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'''Intergan''' is a set of temperaments related to 24edo semiquartal. The term is short for "interseptimal Ganassi," after 12edo's 2.3.17.19 Ganassi temperament. It seeks to cover all of the notable concordances generated by a meantone-range semifourth.
'''Intergan''' is a set of temperaments related to 24edo semiquartal. The term is short for "interseptimal Ganassi," after 12edo's 2.3.17.19 Ganassi temperament. It seeks to cover all of the notable concordances generated by a meantone-range semifourth. It was specified in a [https://groundfault.tumblr.com/post/715804797472800768/the-best-semiquartal-probably 2023 Tumblr post].


* The most basic version is 2.3.11.13/5.17.19 5&24. 4/3 is split into two semifourths most simply represented by 15/13, and 22/17 is equated to 13/10 and 22/19 is equated to 15/13. It contains the 16:19:22 +1+1 sub-diminished triad.
== Semiquartal ==
 
* The most basic version is 2.3.11.13/5.17.19 5&24. 4/3 is split into two semifourths most simply represented by 15/13, and 22/17 is equated to 13/10 and 22/19 is equated to 15/13. It features the remarkably simple 10:13:15 ultramajor triad and its retroversion found in Island temperaments, along with the 16:19:22 +1+1 sub-diminished triad and 16:19:24:27 +3+5+3 minor add6 tetrad. 11:12:17:19 is present, although this adding 13, 14, or 15 to the subgroup will significantly improve its usefulness.
* Tempering out 81/80 is somewhat obvious, resulting in 2.3.5.11.13.17.19 5&24.
* Tempering out 81/80 is somewhat obvious, resulting in 2.3.5.11.13.17.19 5&24.
* Primes 31 and 37 are also obvious to add, equating 37/32 to 22/19 and 37/31 to 19/16, resulting in 2.3.5.11.13.17.19.31.37 5&24. 67edo is roughly optimal due to its very mild meantone fourth being split in half. The 2.31.37 restriction is highly accurate in 43edo.
* Primes 31 and 37 are also obvious to add, equating 37/32 to 22/19 and 37/31 to 19/16, resulting in 2.3.5.11.13.17.19.31.37 5&24. 67edo is roughly optimal due to its very mild meantone fourth being split in half. The 2.31.37 restriction is highly accurate in 43edo.
* Primes 7, 23, and 29 are all more difficult to map due to their complexity, although their difficulty is in ascending order. They are more easily reached by splitting the semifourth generator in half in a manner similar to Negri.
* Primes 7, 23, and 29 are all more difficult to map due to their complexity, although their difficulty is in ascending order. They are more easily reached by splitting the semifourth generator in half in a manner similar to Negri.
{| class="wikitable"
|+Semiquartal Intergan, 14\67 generator, 37-limit patent val, 37-odd-limit
!Gens
!Cents
!JI Interpretations
|-
!33
|1075
|28/15 13/7 54/29 58/31
|-
!32
|824
|21/13 29/18 37/23
|-
!31
|573
|32/23 40/29 7/5 46/33
|-
!30
|322
|23/19 25/21 29/24
|-
!29
|72
|24/23 28/27 30/29 21/20 23/22
|-
!28
|1021
|56/31 52/29 25/14 '''''29/16'''''
