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See [[Ragismic_microtemperaments#Octoid|Ragismic microtemperaments]].
{{Infobox regtemp
| Title = Octoid
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[4375/4374]], [[16875/16807]] (7-limit); <br>[[540/539]], [[1375/1372]], [[4000/3993]] (11-limit)
| Edo join 1 = 72 | Edo join 2 = 80
| Mapping = 8; 3 4 5 3
| Generators = 7/5
| Generators tuning = 583.948
| Optimization method = CWE
| MOS scales = [[8L&nbsp;64s]], [[72L&nbsp;8s]]
| Pergen = (P8/8, P4/3)
| Color name =
| Odd limit 1 = 11 | Mistuning 1 = 1.286 | Complexity 1 = 56
| Odd limit 2 = 11-limit 15 | Mistuning 2 = 1.473 | Complexity 2 = 64
}}
'''Octoid''' is a [[regular temperament]] which takes a [[period]] of [[8edo|1/8 octave]], which represents [[12/11]], and adds a single [[generator]] which represents [[6/5]], [[7/5]], [[9/7]] or [[11/10]]. It [[tempering out|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 {{nowrap|(72 & 224)}} and octopus {{nowrap|(72 & 80)}}.
 
See [[Ragismic microtemperaments #Octoid]] for technical details.
 
== Interval chain ==
{| class="wikitable center-1"
! rowspan="2" | Generator
! colspan="2" | Period 1
! colspan="2" | Period 2
! colspan="2" | Period 3
! colspan="2" | Period 4
! colspan="2" | Period 5
! colspan="2" | Period 6
! colspan="2" | Period 7
! colspan="2" | 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
| style="text-align:right" | 85.791
| [[21/20]]
| 235.791
|
| 385.791
| [[5/4]]
| 535.791
| [[15/11]]
| 685.791
| [[49/33]]
| 835.791
|
| style="text-align:right" | 985.791
|
| 1135.791
| [[27/14]]
|-
| 5
| style="text-align:right" | 69.739
| [[25/24]]
| 219.739
| [[25/22]]
| 369.739
|
| 519.739
| [[27/20]]
| 669.739
|
| 819.739
| [[45/28]]
| style="text-align:right" | 969.739
| [[7/4]]
| 1119.739
| [[21/11]]
|-
| 6
| style="text-align:right" | 53.686
| [[33/32]]
| 203.686
| [[9/8]]
| 353.686
| [[27/22]]
| 503.686
|
| 653.686
| [[35/24]]
| 803.686
| [[35/22]]
| style="text-align:right" | 953.686
|
| 1103.686
|
|-
| 7
| style="text-align:right" | 37.634
| [[45/44]], [[49/48]]
| 187.634
| [[49/44]]
| 337.634
|
| 487.634
|
| 637.634
|
| 787.634
|
| style="text-align:right" | 937.634
|
| 1087.634
| [[15/8]]
|-
| 8
| style="text-align:right" | 21.582
| [[81/80]]
| 171.582
|
| 321.582
|
| 471.582
| [[21/16]]
| 621.582
|
| 771.582
|
| style="text-align:right" | 921.582
|
| 1071.582
|
|}
<nowiki>*</nowiki> in 11-limit CWE tuning
 
== Scales ==
* [[Octoid72]]
* [[Octoid80]]
 
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~7/5 = 583.9418{{c}}
| CWE: ~7/5 = 583.9411{{c}}
| POTE: ~7/5 = 583.9404{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~7/5 = 583.9297{{c}}
| CWE: ~7/5 = 583.9477{{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 ==
* ''Dreyfus'' (archived 2010) by [[Gene Ward Smith]] – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – Octoid[72] in 224edo tuning
 
[[Category:Octoid| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Canopic clan]]
[[Category:Stearnsmic clan]]
Retrieved from "https://en.xen.wiki/w/Octoid"