Octoid: Difference between revisions

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**Imported revision 271583484 - Original comment: **
 
m Text replacement - "Category:Mirkwai clan" to "Category:Canopic clan"
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox regtemp
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| Title = Octoid
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-11-03 13:30:13 UTC</tt>.<br>
| Subgroups = 2.3.5.7, 2.3.5.7.11
: The original revision id was <tt>271583484</tt>.<br>
| Comma basis = [[4375/4374]], [[16875/16807]] (7-limit); <br>[[540/539]], [[1375/1372]], [[4000/3993]] (11-limit)
: The revision comment was: <tt></tt><br>
| Edo join 1 = 72 | Edo join 2 = 80
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| Mapping = 8; 3 4 5 3
<h4>Original Wikitext content:</h4>
| Generators = 7/5
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">See [[Ragismic microtemperaments#Octoid]].</pre></div>
| Generators tuning = 583.948
<h4>Original HTML content:</h4>
| Optimization method = CWE
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Octoid&lt;/title&gt;&lt;/head&gt;&lt;body&gt;See &lt;a class="wiki_link" href="/Ragismic%20microtemperaments#Octoid"&gt;Ragismic microtemperaments&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
| 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]]

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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