Xenharmonic Wiki

Breedsmic temperaments: Difference between revisions

← Older edit

FloraC (talk | contribs)

4 user groups
27,368 edits
VisualWikitext
mNo edit summary
m →Gorgik: - link to dissolved clan
 
(5 intermediate revisions by the same user not shown)
Line 34: Line 34:


== Tertiaseptal ==
== Tertiaseptal ==
{{Main| Tertiaseptal }}
{{See also| Tertiaseptal extensions }}


Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the {{nowrap| 31 & 171 }} temperament, and [[256/245]], [[1029/1024]] less than [[21/20]], serves as its generator. Three of these fall short of [[8/7]] by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. The [[ploidacot]] for this temperament is 20-sheared 22-cot (or pentaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure).  
Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the {{nowrap| 31 & 171 }} temperament, and [[256/245]], [[1029/1024]] less than [[21/20]], serves as its generator. Three of these fall short of [[8/7]] by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. The [[ploidacot]] for this temperament is 20-sheared 22-cot (or pentaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure).  


[[171edo]] makes for an excellent tuning, although [[140edo]] ({{nowrap| {{=}} 171 - 31 }}) also makes sense, and in very high limits [[311edo]] ({{nowrap| {{=}} 140 + 171 }}) is especially notable. The 15- or 16-note [[mos]] can be used to explore no-threes harmony, and the 31-note mos gives plenty of room for those as well.
Tertiaseptal does not extend well to the the [[11-limit]], with three possible mappings for the [[11/1|11th harmonic]]: tertia (109 & 140), tertiaseptia (140 & 171), and septia (171 & 202). Their useful tuning ranges are indicated by the equal-temperament joins. However, since the entire range can use the same mapping for [[13/1|13]] and [[17/1|17]], we can consider tertiaseptal a no-11 13- and 17-limit temperament, in which [[625/624]], [[833/832]], [[1225/1224]], and [[4096/4095]] are tempered out.  


[[171edo]] makes for an excellent tuning, although [[140edo]] ({{nowrap| {{=}} 171 - 31 }}) also makes sense, and for the add-13 add-17 extension [[311edo]] ({{nowrap| {{=}} 140 + 171 }}) is especially notable. The 15- or 16-note [[mos]] can be used to explore no-3's harmony, and the 31-note mos gives plenty of room for those as well.
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 57: Line 60:
[[Badness]] (Sintel): 0.329
[[Badness]] (Sintel): 0.329


=== 11-limit ===
==== 2.3.5.7.13 subgroup ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.13
 
Comma list: 625/624, 2401/2400, 4096/4095


Comma list: 243/242, 441/440, 65625/65536
{{Mapping|legend=2| 1 -19 7 0 43 | 0 22 -5 3 -42 }}


{{Mapping|legend=0| 1 -19 7 0 -48 | 0 22 -5 3 55 }}
{{Mapping|legend=4| 1 -19 7 0 0 43 | 0 22 -5 3 0 -42 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1034{{c}}, ~245/128 = 1122.8694{{c}} (~256/245 = 77.2340{{c}})
* WE: ~2 = 1199.9931{{c}}, ~224/117 = 1122.8203{{c}} (~117/112 = 77.1729{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.7743{{c}} (~256/245 = 77.2257{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.8267{{c}} (~117/112 = 77.1733{{c}})
 
{{Optimal ET sequence|legend=0| 31, 109e, 140e, 171, 202 }}


Badness (Sintel): 1.18
{{Optimal ET sequence|legend=0| 31, 109, 140, 171, 311, 793f }}


==== 13-limit ====
Badness (Sintel): 0.682
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 441/440, 625/624, 3584/3575
==== 2.3.5.7.13.17 subgroup ====
Subgroup: 2.3.5.7.13.17


{{Mapping|legend=0| 1 -19 7 0 -48 43 | 0 22 -5 3 55 -42 }}
Comma list: 625/624, 833/832, 1225/1224, 4096/4095


Optimal tunings:
{{Mapping|legend=2| 1 -19 7 0 43 49 | 0 22 -5 3 -42 -48 }}
* WE: ~2 = 1199.8783{{c}}, ~224/117 = 1122.6835{{c}} (~117/112 = 77.1948{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.7968{{c}} (~117/112 = 77.2032{{c}})


