Porcupine/Chords: Difference between revisions

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Below are listed the [[dyadic chord]]s of 11-limit [[porcupine|porcupine temperament]] that do not have generator steps 7 or 13 as dyads. Typing the chords requires consideration of the fact that porcupine conflates 10/9, 11/10 and 12/11 and also 16/9 and 7/4. If a [[transversal]] can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If a chord is essentially tempered, the chord is analyzed in terms of the transversals 11/10, 6/5, 16/11 and 7/4. Chords that require only [[64/63]] tempering are marked [[archytas chords|archytas]], by [[100/99]] [[ptolemismic chords|ptolemismic]], by [[121/120]] [[biyatismic chords|biyatismic]], by [[176/175]] [[valinorsmic chords|valinorsmic]], and by [[385/384]] [[keenanismic chords|keenanismic]]. Chords that require 64/63 and 176/175 tempering are marked ares, 100/99 and 385/384 tempered chords are [[supermagic chords|supermagic]], and 176/175 and 385/384 tempered chords are marked zeus. Chords that receive tempering by three independent commas above are labeled porcupine.
Below are listed the [[15-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[porcupine|porcupine temperament]] that do not have generator steps 7 or 13 as [[dyad]]s. Typing the chords requires consideration of the fact that porcupine conflates [[10/9]], [[11/10]] and [[12/11]], [[11/9]] with [[6/5]], [[22/15]] with [[16/11]], and [[16/9]] with [[7/4]]. If a [[transversal]] can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If a chord is essentially tempered, the chord is analyzed in terms of the transversals 11/10, 6/5, 16/11 and 7/4. Chords that require only [[64/63]] tempering are marked [[archytas chords|archytas]], by [[100/99]] [[ptolemismic chords|ptolemismic]], by [[121/120]] [[biyatismic chords|biyatismic]], by [[176/175]] [[valinorsmic chords|valinorsmic]], and by [[385/384]] [[keenanismic chords|keenanismic]]. Chords that require any two of 64/63, 100/99 and 176/175 tempering are marked [[ares chords|ares]], that require 100/99 and 385/384 tempering are marked [[keemic chords|keemic]], and that require any two of 121/120, 176/175 and 385/384 are marked [[zeus chords|zeus]]. Chords that receive tempering by three independent commas above are labeled porcupine.


The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in porcupine[7], twice in porcupine[8], and nine times in porcupine[15].
The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in Porcupine[7], twice in Porcupine[8], and nine times in Porcupine[15].


The "As generated" column takes the intervals that were generated and places them in size order. The 1st and 2nd inversion (and so on) columns show the inversions of those generated tones. Note that this gives different results than you might be used to: the major chord (1/1 - 5/4 - 3/2, or 4:5:6) is the second inversion of the generated 0-2-5 chord.  
The "As generated" column takes the intervals that were generated and places them in size order. The 1st and 2nd inversion (and so on) columns show the inversions of those generated tones. Note that this gives different results than you might be used to: the major chord (1–5/4–3/2, or 4:5:6) is the second inversion of the generated 0–2–5 chord.  


Though we're used to thinking of 4:5:6 as the definitive "major chord", with all inversions coming from that, there is nothing definitive about calling these lists below "chord" or "inversion". That's just the way the generators came out.  
Though we are used to thinking of 4:5:6 as the definitive "major chord", with all inversions coming from that, there is nothing definitive about calling these lists below "chord" or "inversion". That is just the way the generators came out.  


The '''bolded''' inversions are named using [[ups and downs]] as described on the [[Pergen|pergens]] page. The pergen is (P8, P4/3) third-of-a-4th, #7 in the [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf notation guide for rank-2 pergens]. One up is -7 generators, octave-reduced, which is a third-sharp. Thus ^<sup>3</sup>C = C# and the enharmonic interval is v<sup>3</sup>A1. The generator is vM2 = 167¢ - c/3, where c = the amount in cents the tempered fifth exceeds 700¢. ^1 = 33¢ + 2.33c. In 22edo, ^1 = 1\22 = 54.5¢.  
The '''bolded''' inversions are named using [[ups and downs]] as described on the [[Pergen]] page. The pergen is (P8, P4/3) third-of-a-4th, #7 in the [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf notation guide for rank-2 pergens]. One up is -7 generators, octave-reduced, which is a third-sharp. Thus ^<sup>3</sup>C = C# and the [[enharmonic unison]] is v<sup>3</sup>A1. The generator is vM2 = 167¢ - ''c''/3, where ''c'' is the amount in cents the tempered fifth exceeds 700¢. ^1 = 33¢ + 2.33''c''. In 22edo, ^1 = 1\22 = 54.5¢.  


In Porcupine, 5/4 = vM3, 7/4 = m7 and 11/8 = ^4. Thus ^1 equals ~81/80 and ~33/32. This may not be true for other (P8, P4/3) temperaments. So the ratios in the table below are specific to Porcupine, but the chord names apply to any (P8, P4/3) temperament.  
In porcupine, 5/4 = vM3, 7/4 = m7 and 11/8 = ^4. Thus ^1 equals ~81/80 and ~33/32. This may not be true for other (P8, P4/3) temperaments. So the ratios in the table below are specific to Porcupine, but the chord names apply to any (P8, P4/3) temperament.  
{| class="wikitable"
 