|-
!27
|770
|14/9 36/23 58/37
|-
!26
|519
|34/25 42/31 23/17 31/23 35/26
|-
!25
|269
|7/6 27/23
|-
!24
|18
|
|-
!23
|967
|40/23 '''''7/4''''' 44/25 50/29 58/33
|-
!22
|716
|56/37 38/25 29/19
|-
!21
|466
|30/23 17/13 21/16 46/35 29/22 33/25 35/27
|-
!20
|215
|26/23 17/15 42/37 35/31
|-
!19
|1164
|35/18
|-
!18
|913
|56/33 22/13 17/10 29/17
|-
!17
|663
|28/19 22/15 19/13 35/24 37/25
|-
!16
|412
|32/25 14/11 34/27 19/15 33/26
|-
!15
|161
|11/10 34/31 21/19 23/21 25/23 35/32
|-
!14
|1110
|48/25 17/9 19/10 21/11 70/37
|-
!13
|860
|28/17 44/27 23/14 58/35 33/20
|-
!12
|609
|36/25 44/31 17/12 38/27 37/26
|-
!11
|358
|16/13 11/9 38/31 21/17 31/25 37/30
|-
!10
|107
|16/15 '''''17/16''''' 19/18 27/25 33/31 35/33
|-
!9
|1057
|24/13 11/6 68/37 35/19 37/20
|-
!8
|806
|8/5 19/12 46/29 35/22
|-
!7
|555
|18/13 '''''11/8''''' 29/21 37/27
|-
!6
|304
|32/27 6/5 44/37 '''''19/16''''' 31/26 37/31
|-
!5
|54
|32/31 26/25 34/33 38/37 27/26 29/28 31/30 33/32 35/34 37/36
|-
!4
|1003
|16/9 9/5 34/19 66/37
|-
!3
|752
|48/31 20/13 17/11 31/20 37/24
|-
!2
|501
|4/3 27/20
|-
!1
|251
|36/31 22/19 15/13 38/33 31/27 '''''37/32'''''
|-
!0
| class="thl" |0
| class="thl" |'''''1/1'''''
|-
!-1
|949
|64/37 26/15 19/11 54/31 31/18 33/19
|-
!-2
|699
|'''''3/2''''' 40/27
|-
!-3
|448
|48/37 40/31 22/17 13/10 31/24
|-
!-4
|197
|10/9 9/8 19/17 37/33
|-
!-5
|1146
|64/33 56/29 72/37 52/27 60/31 68/35 25/13 '''''31/16''''' 33/17 37/19
|-
!-6
|896
|32/19 5/3 52/31 27/16 62/37 37/22
|-
!-7
|645
|16/11 13/9 42/29 54/37
|-
!-8
|394
|24/19 '''''5/4''''' 44/35 29/23
|-
!-9
|143
|12/11 40/37 13/12 38/35 37/34
|-
!-10
|1093
|32/17 36/19 15/8 50/27 62/33 66/35
|-
!-11
|842
|18/11 '''''13/8''''' 60/37 34/21 50/31 31/19
|-
!-12
|591
|24/17 52/37 25/18 27/19 31/22
|-
!-13
|340
|40/33 28/23 17/14 27/22 35/29
|-
!-14
|90
|20/19 18/17 22/21 25/24 37/35
|-
!-15
|1039
|64/35 20/11 38/21 42/23 46/25 31/17
|-
!-16
|788
|11/7 52/33 30/19 25/16 27/17
|-
!-17
|537
|48/35 26/19 15/11 19/14 50/37
|-
!-18
|287
|20/17 13/11 34/29 33/28
|-
!-19
|36
|36/35
|-
!-20
|985
|30/17 23/13 62/35 37/21
|-
!-21
|734
|32/21 44/29 26/17 23/15 50/33 54/35 35/23
|-
!-22
|484
|38/29 25/19 37/28
|-
!-23
|233
|8/7 23/20 25/22 29/25 33/29
|-
!-24
|1182
|
|-
!-25
|931
|12/7 46/27
|-
!-26
|681
|52/35 34/23 46/31 25/17 31/21
|-
!-27
|430
|9/7 23/18 37/29
|-
!-28
|179
|32/29 28/25 29/26 31/28
|-
!-29
|1128
|40/21 44/23 23/12 27/14 29/15
|-
!-30
|878
|48/29 38/23 42/25
|-
!-31
|627
|10/7 '''''23/16''''' 29/20 33/23
|-
!-32
|376
|36/29 26/21 46/37
|-
!-33
|125
|14/13 15/14 29/27 31/29
|}


Like other important temperaments by [[User:Ground]], it may be understood as a relatively low-complexity rank-3 for greater flexibility. 81/80 is an obvious comma to temper out as seen in the standard rank-2 version, but leaving it in creates a significant amount of generator complexity. With this in mind, rank-3 Intergan is constructed as 2.3.5.11.13.17.19.31.37 5&9&24 with the following comma list:
== Rank-3 ==
 