{{Optimal ET sequence|legend=0| 31, 140e, 171, 373ef }}
{{Mapping|legend=4| 1 -19 7 0 0 43 49 | 0 22 -5 3 0 -42 -48 }}
 
Badness (Sintel): 1.52
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
 
{{Mapping|legend=0| 1 -19 7 0 -48 43 49 | 0 22 -5 3 55 -42 -48 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8677{{c}}, ~65/34 = 1122.6748{{c}} (~68/65 = 77.1929{{c}})
* WE: ~2 = 1200.0139{{c}}, ~65/34 = 1122.8361{{c}} (~68/65 = 77.1779{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.7985{{c}} (~68/65 = 77.2015{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8230{{c}} (~68/65 = 77.1770{{c}})


{{Optimal ET sequence|legend=0| 31, 140e, 171 }}
{{Optimal ET sequence|legend=0| 31, 109g, 140, 171, 311, 482 }}


Badness (Sintel): 1.40
Badness (Sintel): 0.574


=== Tertia ===
=== Tertia ===
Subgroup:2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 1331/1323, 1375/1372
Comma list: 385/384, 1331/1323, 1375/1372
Line 146: Line 138:


Badness (Sintel): 1.14
Badness (Sintel): 1.14
=== Septia ===
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 65625/65536
{{Mapping|legend=0| 1 -19 7 0 -48 | 0 22 -5 3 55 }}
Optimal tunings:
* WE: ~2 = 1200.1034{{c}}, ~245/128 = 1122.8694{{c}} (~256/245 = 77.2340{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.7743{{c}} (~256/245 = 77.2257{{c}})
{{Optimal ET sequence|legend=0| 31, 109e, 140e, 171, 202 }}
Badness (Sintel): 1.18
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 441/440, 625/624, 3584/3575
{{Mapping|legend=0| 1 -19 7 0 -48 43 | 0 22 -5 3 55 -42 }}
Optimal tunings:
* WE: ~2 = 1199.8783{{c}}, ~224/117 = 1122.6835{{c}} (~117/112 = 77.1948{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.7968{{c}} (~117/112 = 77.2032{{c}})
{{Optimal ET sequence|legend=0| 31, 140e, 171, 373ef }}
Badness (Sintel): 1.52
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
{{Mapping|legend=0| 1 -19 7 0 -48 43 49 | 0 22 -5 3 55 -42 -48 }}
Optimal tunings:
* WE: ~2 = 1199.8677{{c}}, ~65/34 = 1122.6748{{c}} (~68/65 = 77.1929{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.7985{{c}} (~68/65 = 77.2015{{c}})
{{Optimal ET sequence|legend=0| 31, 140e, 171 }}
Badness (Sintel): 1.40


=== Tertiaseptia ===
=== Tertiaseptia ===
This extension was considered by [[Gene Ward Smith]] as a 41-limit temperament<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_9274.html Yahoo! Tuning Group | ''A 41-limit temperament'']</ref>. It can be extended as such by tempering out 875/874, 714/713, 703/702 and 697/696, and mapping 19, 31, 37 and 41 to 94, 105, -81 and +10 steps, respectively.  
This extension is enormously complex, but preserves much of the accuracy. It was considered by [[Gene Ward Smith]] as a 41-limit temperament<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_9274.html Yahoo! Tuning Group | ''A 41-limit temperament'']</ref>. It can be extended as such by tempering out 875/874, 714/713, 703/702 and 697/696, and mapping 19, 31, 37 and 41 to 94, 105, -81 and +10 steps, respectively.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,071: Line 1,108:


== Gorgik ==
== Gorgik ==
{{See also| Llywelynsmic clan }}
Gorgik may be described as the {{nowrap| 21 & 37 }} temperament, with a [[ploidacot]] of 14-sheared 18-cot (or alpha-heptaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] [[restriction]]). [[58edo]] makes for a strong tuning for this temperament.  
 
Gorgik may be described as the {{nowrap| 21 & 37 }} temperament, with a [[ploidacot]] of 14-sheared 18-cot (or alpha-heptaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] [[restriction]]). [[58edo]] makes for a strong tuning for this temperament.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Retrieved from "https://en.xen.wiki/w/Breedsmic_temperaments"