{| class="wikitable center-all"
|+Porcupine's genchain
|+Porcupine's genchain
!Genspan
! Genspan
!0
! 0
!1
! 1
!2
! 2
!3
! 3
!4
! 4
!5
! 5
!6
! 6
!7
! 7
!8
! 8
!9
! 9
!10
! 10
!11
! 11
!12
! 12
!13
! 13
!14
! 14
|-
|-
!Cents (22edo)
! Cents (22edo)
|0
| 0
|164
| 164
|327
| 327
|491
| 491
|655
| 655
|818
| 818
|982
| 982
|1145
| 1145
|109
| 109
|273
| 273
|436
| 436
|600
| 600
|764
| 764
|927
| 927
|1091
| 1091
|-
|-
!Ratio
! Ratio
|1/1
| 1/1
|10/9
| 10/9<br>11/10
11/10
| 6/5<br>11/9
|6/5
| 4/3
11/9
| 16/11
|4/3
| 8/5
|16/11
| 16/9<br>7/4
|8/5
| 48/25<br>160/81
|16/9
| 16/15<br>21/20
7/4
| 7/6
|14/25
| 14/11
160/81
| 7/5
|16/15
| 14/9
21/20
|  
|7/6
| 28/15
|14/11
|7/5
|14/9
|
|28/15
|-
|-
!Interval
! Interval
|'''P1'''
| '''P1'''
|vM2
| vM2
|^m3
| ^m3
|'''P4'''
| '''P4'''
|v5
| v5
|^m6
| ^m6
|'''m7'''
| '''m7'''
|v8
| v8
|^m2
| ^m2
|'''m3'''
| '''m3'''
|v4
| v4
|^b5
| ^b5
|'''m6'''
| '''m6'''
|vm7
| vm7
|^d8
| ^d8
|-
|-
!Note (in C)
! Note (in C)
|'''C'''
| '''C'''
|vD
| vD
|^Eb
| ^Eb
|'''F'''
| '''F'''
|vG
| vG
|^Ab
| ^Ab
|'''Bb'''
| '''Bb'''
|vC
| vC
|^Db
| ^Db
|'''Eb'''
| '''Eb'''
|vF
| vF
|^Gb
| ^Gb
|'''Ab'''
| '''Ab'''
|vBb
| vBb
|^Cb
| ^Cb
|}
|}
'''''TODO: complete the tables'''''
{{Todo|inline=1|complete table|research|comment=Both tetrads and pentads are incomplete. Add the missing chords.}}