Like other well-documented temperaments by [[User:Ground]], it may be understood as a relatively low-complexity rank-3 for greater flexibility. 81/80 is an obvious comma to temper out as seen in the standard rank-2 version, but leaving it in creates a significant amount of generator complexity. With this in mind, rank-3 Intergan is constructed as 2.3.5.11.13.17.19.31.37 5&9&24 with the following comma list:
* 4/3 / (15/13)^2 = 676/675
* 4/3 / (15/13)^2 = 676/675
* 22/19 / (15/13) = 286/285
* 22/19 / (15/13) = 286/285
Line 16: Line 294:
{| class="wikitable"
{| class="wikitable"
|+Rank-3 Intergan, 2.3.5.11.13.17.19.31.37 5&9&24 CE tuning; defining primes highlighted, along with the two most likely mappings of 7, 23, and 29
|+Rank-3 Intergan, 2.3.5.11.13.17.19.31.37 5&9&24 CE tuning; defining primes highlighted, along with the two most likely mappings of 7, 23, and 29
|
!
| -5
! -5
| -4
! -4
| -3
! -3
| -2
! -2
| -1
! -1
|0
!0
|1
!1
|2
!2
|3
!3
|4
!4
|5
!5
|-
|-
| -5
! -5
|482
|482
|877
|877
Line 42: Line 320:
|827
|827
|-
|-
| -4
! -4
|231
|231
| class="thl" |'''626'''
| class="thl" |'''626'''
Line 55: Line 333:
|576
|576
|-
|-
| -3
! -3
|1180
|1180
|375
|375
Line 68: Line 346:
|325
|325
|-
|-
| -2
! -2
|929
|929
|124
|124
Line 81: Line 359:
|75
|75
|-
|-
| -1
! -1
|678
|678
|1073
|1073
Line 94: Line 372:
| class="thl" |'''1024'''
| class="thl" |'''1024'''
|-
|-
|0
!0
|427
|427
|822
|822
Line 107: Line 385:
|773
|773
|-
|-
|1
!1
|176
|176
|571
|571
Line 120: Line 398:
|522
|522
|-
|-
|2
!2
|1125
|1125
|320
|320
Line 133: Line 411:
|271
|271
|-
|-
|3
!3
|875
|875
|69
|69
Line 146: Line 424:
|20
|20
|-
|-
|4
!4
|624
|624
|1018
|1018
Line 159: Line 437:
|969
|969
|-
|-
|5
!5
|373
|373
|767
|767
Line 173: Line 451:
|}
|}
Being so close to 24edo means that 1/2, 1/3, 1/4, 1/6, and 1/8-octave temperaments are all available. 12edo's remarkable accuracy in 2.3.17.19 provides many ways to temper the subgroup and its extensions, but this requires more effort to produce a temperament that isn't virtually 12edo; Intergan's interseptimals are one option. The rank-3 temperament may be attached to one of numerous often multi-period temperaments in this subgroup to reduce it to rank-2.
Being so close to 24edo means that 1/2, 1/3, 1/4, 1/6, and 1/8-octave temperaments are all available. 12edo's remarkable accuracy in 2.3.17.19 provides many ways to temper the subgroup and its extensions, but this requires more effort to produce a temperament that isn't virtually 12edo; Intergan's interseptimals are one option. The rank-3 temperament may be attached to one of numerous often multi-period temperaments in this subgroup to reduce it to rank-2.
* Third-octave: 121172^-11 = 2057/2048 and (27/17)^3/4 = 19683/19652. The former is called the Blume comma, and it is obscure despite being both accurate and low-complexity. The third-octave version of Intergan is especially useful when adding the elusive prime 29 because 29/23 is 401¢.
* Third-octave: Add 121172^-11 = 2057/2048, called the Blume comma, and it is obscure despite being both accurate and low-complexity. The third-octave version of Intergan is especially useful when adding the elusive prime 29 because 29/23 is 401¢. It is equivalent to adding the 2/(5/4)^3 = 128/125 Augmented comma. (27/17)^3/4 = 19683/19652 is a typical comma for third-octave temperaments but it has significantly more complex result.
* Quarter-octave: add (19/16)^4/2=130321/131072. This is one of very few useful commas dividing the octave into four equal parts.
* Quarter-octave: Add the (6/5)^4/2 = 648/625 Diminished comma or the smaller 2/(19/16)^4 = 131072/130321. These are the two most useful commas dividing the octave into four equal parts besides the (25/21)^4/2 = 390625/388962 Dimcomp comma.
The different versions of Intergan may be defined by the sheer number of intervals it maps to the ~19/16 interval: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37, and the otonal semifourth stack 26:31:37:44.
2.3.5.11.13.17.19.31.37 edos:
* Meantone: 24, 19, 43, 5, 67, 110[+5, +19], 91[+5], 115[+5]
* Augmented: 24, 9, 15, 33[+31], 57[+5, +31]
* Diminished: 24, 52, 76[+5]
The different versions of Intergan may be defined by the sheer number of intervals it maps to ~19/16: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37. The semiquartal temperament equates all of them besides 25/21 due to its sharp 5, instead equating it to 6/5 in the 7-limit. This includes an accurate otonal approximation of 0-1-2-3\4, 26:31:37:44.


