No-fives subgroup temperaments: Difference between revisions

{{See also| Whitewood }}

[[Comma list]]: [[4096/3993]]

=== Aerophore ===

[[Subgroup]]: 2.3.11.19

[[Comma list]]: 363/361, 729/704

{{Mapping|legend=2| 1 0 -6 -6 | 0 2 12 13 }}

: mapping generators: ~2, ~19/11

* [[WE]]: ~2 = 1202.6380{{c}}, ~19/11 = 947.4782{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~19/11 = 945.7779{{c}}

{{Optimal ET sequence|legend=1| 14, 19, 33 }}

[[Badness]] (Sintel): 1.59

== Temperaments with a 2.3.13 gene ==

=== Superflat ===

Superflat is a diatonic-based temperament that makes [[1053/1024]] vanish, so [[13/8]] is a minor sixth, and [[16/13]] is a major third. The more accurate tunings for this temperament are generated by a fifth at least as flat as those of [[flattone]], although often even flatter (such as [[40edo]]'s fifth). Superflat can be viewed as a 2.3.13 subgroup analogue of [[meantone]] and [[archy]]. Superflat diatonic scales have a character somewhere between neutral third scales (or [[3L 4s|mosh]]) and meantone diatonic scales.

[[Subgroup]]: 2.3.13

[[Comma list]]: [[1053/1024]]

{{Mapping|legend=2|1 1 6|0 1 -4}}

[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.939

[[Tp tuning #T2 tuning|RMS error]]: 1.591 cents

[[Subgroup]]: 2.3.11.13

[[Comma list]]: [[144/143]], [[729/704]]

[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.247

{{Optimal ET sequence|legend=1|7, 19, 26, 59b }}

=== Ultraflat ===

Ultraflat is the much more inaccurate cousin of superflat, with even flatter fifths. [[27/26]] is tempered out rather than [[1053/1024]], so [[13/8]] is a major sixth. These temperaments intersect in [[7edo]], where major sixths and minor sixths are not distinguished.

[[Subgroup]]: 2.3.13

 

 

 

[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[3/2]] = 688.391

 

 

[[Tp tuning #T2 tuning|RMS error]]: 4.367 cents

[[Comma list]]: [[2197/2187]]

{{Mapping|legend=2|1 0 0|0 3 7}}

[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/9]] = 634.173

{{Optimal ET sequence|legend=1|11bff, 13f, 15, 17, 36, 53, 70, 123, 193, 316, 755f }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2054 cents

=== Shoal ===

The 2.3.13.23 subgroup is remarkable for containing not one but two [[superparticular]] intervals as small as [[3888/3887]] and [[12168/12167]]. Tempering out both of them gives us a rank-2 temperament where a sharp whole tone of [[26/23]] is the generator, two of which stack to a [[23/18]] supermajor third, and eight of which stack to a [[8/3]] perfect eleventh. [[17edo]] is a trivial tuning where 26/23 is equated to [[9/8]], tempering out the comma [[208/207]]. More accurate tunings of shoal create a 17-note [[MOS]] scale, serving as a [[circulating temperament]] of 17edo, where 208/207 is the [[chroma]] between large and small steps.

[[Subgroup]]: 2.3.13.23

[[Comma list]]: 3888/3887, 12168/12167

{{Mapping|legend=2|1 3 6 7|0 -8 -13 -14}}

[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~26/23 = 212.261

{{Optimal ET sequence|legend=1|11fi, 17, 79, 96, 113, 130, 147, 424, 571, 1289, 1860, 3149 }}

Badness (Sintel): 0.021

* [[Shoal17]]

 

* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] - [[Budjarn Lambeth]] (2025)

 

The 2.3.13 gene subgroup is not nearly as good as Shoal, but it can extend extremely well to other no-5 subgroups. It is very well represented in [[26edo]], where a pure 13/12 can serve as the generator, but [[94edo]] provides a much better tuning in higher subgroups.

[[Comma list]]: [[373248/371293]]

{{Mapping|legend=2|1 1 3|0 5 6}}

[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.360

Badness (Sintel): 0.383

[[Subgroup]]: 2.3.11.13

[[Comma list]]: [[352/351]], [[531674/531441]]

[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.537

{{Optimal ET sequence|legend=1|17, 77, 94, 111, 128, 145, 205}}

[[Subgroup]]: 2.3.7.11.13

[[Comma list]]: [[352/351]], [[729/728]], [[1573/1568]]

{{Mapping|legend=2|1 1 0 1 3|0 5 24 21 6}}

[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.429

{{Optimal ET sequence|legend=1|17, 77, 94, 111, 128, 145, 205}}

Badness (Sintel): 0.415

===== 2.3.7.11.13.23 =====

[[Comma list]]: [[352/351]], [[729/728]], [[253/252]], [[1288/1287]]

{{Mapping|legend=2|1 1 0 1 3 3|0 5 24 21 6 13}}

[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.426

{{Optimal ET sequence|legend=1|17, 77, 94, 111, 205}}

Badness (Sintel): 0.452

===== 2.3.7.11.13.23.29 =====

[[Comma list]]: [[352/351]], [[729/728]], [[253/252]], [[1288/1287]], [[378/377]]

[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.426

{{Optimal ET sequence|legend=1|17, 77, 94, 111j}}

Badness (Sintel): 0.511

[[Subgroup]]: 2.3.7.11.13.19.23.29

[[Comma list]]: [[352/351]], [[729/728]], [[209/208]], [[253/252]], [[1288/1287]], [[378/377]]

{{Mapping|legend=2|1 1 0 1 3 -6 3 1|0 5 24 21 6 -15 13 33}}

[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/12]] = 140.426

 