== Triads ==
== Triads ==
Line 112: Line 108:
! Type
! Type
! As generated
! As generated
! 1st Inversion
! 1st inversion
! 2nd Inversion
! 2nd inversion
!Name
! Name
|-
|-
| 0-1-2
| 0-1-2
Line 122: Line 118:
| 1/1-12/11-20/11
| 1/1-12/11-20/11
| 1/1-5/3-11/6
| 1/1-5/3-11/6
|C^mv9no5
| C^mv9no5
|-
|-
| 0-1-3
| 0-1-3
Line 130: Line 126:
| '''1/1-6/5-9/5'''
| '''1/1-6/5-9/5'''
| 1/1-3/2-5/3
| 1/1-3/2-5/3
|C^m7no5
| C^m7no5
|-
|-
| 0-2-3
| 0-2-3
Line 138: Line 134:
| 1/1-12/11-18/11
| 1/1-12/11-18/11
| '''1/1-3/2-11/6'''
| '''1/1-3/2-11/6'''
|C^m7no3
| C^m7no3
|-
|-
| 0-1-4
| 0-1-4
Line 146: Line 142:
| 1/1-4/3-11/6
| 1/1-4/3-11/6
| '''1/1-11/8-3/2'''
| '''1/1-11/8-3/2'''
|C^4
| C^4
|-
|-
| 0-2-4
| 0-2-4
Line 154: Line 150:
| 1/1-11/9-5/3
| 1/1-11/9-5/3
| 1/1-15/11-18/11
| 1/1-15/11-18/11
|C^m(v5)
| C^m(v5)
|-
|-
| 0-3-4
| 0-3-4
Line 162: Line 158:
| '''1/1-11/10-3/2'''
| '''1/1-11/10-3/2'''
| 1/1-15/11-20/11
| 1/1-15/11-20/11
|Cv2
| Cv2
|-
|-
| 0-1-5
| 0-1-5
Line 170: Line 166:
| '''1/1-16/11-9/5'''
| '''1/1-16/11-9/5'''
| 1/1-5/4-11/8
| 1/1-5/4-11/8
|C^7(v5)no3
| C^7(v5)no3
|-
|-
| 0-2-5
| 0-2-5
Line 178: Line 174:
| 1/1-4/3-5/3
| 1/1-4/3-5/3
| '''1/1-5/4-3/2'''
| '''1/1-5/4-3/2'''
|Cv
| Cv
|-
|-
| 0-3-5
| 0-3-5
Line 186: Line 182:
| '''1/1-6/5-3/2'''
| '''1/1-6/5-3/2'''
| 1/1-5/4-5/3
| 1/1-5/4-5/3
|C^m
| C^m
|-
|-
| 0-4-5
| 0-4-5
Line 194: Line 190:
| 1/1-12/11-15/11
| 1/1-12/11-15/11
| '''1/1-5/4-11/6'''
| '''1/1-5/4-11/6'''
|Cv^7no5
| Cv^7no5
|-
|-
| 0-1-6
| 0-1-6
Line 202: Line 198:
| 1/1-8/5-9/5
| 1/1-8/5-9/5
| '''1/1-9/8-5/4'''
| '''1/1-9/8-5/4'''
|Cv,9no5
| Cv,9no5
|-
|-
| 0-2-6
| 0-2-6
Line 210: Line 206:
| 1/1-16/11-5/3
| 1/1-16/11-5/3
| '''1/1-9/8-11/8'''
| '''1/1-9/8-11/8'''
|Csus2(v5)
| Csus2(v5)
|-
|-
| 0-3-6
| 0-3-6
Line 218: Line 214:
| '''1/1-4/3-3/2'''
| '''1/1-4/3-3/2'''
| '''1/1-9/8-3/2'''
| '''1/1-9/8-3/2'''
|C4 <u>or</u> C2
| C4 ''or'' C2
|-
|-
| 0-4-6
| 0-4-6
Line 226: Line 222:
| '''1/1-11/9-11/8'''
| '''1/1-11/9-11/8'''
| 1/1-9/8-5/3
| 1/1-9/8-5/3
|C^m^4no5
| C^m^4no5
|-
|-
| 0-5-6
| 0-5-6
Line 234: Line 230:
| 1/1-10/9-5/4
| 1/1-10/9-5/4
| '''1/1-9/8-9/5'''
| '''1/1-9/8-9/5'''
|C^m9no35
| C^m9no35
|-
|-
| 0-2-8
| 0-2-8
Line 242: Line 238:
| '''1/1-9/8-15/8'''
| '''1/1-9/8-15/8'''
| 1/1-5/3-16/9
| 1/1-5/3-16/9
|CvM9no35
| CvM9no35
|-
|-
| 0-3-8
| 0-3-8
Line 250: Line 246:
| '''1/1-5/4-15/8'''
| '''1/1-5/4-15/8'''
| 1/1-3/2-8/5
| 1/1-3/2-8/5
|CvM7no5
| CvM7no5
|-
|-
| 0-4-8
| 0-4-8
Line 258: Line 254:
| 1/1-11/8-15/8
| 1/1-11/8-15/8
| '''1/1-15/11-16/11'''
| '''1/1-15/11-16/11'''
|C^4(v5)
| C^4(v5)
|-
|-
| 0-5-8
| 0-5-8
Line 266: Line 262:
| '''1/1-3/2-15/8'''
| '''1/1-3/2-15/8'''
| 1/1-5/4-4/3
| 1/1-5/4-4/3
|CvM7no3
| CvM7no3
|-
|-
| 0-6-8
| 0-6-8
Line 274: Line 270:
| 1/1-5/3-15/8
| 1/1-5/3-15/8
| '''1/1-9/8-6/5'''
| '''1/1-9/8-6/5'''
|C^m,9no5
| C^m,9no5
|-
|-
| 0-1-9
| 0-1-9
Line 282: Line 278:
| 1/1-16/15-20/11
| 1/1-16/15-20/11
| 1/1-12/7-15/8
| 1/1-12/7-15/8
|Cmv9no5
| Cmv9no5
|-
|-
| 0-3-9
| 0-3-9
Line 290: Line 286:
| 1/1-8/7-12/7
| 1/1-8/7-12/7
| '''1/1-3/2-7/4'''
| '''1/1-3/2-7/4'''
|C7no3
| C7no3
|-
|-
| 0-4-9
| 0-4-9
Line 298: Line 294:
| '''1/1-5/4-12/7'''
| '''1/1-5/4-12/7'''
| 1/1-11/8-8/5
| 1/1-11/8-8/5
|Cv,6no5
| Cv,6no5
|-
|-
| 0-5-9
| 0-5-9
Line 306: Line 302:
| 1/1-11/8-12/7
| 1/1-11/8-12/7
| '''1/1-5/4-16/11'''
| '''1/1-5/4-16/11'''
|Cv(v5)
| Cv(v5)
|-
|-
| 0-6-9
| 0-6-9
Line 314: Line 310:
| 1/1-3/2-12/7
| 1/1-3/2-12/7
| 1/1-8/7-4/3
| 1/1-8/7-4/3
|Cm7no5
| Cm7no5
|-
|-
| 0-8-9
| 0-8-9
Line 322: Line 318:
| 1/1-11/10-15/8
| 1/1-11/10-15/8
| 1/1-12/7-20/11
| 1/1-12/7-20/11
|Cm^b9no5
| Cm^b9no5
|-
|-
| 0-1-10
| 0-1-10
Line 330: Line 326:
| '''1/1-7/6-11/6'''
| '''1/1-7/6-11/6'''
| 1/1-11/7-12/7
| 1/1-11/7-12/7
|Cm^7no5
| Cm^7no5
|-
|-
| 0-2-10
| 0-2-10
Line 338: Line 334:
| 1/1-16/15-5/3
| 1/1-16/15-5/3
| '''1/1-11/7-15/8'''
| '''1/1-11/7-15/8'''
|CvM7(^5)no3
| CvM7(^5)no3
|-
|-
| 0-4-10
| 0-4-10
Line 346: Line 342:
| 1/1-8/7-11/7
| 1/1-8/7-11/7
| '''1/1-11/8-7/4'''
| '''1/1-11/8-7/4'''
|C7(^4)no5
| C7(^4)no5
|-
|-
| 0-5-10
| 0-5-10
Line 354: Line 350:
| 1/1-5/4-11/7
| 1/1-5/4-11/7
| '''1/1-5/4-8/5'''
| '''1/1-5/4-8/5'''
|Cv^b6
| Cv^b6
|-
|-
| 0-6-10
| 0-6-10
Line 362: Line 358:
| 1/1-11/8-11/7
| 1/1-11/8-11/7
| 1/1-8/7-16/11
| 1/1-8/7-16/11
|C7(v4)no5
| C7(v4)no5
|-
|-
| 0-8-10
| 0-8-10
Line 370: Line 366:
| '''1/1-6/5-15/8'''
| '''1/1-6/5-15/8'''
| 1/1-11/7-5/3
| 1/1-11/7-5/3
|C^mvM7
| C^mvM7
|-
|-
| 0-9-10
| 0-9-10
Line 378: Line 374:
| 1/1-12/11-12/7
| 1/1-12/11-12/7
| 1/1-11/7-11/6
| 1/1-11/7-11/6
|Cm,v11no5
| Cm,v11no5
|-
|-
| 0-1-11
| 0-1-11
Line 386: Line 382:
| '''1/1-14/11-20/11'''
