Mint temperaments: Difference between revisions

This is a collection of low [[complexity]], high [[error]], [[regular temperament|temperaments]] which [[temper out]] the septimal quarter-tone, [[36/35]]. 36 is [[Wikipedia: Square triangular number|both]] a [http://mathworld.wolfram.com/SquareNumber.html square] and a [[triangular number]], and this helps make 36/35 a septimal interval of considerable significance. These temperaments equate [[6/5]] with [[7/6]], [[5/4]] with [[9/7]], and [[7/4]] with [[9/5]], so minor and major thirds and sixths are intervals of 5 and 7 at the same time.  

* [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]

* ''[[Whitewood]]'' (+2187/2048) → [[Whitewood family #Septimal whitewood|Whitewood family]]

: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 28/25, 36/35

{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}

: mapping generators: ~2, ~7/6

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1186.8093{{c}}, ~7/6 = 237.3390{{c}}

: [[error map]]: {{val| -13.191 -3.129 +61.983 -45.851 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 231.845{{c}}

: error map: {{val| 0.000 +6.291 +85.850 -24.498 }}

{{Optimal ET sequence|legend=1| 1bd, …, 4bcd, 5 }}

[[Badness]] (Sintel): 1.19

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lafayette]].''

Named by [[Gene Ward Smith]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18891.html Yahoo! Tuning Group | ''Progression temperament'']</ref>, progression can be described as the {{nowrap| 8d & 9 }} temperament. It has a generator that is a somewhat flat neutral second, three make [[5/4]], five make [[3/2]], and seven make [[7/4]], with a [[ploidacot]] signature of pentacot. [[17edo]] is an obvious tuning for it.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 36/35, 392/375

{{Mapping|legend=1| 1 1 2 2 | 0 5 3 7 }}

: mapping generators: ~2, ~15/14

[[Optimal tuning]]s:

: [[error map]]: {{val| -6.046 -6.916 +22.246 +0.601 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~15/14 = 139.8991{{c}}

: error map: {{val| 0.000 -2.460 +33.384 +10.468 }}

{{Optimal ET sequence|legend=1| 8d, 9, 17c }}

[[Badness]] (Sintel): 1.22

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 77/75

 

Mapping: {{mapping| 1 1 2 2 3 | 0 5 3 7 4 }}

 

Optimal tunings:  

* WE: ~2 = 1194.7089{{c}}, ~12/11 = 140.1262{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.8776{{c}}

 

{{Optimal ET sequence|legend=0| 8d, 9, 17c }}

 

Badness (Sintel): 0.861

 

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

 

Comma list: 26/25, 36/35, 56/55, 66/65

 

 

* WE: ~2 = 1195.3694{{c}}, ~13/12 = 140.2080{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.9749{{c}}

 

{{Optimal ET sequence|legend=0| 8d, 9, 17c }}

 

Badness (Sintel): 0.750

 

=== 17-limit ===

 

Comma list: 26/25, 36/35, 51/50, 56/55, 66/65

 

 

* WE: ~2 = 1193.8350{{c}}, ~13/12 = 140.6779{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.5885{{c}}

 

{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}

 

Badness (Sintel): 0.853

 

=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

 

Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65

 

Mapping: {{mapping| 1 1 2 2 3 3 4 4 | 0 5 3 7 4 6 1 2 }}

 

Optimal tunings:  

* WE: ~2 = 1196.1446{{c}}, ~13/12 = 140.0276{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.0301{{c}}

 

 

Badness (Sintel): 1.02

 

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Delorean]].''

 

Subklei is likely named for its flatter minor third generator than [[kleismic]]. It tempers out [[1029/1000]] as well as [[2401/2400]] and can be described as {{nowrap| 17c & 21 }}. Its [[ploidacot]] is delta-hexacot. Note that in the data below, the generator is the [[5/3]][[~]][[12/7]] major sixth, so that six generators minus four octaves give the [[3/2|perfect fifth]].

 

[[Subgroup]]: 2.3.5.7

 

[[Comma list]]: 36/35, 1029/1000

 

{{Mapping|legend=1| 1 -3 -3 -1 | 0 6 7 5 }}

: mapping generators: ~2, ~5/3

 

[[Optimal tuning]]s:  

: [[error map]]: {{val| -2.471 -11.171 +18.366 +3.121 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 915.4228{{c}}

: error map: {{val| 0.000 -9.418 +21.646 +8.288 }}

 

{{Optimal ET sequence|legend=1| 4, 13cd, 17c, 21, 38c }}

 

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

Comma list: 36/35, 77/75, 352/343

 

Mapping: {{mapping| 1 -3 -3 -1 -8 | 0 6 7 5 15 }}

 

* WE: ~2 = 1196.3345{{c}}, ~5/3 = 913.9469{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.2970{{c}}

 

{{Optimal ET sequence|legend=0| 4e, …, 13cdee, 17c }}

 

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

Comma list: 26/25, 36/35, 66/65, 352/343

 

Mapping: {{mapping| 1 3 4 4 7 7 | 0 6 7 5 15 14 }}

 

* WE: ~2 = 1196.0784{{c}}, ~5/3 = 914.1466{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.6712{{c}}

 

{{Optimal ET sequence|legend=0| 4ef, …, 13cdeef, 17c }}

 

Badness (Sintel): 1.34

 

=== Subkla ===

Subgroup: 2.3.5.7.11

 

Comma list: 36/35, 56/55, 1029/1000

 

Mapping: {{mapping| 1 -3 -3 -1 5 | 0 6 7 5 -2 }}

 

* WE: ~2 = 1196.3809{{c}}, ~5/3 = 913.4153{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.8804{{c}}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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