{{Navbox regtemp}}
{{Navbox regtemp}}
{{Cat|temperaments}}
{{Cat|temperaments}}

Latest revision as of 05:22, 12 July 2026

Intergan is a set of temperaments related to 24edo semiquartal. The term is short for "interseptimal Ganassi," after 12edo's 2.3.17.19 Ganassi temperament. It seeks to cover all of the notable concordances generated by a meantone-range semifourth. It was specified in a 2023 Tumblr post.

Semiquartal

  • The most basic version is 2.3.11.13/5.17.19 5&24. 4/3 is split into two semifourths most simply represented by 15/13, and 22/17 is equated to 13/10 and 22/19 is equated to 15/13. It features the remarkably simple 10:13:15 ultramajor triad and its retroversion found in Island temperaments, along with the 16:19:22 +1+1 sub-diminished triad and 16:19:24:27 +3+5+3 minor add6 tetrad. 11:12:17:19 is present, although this adding 13, 14, or 15 to the subgroup will significantly improve its usefulness.
  • Tempering out 81/80 is somewhat obvious, resulting in 2.3.5.11.13.17.19 5&24.
  • Primes 31 and 37 are also obvious to add, equating 37/32 to 22/19 and 37/31 to 19/16, resulting in 2.3.5.11.13.17.19.31.37 5&24. 67edo is roughly optimal due to its very mild meantone fourth being split in half. The 2.31.37 restriction is highly accurate in 43edo.
  • Primes 7, 23, and 29 are all more difficult to map due to their complexity, although their difficulty is in ascending order. They are more easily reached by splitting the semifourth generator in half in a manner similar to Negri.
Semiquartal Intergan, 14\67 generator, 37-limit patent val, 37-odd-limit
Gens Cents JI Interpretations
33 1075 28/15 13/7 54/29 58/31
32 824 21/13 29/18 37/23
31 573 32/23 40/29 7/5 46/33
30 322 23/19 25/21 29/24
29 72 24/23 28/27 30/29 21/20 23/22
28 1021 56/31 52/29 25/14 29/16
27 770 14/9 36/23 58/37
26 519 34/25 42/31 23/17 31/23 35/26
25 269 7/6 27/23
24 18
23 967 40/23 7/4 44/25 50/29 58/33
22 716 56/37 38/25 29/19
21 466 30/23 17/13 21/16 46/35 29/22 33/25 35/27
20 215 26/23 17/15 42/37 35/31
19 1164 35/18
18 913 56/33 22/13 17/10 29/17
17 663 28/19 22/15 19/13 35/24 37/25
16 412 32/25 14/11 34/27 19/15 33/26
15 161 11/10 34/31 21/19 23/21 25/23 35/32
14 1110 48/25 17/9 19/10 21/11 70/37
13 860 28/17 44/27 23/14 58/35 33/20