| '''1/1-14/11-20/11'''
| 1/1-10/7-11/7
| 1/1-10/7-11/7
|C^7(v4)no5
| C^7(v4)no5
|-
|-
| 0-2-11
| 0-2-11
Line 394: Line 390:
| 1/1-7/6-5/3
| 1/1-7/6-5/3
| 1/1-10/7-12/7
| 1/1-10/7-12/7
|C^m(vv5)
| C^m(vv5)
|-
|-
| 0-3-11
| 0-3-11
Line 402: Line 398:
| '''1/1-16/15-3/2'''
| '''1/1-16/15-3/2'''
| 1/1-10/7-15/8
| 1/1-10/7-15/8
|C^b2
| C^b2
|-
|-
| 0-5-11
| 0-5-11
Line 410: Line 406:
| 1/1-8/7-10/7
| 1/1-8/7-10/7
| '''1/1-5/4-7/4'''
| '''1/1-5/4-7/4'''
|Cv,7no5
| Cv,7no5
|-
|-
| 0-6-11
| 0-6-11
Line 418: Line 414:
| 1/1-5/4-10/7
| 1/1-5/4-10/7
| 1/1-8/7-8/5
| 1/1-8/7-8/5
|C7(vv5)no3
| C7(vv5)no3
|-
|-
| 0-8-11
| 0-8-11
Line 426: Line 422:
| '''1/1-4/3-15/8'''
| '''1/1-4/3-15/8'''
| 1/1-10/7-3/2
| 1/1-10/7-3/2
|CvM7(4)
| CvM7(4)
|-
|-
| 0-9-11
| 0-9-11
Line 434: Line 430:
| 1/1-6/5-12/7
| 1/1-6/5-12/7
| 1/1-10/7-5/3
| 1/1-10/7-5/3
|Cm(vv5)
| Cm(vv5)
|-
|-
| 0-10-11
| 0-10-11
Line 442: Line 438:
| 1/1-11/10-11/7
| 1/1-11/10-11/7
| 1/1-10/7-20/11
| 1/1-10/7-20/11
|Cv4(vv5)
| Cv4(vv5)
|-
|-
| 0-1-12
| 0-1-12
Line 450: Line 446:
| 1/1-7/5-9/5
| 1/1-7/5-9/5
| '''1/1-9/7-10/7'''
| '''1/1-9/7-10/7'''
|C,^^11no5
| C,^^11no5
|-
|-
| 0-2-12
| 0-2-12
Line 458: Line 454:
| 1/1-14/11-18/11
| 1/1-14/11-18/11
| '''1/1-9/7-11/7'''
| '''1/1-9/7-11/7'''
|C(^5)
| C(^5)
|-
|-
| 0-3-12
| 0-3-12
Line 466: Line 462:
| '''1/1-7/6-3/2'''
| '''1/1-7/6-3/2'''
| 1/1-9/7-12/7
| 1/1-9/7-12/7
|Cm
| Cm
|-
|-
| 0-4-12
| 0-4-12
Line 474: Line 470:
| 1/1-16/15-11/8
| 1/1-16/15-11/8
| '''1/1-9/7-15/8'''
| '''1/1-9/7-15/8'''
|C,vM7no5
| C,vM7no5
|-
|-
| 0-6-12
| 0-6-12
Line 482: Line 478:
| 1/1-8/7-9/7
| 1/1-8/7-9/7
| '''1/1-9/8-7/4'''
| '''1/1-9/8-7/4'''
|C9no35
| C9no35
|-
|-
| 0-8-12
| 0-8-12
Line 490: Line 486:
| 1/1-16/11-15/8
| 1/1-16/11-15/8
| '''1/1-9/7-11/8'''
| '''1/1-9/7-11/8'''
|C,^11no5
| C,^11no5
|-
|-
| 0-9-12
| 0-9-12
Line 498: Line 494:
| 1/1-4/3-12/7
| 1/1-4/3-12/7
| '''1/1-9/7-3/2'''
| '''1/1-9/7-3/2'''
|C
| C
|-
|-
| 0-10-12
| 0-10-12
Line 506: Line 502:
| 1/1-11/9-11/7
| 1/1-11/9-11/7
| '''1/1-9/7-18/11'''
| '''1/1-9/7-18/11'''
|C,v6no5
| C,v6no5
|-
|-
| 0-11-12
| 0-11-12
Line 514: Line 510:
| 1/1-10/9-10/7
| 1/1-10/9-10/7
| '''1/1-9/7-9/5'''
| '''1/1-9/7-9/5'''
|C,^7no5
| C,^7no5
|-
|-
| 0-2-14
| 0-2-14
Line 608: Line 604:
! Second inversion
! Second inversion
! Third inversion
! Third inversion
!Name
! Name
|-
|-
| 0-1-2-3
| 0-1-2-3
Line 617: Line 613:
| 1/1-12/11-18/11-20/11  
| 1/1-12/11-18/11-20/11  
| '''1/1-3/2-5/3-11/6'''
| '''1/1-3/2-5/3-11/6'''
|Cv6^7no3
| Cv6^7no3
|-
|-
| 0-1-2-4
| 0-1-2-4
Line 626: Line 622:
| 1/1-6/5-18/11-9/5   
| 1/1-6/5-18/11-9/5   
| '''1/1-15/11-3/2-5/3'''
| '''1/1-15/11-3/2-5/3'''
|Cv6(^4)
| Cv6(^4)
|-
|-
| 0-1-3-4
| 0-1-3-4
Line 635: Line 631:
| '''1/1-11/10-3/2-5/3'''
| '''1/1-11/10-3/2-5/3'''
| 1/1-15/11-3/2-20/11  
| 1/1-15/11-3/2-20/11  
|Cv2v6
| Cv2v6
|-
|-
| 0-1-2-5
| 0-1-2-5
Line 644: Line 640:
| 1/1-4/3-5/3-11/6   
| 1/1-4/3-5/3-11/6   
| '''1/1-5/4-11/8-3/2'''
| '''1/1-5/4-11/8-3/2'''
|Cv^4
| Cv^4
|-
|-
| 0-1-3-5
| 0-1-3-5
Line 653: Line 649:
| '''1/1-6/5-3/2-5/3'''
| '''1/1-6/5-3/2-5/3'''
| 1/1-5/4-11/8-5/3   
| 1/1-5/4-11/8-5/3   
|C^mv6
| C^mv6
|-
|-
| 0-1-4-5
| 0-1-4-5
Line 662: Line 658:
| '''1/1-11/10-11/8-3/2'''
| '''1/1-11/10-11/8-3/2'''
| 1/1-5/4-11/8-20/11  
| 1/1-5/4-11/8-20/11  
|C^4v9
| C^4v9
|-
|-
| 0-2-3-5
| 0-2-3-5
Line 671: Line 667:
| '''1/1-6/5-3/2-9/5'''
| '''1/1-6/5-3/2-9/5'''
| '''1/1-5/4-3/2-5/3'''
| '''1/1-5/4-3/2-5/3'''
|Cv6 <u>or</u> C^m7
| Cv6 ''or'' C^m7
|-
|-
|0-2-4-5
| 0-2-4-5
|1-6/5-16/11-8/5
| 1-6/5-16/11-8/5
|
| ptolemismic
|1/1-6/5-16/11-8/5
| 1/1-6/5-16/11-8/5
|
| 1/1-6/5-4/3-5/3
|
| 1/1-11/10-11/8-5/3
|'''1/1-5/4-3/2-9/5'''
| '''1/1-5/4-3/2-9/5'''
|Cv^7
| Cv^7
|-
|-
| 0-2-4-6
| 0-2-4-6
| 1-6/5-16/11-7/4
| 1-6/5-16/11-7/4
| supermagic
| keemic
| '''1/1-6/5-16/11-7/4'''
| '''1/1-6/5-16/11-7/4'''
| 1/1-6/5-16/11-5/3
| 1/1-6/5-16/11-5/3
| '''1/1-6/5-11/8-5/3'''
| '''1/1-6/5-11/8-5/3'''
| 1/1-8/7-11/8-5/3
| 1/1-8/7-11/8-5/3
|C^m,7(v5) <u>or</u>
| C^m,7(v5) ''or''<br>C^mv6^11no5
C^mv6^11no5
|-
|-
| 0-3-6-9
| 0-3-6-9
Line 699: Line 694:
| '''1/1-4/3-3/2-7/4'''
| '''1/1-4/3-3/2-7/4'''
| 1/1-8/7-4/3-3/2
| 1/1-8/7-4/3-3/2
|C7sus4
| C7sus4
|-
|-
|0-3-9-12
| 0-3-9-12
|1-4/3-7/6-14/9
| 1-4/3-7/6-14/9
|archytas
| archytas
|1/1-7/6-4/3-14/9
| 1/1-7/6-4/3-14/9
|'''1/1-7/6-3/2-7/4'''
| '''1/1-7/6-3/2-7/4'''
|1/1-9/8-4/3-12/7
| 1/1-9/8-4/3-12/7
|1/1-9/7-3/2-12/7
| 1/1-9/7-3/2-12/7
|Cm7 <u>or</u> C6
| Cm7 ''or'' C6
|-
|-
| 0-4-8-12
| 0-4-8-12
Line 717: Line 712:
| 1/1-16/15-11/8-22/15
| 1/1-16/15-11/8-22/15
| '''1/1-9/7-11/8-15/8'''
| '''1/1-9/7-11/8-15/8'''
|C,vM7^11no5
| C,vM7^11no5
|}
|}


