: ~7/5 = 583.9411 ¢ POTE: ~7/5 = 583.9404 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~7/5 = 583.9297 ¢ CWE: ~7/5 = 583.9477 ¢ POTE: ~7/5 = 583.9622 ¢

Tuning spectrum

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
1\8 150.000 8d val, lower bound of 7-odd-limit diamond monotone
12\88 163.636 88bcde val, lower bound of 9- and 11-odd-limit diamond monotone
9/7 164.916
11\80 165.000
11/10 165.004
32\232 165.517 232d val
5/3 165.641
21\152 165.789
11/9 165.803
5/4 165.922 5-odd-limit minimax
52\376 165.957
3/2 166.015 11-limit 15-odd-limit minimax
31\224 166.071
9/5 166.202 9- and 11-odd-limit minimax
41\296 166.216
11/8 166.227
7/4 166.235 7-odd-limit minimax
11/7 166.246
7/6 166.565
10\72 166.667
7/5 167.488
9\64 168.750 64cd val, upper bound of 9- and 11-odd-limit diamond monotone
8\56 171.429 56bccdde val, upper bound of 7-odd-limit diamond monotone

* Besides the octave

Music

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