12 609 36/25 44/31 17/12 38/27 37/26
11 358 16/13 11/9 38/31 21/17 31/25 37/30
10 107 16/15 17/16 19/18 27/25 33/31 35/33
9 1057 24/13 11/6 68/37 35/19 37/20
8 806 8/5 19/12 46/29 35/22
7 555 18/13 11/8 29/21 37/27
6 304 32/27 6/5 44/37 19/16 31/26 37/31
5 54 32/31 26/25 34/33 38/37 27/26 29/28 31/30 33/32 35/34 37/36
4 1003 16/9 9/5 34/19 66/37
3 752 48/31 20/13 17/11 31/20 37/24
2 501 4/3 27/20
1 251 36/31 22/19 15/13 38/33 31/27 37/32
0 0 1/1
-1 949 64/37 26/15 19/11 54/31 31/18 33/19
-2 699 3/2 40/27
-3 448 48/37 40/31 22/17 13/10 31/24
-4 197 10/9 9/8 19/17 37/33
-5 1146 64/33 56/29 72/37 52/27 60/31 68/35 25/13 31/16 33/17 37/19
-6 896 32/19 5/3 52/31 27/16 62/37 37/22
-7 645 16/11 13/9 42/29 54/37
-8 394 24/19 5/4 44/35 29/23
-9 143 12/11 40/37 13/12 38/35 37/34
-10 1093 32/17 36/19 15/8 50/27 62/33 66/35
-11 842 18/11 13/8 60/37 34/21 50/31 31/19
-12 591 24/17 52/37 25/18 27/19 31/22
-13 340 40/33 28/23 17/14 27/22 35/29
-14 90 20/19 18/17 22/21 25/24 37/35
-15 1039 64/35 20/11 38/21 42/23 46/25 31/17
-16 788 11/7 52/33 30/19 25/16 27/17
-17 537 48/35 26/19 15/11 19/14 50/37
-18 287 20/17 13/11 34/29 33/28
-19 36 36/35
-20 985 30/17 23/13 62/35 37/21
-21 734 32/21 44/29 26/17 23/15 50/33 54/35 35/23
-22 484 38/29 25/19 37/28
-23 233 8/7 23/20 25/22 29/25 33/29
-24 1182
-25 931 12/7 46/27
-26 681 52/35 34/23 46/31 25/17 31/21
-27 430 9/7 23/18 37/29
-28 179 32/29 28/25 29/26 31/28
-29 1128 40/21 44/23 23/12 27/14 29/15
-30 878 48/29 38/23 42/25
-31 627 10/7 23/16 29/20 33/23
-32 376 36/29 26/21 46/37
-33 125 14/13 15/14 29/27 31/29

Rank-3

Like other well-documented temperaments by User:Ground, it may be understood as a relatively low-complexity rank-3 for greater flexibility. 81/80 is an obvious comma to temper out as seen in the standard rank-2 version, but leaving it in creates a significant amount of generator complexity. With this in mind, rank-3 Intergan is constructed as 2.3.5.11.13.17.19.31.37 5&9&24 with the following comma list:

  • 4/3 / (15/13)^2 = 676/675
  • 22/19 / (15/13) = 286/285
  • 13/10 / (22/17) = 221/220
  • 16/15 / (17/16) = 256/255
  • 37/32 / (15/13) = 481/480
  • 37/31 / (19/16) = 592/589

It seems obvious to add (22/17)^2/(5/3) = 1452/1445, but that's equivalent to adding 81/80, as is 5/4/(19/17)^2 = 1445/1444.