== Pentads ==
== Pentads ==
{{todo|inline=1| clarify | comment=investigate what's missing there }}
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 728: Line 721:
! Transversal
! Transversal
! Type
! Type
!Name
! Name
|-
|
|
|
|
|-
|-
| 0-1-2-3-6
| 0-1-2-3-6
| 1-10/9-11/9-4/3-16/9
| 1-10/9-11/9-4/3-16/9
| otonal
| otonal
|Cv,9^11
| Cv,9^11
|-
|-
| 0-2-3-4-6
| 0-2-3-4-6
|  
| 1-6/5-4/3-16/11-16/9
|  
| keemic
|C^m,7,11(v5) <u>or</u>
| C^m,7,11(v5) ''or''<br>C4^7v9 ''or'' C^4v6,9
C4^7v9 <u>or</u> C^4v6,9
|-
|-
| 0-3-4-5-6
| 0-3-4-5-6
|  
| 1-4/3-16/11-8/5-16/9
|  
| utonal
|C^mv9,11
| C^mv9,11
|-
|-
| 0-2-4-6-8
| 0-2-4-6-8
| 1-6/5-16/11-7/4-16/15
| 1-6/5-16/11-7/4-16/15
| porcupine
| porcupine
|C^m,7(v5) <u>or</u>
| C^m,7(v5) ''or''<br>C^mv6^11no5
C^mv6^11no5
|-
|-
| 0-3-6-9-12
| 0-3-6-9-12
| 1-4/3-7/4-7/6-14/9
| 1-4/3-7/4-7/6-14/9
| archytas
| archytas
|C9(4) <u>or</u> C6,9 <u>or</u> Cm7,11
| C9(4) ''or'' C6,9 ''or'' Cm7,11
|}
|}