Badness (Sintel): 0.699

: sval mapping generators: ~17/12, ~3

: [[gencom]]: [17/12 3; 289/288]

[[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)

{{Optimal ET sequence|legend=1| 12, 58, 70, 82, 94, 106, 118, 224g }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2247 cents

[[Subgroup]]: 2.3.85

[[Comma list]]: 2.3.85 {{val|-40 1 6 }}

 

 

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~85/64 = 483.034

 

[[Support]]ing [[ET]]s: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159**

 

==== 2.3.7.85 subgroup ====

[[Subgroup]]: 2.3.7.85

[[Comma list]]: 1029/1024, 7225/7203

{{mapping|legend=2| 1 4 2 6 | 0 -6 2 1}}

 

 

 

 

 

=== Dog ===

The dog temperament is based by [[2L 5s]] or [[7L 2s]] scale that makes [[81/76]] vanish, so [[19/16]] is a major third. It can be viewed as a 2.3.19 subgroup analogue of [[Pelogic family|mavila]].

 

 

 

 

[[Mapping|Sval mapping]]: [{{val|1 2 6}}, {{val|0 -1 -4}}]

 

 

 

 

Boethian is a [[5L 2s|diatonic-based]] temperament that makes [[513/512]] vanish, so that the major third (C–E) is ~[[24/19]] and the minor third (C–E♭) is ~[[19/16]]. As such, it functions as a 2.3.19-subgroup analogue of [[meantone]], though the small size of the comma puts it at [[schismic]] level of accuracy. In particular, the equal temperaments in the tuning spectrum up to 1/2-comma (flattened) boethian temperament (very close to [[12edo]]) are included in the schismic tuning spectrum in the 5-limit, so boethian intersects with schismic in the prime-5 infill extension thereof, called [[nestoria]], which also tempers out [[361/360]], the difference between 19/18 and 20/19 or between 19/15 and 24/19.  

[[Comma list]]: [[513/512]]

{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}

: mapping generators: ~2, ~3

[[Optimal tuning]] (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.3288{{c}}

{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296 }}

[[Badness]] (Smith): 0.000374

Lipsett temperament is a pleasantly melodic little temperament with a highly useable 5-tone and 9-tone mos. It is audibly similar to [[semaphore]] temperament, so it could be thought of as semaphore but for the 23rd harmonic instead of the 7th. It is named for Arthur Lipsett, the director off the Canadian short film ’21-87’. Leia’s prison cell in Star Wars is numbered ‘2187’, as a nod to the influence the film had on George Lucas.

[[Comma list]]: [[2187/2116]]

{{Mapping|legend=2| 1 0 -1 | 0 2 7}}

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~46/27 = 948.526

[[Badness]] (Smith): 8.998 × 10^-3

[[Mapping]]: [{{val|1 2 5}}, {{val|0 3 -1}}]

[[CTE tuning|CTE generator]]: ~32/29 = 166.067

This temperament tempers out 4107/4096 and thus equates 2 [[37/32]]'s with [[4/3]].

[[Gencom]]: [2 37/32; 4107/4096]

[[Mapping]]: [{{val|1 1 5}}, {{val|0 -2 1}}]

 

[[POTE generator]]: ~[[37/32]] = 249.075

 

 

==== 2.3.7.37 subgroup ====

[[Subgroup]]: 2.3.7.37

 

 

 

[[Mapping]]: [{{val|1 1 1 5}}, {{val|0 -2 -1 1}}]

 

[[POTE generator]]: ~[[37/32]] = 247.782

 

{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 53d }}

 

 

 

 

[[Gencom]]: [2 37/32; 4107/4096 17592186044416/17562397269605]

 

[[Mapping]]: [{{val|1 1 4 5}}, {{val|0 -2 -8 1}}]

 

[[POTE generator]]: ~[[37/32]] = 251.393

 

{{Optimal ET sequence|legend=1| 5, 14c, 19, 43, 62 }}

 

===== 2.3.5.7.37 subgroup =====

[[Subgroup]]: 2.3.5.7.37

 

 

 

[[Mapping]]: [{{val|1 1 4 1 5}}, {{val|0 -2 -8 -1 1}}]

[[POTE generator]]: ~[[37/32]] = 251.204

[[Gencom]]: [2 4/3; 81/74]

[[Mapping]]: [{{val|1 1 0}}, {{val|0 -1 12}}]

[[POTE generator]]: ~4/3 = 521.397

Subgroup: 2.3.41

[[Gencom]]: [2 4/3; 82/81]

[[Mapping|Sval mapping]]: [{{val|1 2 7}}, {{val|0 -1 -4}}]

 

[[Tp tuning|POL2 generator]]: ~4/3 = 494.509

 

{{Optimal ET sequence|legend=1| 5, 12, 17 }}

 

==== 2.3.7.41 subgroup ====

Subgroup: 2.3.7.41

 

[[Comma list]]: 64/63, 82/81

 

 

[[Mapping|Sval mapping]]: [{{val| 1 2 2 7 }}, {{val| 0 -1 2 -4 }}]

 

[[POTE generator]]: ~4/3 = 490.0323

 

[[TOP tuning|TOP generator]]s: ~2 = 1197.2342, ~4/3 = 488.9029

 

{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 49 }}

 

 

 

[[Gencom]]: [2 4/3; 64/63 82/81 99/98]

 

[[Mapping|Sval mapping]]: [{{val| 1 2 2 1 7 }}, {{val| 0 -1 2 6 -4 }}]

 

[[TOP tuning|TOP generator]]s: ~2 = 1197.9683, ~4/3 = 491.3454