Rank-3 Intergan, 2.3.5.11.13.17.19.31.37 5&9&24 CE tuning; defining primes highlighted, along with the two most likely mappings of 7, 23, and 29
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 482 877 71 466 860 55 449 844 38 433 827
-4 231 626 1020 215 609 1004 198 593 987 182 576
-3 1180 375 769 1164 358 753 1147 342 736 1131 325
-2 929 124 518 913 107 502 896 91 485 880 75
-1 678 1073 267 662 1056 251 645 1040 235 629 1024
0 427 822 16 411 805 0 395 789 1184 378 773
1 176 571 965 160 555 949 144 538 933 127 522
2 1125 320 715 1109 304 698 1093 287 682 1076 271
3 875 69 464 858 53 447 842 36 431 825 20
4 624 1018 213 607 1002 196 591 985 180 574 969
5 373 767 1162 356 751 1145 340 734 1129 323 718

Being so close to 24edo means that 1/2, 1/3, 1/4, 1/6, and 1/8-octave temperaments are all available. 12edo's remarkable accuracy in 2.3.17.19 provides many ways to temper the subgroup and its extensions, but this requires more effort to produce a temperament that isn't virtually 12edo; Intergan's interseptimals are one option. The rank-3 temperament may be attached to one of numerous often multi-period temperaments in this subgroup to reduce it to rank-2.

  • Third-octave: Add 121172^-11 = 2057/2048, called the Blume comma, and it is obscure despite being both accurate and low-complexity. The third-octave version of Intergan is especially useful when adding the elusive prime 29 because 29/23 is 401¢. It is equivalent to adding the 2/(5/4)^3 = 128/125 Augmented comma. (27/17)^3/4 = 19683/19652 is a typical comma for third-octave temperaments but it has significantly more complex result.
  • Quarter-octave: Add the (6/5)^4/2 = 648/625 Diminished comma or the smaller 2/(19/16)^4 = 131072/130321. These are the two most useful commas dividing the octave into four equal parts besides the (25/21)^4/2 = 390625/388962 Dimcomp comma.

2.3.5.11.13.17.19.31.37 edos:

  • Meantone: 24, 19, 43, 5, 67, 110[+5, +19], 91[+5], 115[+5]
  • Augmented: 24, 9, 15, 33[+31], 57[+5, +31]
  • Diminished: 24, 52, 76[+5]

The different versions of Intergan may be defined by the sheer number of intervals it maps to ~19/16: 19/16, 25/21, 31/26, 32/27, 37/31, 44/37. The semiquartal temperament equates all of them besides 25/21 due to its sharp 5, instead equating it to 6/5 in the 7-limit. This includes an accurate otonal approximation of 0-1-2-3\4, 26:31:37:44.


View • Talk • EditRegular temperaments
Rank-2
Acot Blackwood (1/5-octave) • Whitewood (1/7-octave) • Compton (1/12-octave)
Monocot Meantone • Schismic • Gentle-fifth temperaments • Archy
Complexity 2 Diaschismic (diploid monocot) • Pajara (diploid monocot) • Injera (diploid monocot) • Rastmatic (dicot) • Mohajira (dicot) • Intertridecimal (dicot) • Interseptimal (alpha-dicot)
Complexity 3 Augmented (triploid) • Misty (triploid) • Slendric (tricot) • Porcupine (omega-tricot) • Hedgehog (diploid alpha-tricot)
Complexity 4 Diminished (tetraploid) • Tetracot (tetracot) • Buzzard (alpha-tetracot) • Squares (beta-tetracot) • Negri (omega-tetracot)
Complexity 5-6 Magic (alpha-pentacot) • Amity (gamma-pentacot) • Kleismic (alpha-hexacot) • Miracle (hexacot)
Higher complexity Orwell (alpha-heptacot) • Sensi (beta-heptacot) • Octacot (octacot) • Wurschmidt (beta-octacot) • Valentine (enneacot) • Ammonite (delta-enneacot) • Myna (beta-decacot) • Ennealimmal (enneaploid dicot)
Straddle-3 A-Team (alter-tricot) • Machine (alter-monocot)
No-3 Mabilic (alpha-triseph) • Orgone (trimech) • Didacus (diseph) • Sidewalk (epsilon-heptaseph)
No-octaves Sensamagic (monogem)
Exotemperaments Dicot • Mavila • Father
Higher-rank
Rank-3 Aberschismic • Marvel • Parapyth • Pentagoth • Hibernal • Intergan
Retrieved from "https://xenreference.com/wiki/index.php?title=Intergan&oldid=7768"