Latest revision as of 23:50, 18 February 2026

Below are listed the 15-odd-limit dyadic chords of 11-limit porcupine temperament that do not have generator steps 7 or 13 as dyads. Typing the chords requires consideration of the fact that porcupine conflates 10/9, 11/10 and 12/11, 11/9 with 6/5, 22/15 with 16/11, and 16/9 with 7/4. If a transversal can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If a chord is essentially tempered, the chord is analyzed in terms of the transversals 11/10, 6/5, 16/11 and 7/4. Chords that require only 64/63 tempering are marked archytas, by 100/99 ptolemismic, by 121/120 biyatismic, by 176/175 valinorsmic, and by 385/384 keenanismic. Chords that require any two of 64/63, 100/99 and 176/175 tempering are marked ares, that require 100/99 and 385/384 tempering are marked keemic, and that require any two of 121/120, 176/175 and 385/384 are marked zeus. Chords that receive tempering by three independent commas above are labeled porcupine.

The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in Porcupine[7], twice in Porcupine[8], and nine times in Porcupine[15].

The "As generated" column takes the intervals that were generated and places them in size order. The 1st and 2nd inversion (and so on) columns show the inversions of those generated tones. Note that this gives different results than you might be used to: the major chord (1–5/4–3/2, or 4:5:6) is the second inversion of the generated 0–2–5 chord.

Though we are used to thinking of 4:5:6 as the definitive "major chord", with all inversions coming from that, there is nothing definitive about calling these lists below "chord" or "inversion". That is just the way the generators came out.

The bolded inversions are named using ups and downs as described on the Pergen page. The pergen is (P8, P4/3) third-of-a-4th, #7 in the notation guide for rank-2 pergens. One up is -7 generators, octave-reduced, which is a third-sharp. Thus ^3C = C# and the enharmonic unison is v3A1. The generator is vM2 = 167¢ - c/3, where c is the amount in cents the tempered fifth exceeds 700¢. ^1 = 33¢ + 2.33c. In 22edo, ^1 = 1\22 = 54.5¢.

In porcupine, 5/4 = vM3, 7/4 = m7 and 11/8 = ^4. Thus ^1 equals ~81/80 and ~33/32. This may not be true for other (P8, P4/3) temperaments. So the ratios in the table below are specific to Porcupine, but the chord names apply to any (P8, P4/3) temperament.

Porcupine's genchain
Genspan 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Cents (22edo) 0 164 327 491 655 818 982 1145 109 273 436 600 764 927 1091
Ratio 1/1 10/9
11/10 		6/5
11/9 		4/3 16/11 8/5 16/9
7/4 		48/25
160/81 		16/15
21/20 		7/6 14/11 7/5 14/9 28/15
Interval P1 vM2 ^m3 P4 v5 ^m6 m7 v8 ^m2 m3 v4 ^b5 m6 vm7 ^d8
Note (in C) C vD ^Eb F vG ^Ab Bb vC ^Db Eb vF ^Gb Ab vBb ^Cb
Todo: complete table, research

Both tetrads and pentads are incomplete. Add the missing chords.

Triads

Chord Transversal Type As generated 1st inversion 2nd inversion Name
0-1-2 1-11/10-6/5 otonal 1/1-11/10-6/5 1/1-12/11-20/11 1/1-5/3-11/6 C^mv9no5
0-1-3 1-10/9-4/3 otonal 1/1-10/9-4/3 1/1-6/5-9/5 1/1-3/2-5/3 C^m7no5
0-2-3 1-11/9-4/3 otonal 1/1-11/9-4/3 1/1-12/11-18/11 1/1-3/2-11/6 C^m7no3
0-1-4 1-12/11-16/11 otonal 1/1-12/11-16/11 1/1-4/3-11/6 1/1-11/8-3/2 C^4
0-2-4 1-6/5-22/15 otonal 1/1-6/5-22/15 1/1-11/9-5/3 1/1-15/11-18/11 C^m(v5)
0-3-4 1-4/3-22/15 otonal 1/1-4/3-22/15 1/1-11/10-3/2 1/1-15/11-20/11 Cv2
0-1-5 1-11/10-8/5 otonal 1/1-11/10-8/5 1/1-16/11-9/5 1/1-5/4-11/8 C^7(v5)no3
0-2-5 1-6/5-8/5 otonal 1/1-6/5-8/5 1/1-4/3-5/3 1/1-5/4-3/2 Cv
0-3-5 1-4/3-8/5 utonal 1/1-4/3-8/5 1/1-6/5-3/2 1/1-5/4-5/3 C^m
0-4-5 1-22/15-8/5 otonal 1/1-22/15-8/5 1/1-12/11-15/11 1/1-5/4-11/6 Cv^7no5
0-1-6 1-10/9-16/9 otonal 1/1-10/9-16/9 1/1-8/5-9/5 1/1-9/8-5/4 Cv,9no5
0-2-6 1-11/9-16/9 otonal 1/1-11/9-16/9 1/1-16/11-5/3 1/1-9/8-11/8 Csus2(v5)
0-3-6 1-4/3-16/9 ambitonal 1/1-4/3-16/9 1/1-4/3-3/2 1/1-9/8-3/2 C4 or C2
0-4-6 1-16/11-16/9 utonal 1/1-16/11-16/9 1/1-11/9-11/8 1/1-9/8-5/3 C^m^4no5
0-5-6 1-8/5-16/9 utonal 1/1-8/5-16/9 1/1-10/9-5/4 1/1-9/8-9/5 C^m9no35
0-2-8 1-6/5-16/15 otonal 1/1-16/15-6/5 1/1-9/8-15/8 1/1-5/3-16/9 CvM9no35
0-3-8 1-4/3-16/15 ambitonal 1/1-16/15-4/3 1/1-5/4-15/8 1/1-3/2-8/5 CvM7no5
0-4-8 1-22/15-16/15 otonal 1/1-16/15-22/15 1/1-11/8-15/8 1/1-15/11-16/11 C^4(v5)
0-5-8 1-8/5-16/15 ambitonal 1/1-16/15-8/5 1/1-3/2-15/8 1/1-5/4-4/3 CvM7no3
0-6-8 1-16/9-16/15 utonal 1/1-16/15-16/9 1/1-5/3-15/8 1/1-9/8-6/5 C^m,9no5
0-1-9 1-11/10-7/6 valinorsmic 1/1-11/10-7/6 1/1-16/15-20/11 1/1-12/7-15/8 Cmv9no5
0-3-9 1-4/3-7/6 otonal 1/1-7/6-4/3 1/1-8/7-12/7 1/1-3/2-7/4 C7no3
0-4-9 1-16/11-7/6 keenanismic 1/1-7/6-16/11 1/1-5/4-12/7 1/1-11/8-8/5 Cv,6no5
0-5-9 1-8/5-7/6 keenanismic 1/1-7/6-8/5 1/1-11/8-12/7 1/1-5/4-16/11 Cv(v5)
0-6-9 1-7/4-7/6 utonal 1/1-7/6-7/4 1/1-3/2-12/7 1/1-8/7-4/3 Cm7no5
0-8-9 1-16/15-7/6 valinorsmic 1/1-16/15-7/6 1/1-11/10-15/8 1/1-12/7-20/11 Cm^b9no5
0-1-10 1-12/11-14/11 otonal 1/1-12/11-14/11 1/1-7/6-11/6 1/1-11/7-12/7 Cm^7no5
0-2-10 1-6/5-14/11 valinorsmic 1/1-6/5-14/11 1/1-16/15-5/3 1/1-11/7-15/8 CvM7(^5)no3
0-4-10 1-16/11-14/11 otonal 1/1-14/11-16/11 1/1-8/7-11/7 1/1-11/8-7/4 C7(^4)no5
0-5-10 1-8/5-14/11 valinorsmic 1/1-14/11-8/5 1/1-5/4-11/7 1/1-5/4-8/5 Cv^b6
0-6-10 1-7/4-14/11 utonal 1/1-14/11-7/4 1/1-11/8-11/7 1/1-8/7-16/11 C7(v4)no5
0-8-10 1-16/15-14/11 valinorsmic 1/1-16/15-14/11 1/1-6/5-15/8 1/1-11/7-5/3 C^mvM7
0-9-10 1-7/6-14/11 utonal 1/1-7/6-14/11 1/1-12/11-12/7 1/1-11/7-11/6 Cm,v11no5
0-1-11 1-11/10-7/5 otonal 1/1-11/10-7/5 1/1-14/11-20/11 1/1-10/7-11/7 C^7(v4)no5
0-2-11 1-6/5-7/5 otonal 1/1-6/5-7/5 1/1-7/6-5/3 1/1-10/7-12/7 C^m(vv5)
0-3-11 1-4/3-7/5 archytas 1/1-4/3-7/5 1/1-16/15-3/2 1/1-10/7-15/8 C^b2
0-5-11 1-8/5-7/5 otonal 1/1-7/5-8/5 1/1-8/7-10/7 1/1-5/4-7/4 Cv,7no5
0-6-11 1-7/4-7/5 utonal 1/1-7/5-7/4 1/1-5/4-10/7 1/1-8/7-8/5 C7(vv5)no3
0-8-11 1-16/15-7/5 archytas 1/1-16/15-7/5 1/1-4/3-15/8 1/1-10/7-3/2 CvM7(4)
0-9-11 1-7/6-7/5 utonal 1/1-7/6-7/5 1/1-6/5-12/7 1/1-10/7-5/3 Cm(vv5)
0-10-11 1-14/11-7/5 utonal 1/1-14/11-7/5 1/1-11/10-11/7 1/1-10/7-20/11 Cv4(vv5)
0-1-12 1-10/9-14/9 otonal 1/1-10/9-14/9 1/1-7/5-9/5 1/1-9/7-10/7 C,^^11no5
0-2-12 1-11/9-14/9 otonal 1/1-11/9-14/9 1/1-14/11-18/11 1/1-9/7-11/7 C(^5)
0-3-12 1-4/3-14/9 otonal 1/1-4/3-14/9 1/1-7/6-3/2 1/1-9/7-12/7 Cm
0-4-12 1-16/11-14/9 keenanismic 1/1-16/11-14/9 1/1-16/15-11/8 1/1-9/7-15/8 C,vM7no5
0-6-12 1-16/9-14/9 otonal 1/1-14/9-16/9 1/1-8/7-9/7 1/1-9/8-7/4 C9no35
0-8-12 1-16/15-14/9 keenanismic 1/1-16/15-14/9 1/1-16/11-15/8 1/1-9/7-11/8 C,^11no5
0-9-12 1-7/6-14/9 utonal 1/1-7/6-14/9 1/1-4/3-12/7 1/1-9/7-3/2 C
0-10-12 1-14/11-14/9 utonal 1/1-14/11-14/9 1/1-11/9-11/7 1/1-9/7-18/11 C,v6no5
0-11-12 1-7/5-14/9 utonal 1/1-7/5-14/9 1/1-10/9-10/7 1/1-9/7-9/5 C,^7no5
0-2-14 1-6/5-28/15 otonal 1/1-6/5-28/15 1/1-14/9-5/3 1/1-15/14-9/7
0-3-14 1-4/3-28/15 otonal 1/1-4/3-28/15 1/1-7/5-3/2 1/1-15/14-10/7
0-4-14 1-22/15-28/15 otonal 1/1-22/15-28/15 1/1-14/11-15/11 1/1-15/14-11/7
0-5-14 1-8/5-28/15 otonal 1/1-8/5-28/15 1/1-7/6-5/4 1/1-15/14-12/7
0-6-14 1-7/4-28/15 utonal 1/1-7/4-28/15 1/1-16/15-8/7 1/1-15/14-15/8
0-8-14 1-16/15-28/15 otonal 1/1-16/15-28/15 1/1-7/4-15/8 1/1-15/14-8/7
0-9-14 1-7/6-28/15 utonal 1/1-7/6-28/15 1/1-8/5-12/7 1/1-15/14-5/4
0-10-14 1-14/11-28/15 utonal 1/1-14/11-28/15 1/1-22/15-11/7 1/1-15/14-15/11
0-11-14 1-7/5-28/15 utonal 1/1-7/5-28/15 1/1-4/3-10/7 1/1-15/14-3/2
0-12-14 1-14/9-28/15 utonal 1/1-14/9-28/15 1/1-6/5-9/7 1/1-15/14-5/3

Tetrads

Chord Transversal Type As generated First inversion Second inversion Third inversion Name
0-1-2-3 1-10/9-11/9-4/3 otonal 1/1-10/9-11/9-4/3 1/1-11/10-6/5-9/5 1/1-12/11-18/11-20/11 1/1-3/2-5/3-11/6 Cv6^7no3
0-1-2-4 1-11/10-11/9-22/15 utonal 1/1-11/10-11/9-22/15 1/1-10/9-4/3-20/11 1/1-6/5-18/11-9/5 1/1-15/11-3/2-5/3 Cv6(^4)
0-1-3-4 1-10/9-4/3-22/15 ptolemismic 1/1-10/9-4/3-22/15 1/1-6/5-4/3-9/5 1/1-11/10-3/2-5/3 1/1-15/11-3/2-20/11 Cv2v6
0-1-2-5 1-11/10-6/5-8/5 otonal 1/1-11/10-6/5-8/5 1/1-12/11-16/11-20/11 1/1-4/3-5/3-11/6 1/1-5/4-11/8-3/2 Cv^4
0-1-3-5 1-11/10-4/3-8/5 ptolemismic 1/1-11/10-4/3-8/5 1/1-6/5-16/11-20/11 1/1-6/5-3/2-5/3 1/1-5/4-11/8-5/3 C^mv6
0-1-4-5 1-11/10-16/11-8/5 biyatismic 1/1-11/10-16/11-8/5 1/1-4/3-16/11-20/11 1/1-11/10-11/8-3/2 1/1-5/4-11/8-20/11 C^4v9
0-2-3-5 1-6/5-4/3-8/5 ambitonal 1/1-6/5-4/3-8/5 1/1-10/9-4/3-5/3 1/1-6/5-3/2-9/5 1/1-5/4-3/2-5/3 Cv6 or C^m7
0-2-4-5 1-6/5-16/11-8/5 ptolemismic 1/1-6/5-16/11-8/5 1/1-6/5-4/3-5/3 1/1-11/10-11/8-5/3 1/1-5/4-3/2-9/5 Cv^7
0-2-4-6 1-6/5-16/11-7/4 keemic 1/1-6/5-16/11-7/4 1/1-6/5-16/11-5/3 1/1-6/5-11/8-5/3 1/1-8/7-11/8-5/3 C^m,7(v5) or
C^mv6^11no5
0-3-6-9 1-4/3-7/4-7/6 archytas 1/1/1-7/6-4/3-7/4 1/1-8/7-3/2-12/7 1/1-4/3-3/2-7/4 1/1-8/7-4/3-3/2 C7sus4
0-3-9-12 1-4/3-7/6-14/9 archytas 1/1-7/6-4/3-14/9 1/1-7/6-3/2-7/4 1/1-9/8-4/3-12/7 1/1-9/7-3/2-12/7 Cm7 or C6
0-4-8-12 1-16/11-16/15-14/9 zeus 1/1-16/15-16/11-14/9 1/1-15/11-16/11-15/8 1/1-16/15-11/8-22/15 1/1-9/7-11/8-15/8 C,vM7^11no5

Pentads

Chord Transversal Type Name
0-1-2-3-6 1-10/9-11/9-4/3-16/9 otonal Cv,9^11
0-2-3-4-6 1-6/5-4/3-16/11-16/9 keemic C^m,7,11(v5) or
C4^7v9 or C^4v6,9
0-3-4-5-6 1-4/3-16/11-8/5-16/9 utonal C^mv9,11
0-2-4-6-8 1-6/5-16/11-7/4-16/15 porcupine C^m,7(v5) or
C^mv6^11no5
0-3-6-9-12 1-4/3-7/4-7/6-14/9 archytas C9(4) or C6,9 or Cm7,11
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