Breedsmic temperaments: Difference between revisions

This page discusses miscellaneous [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[breedsma]] ({{monzo|legend=1| -5 -1 -2 4 }}, [[ratio]]: 2401/2400). This is the amount by which two [[49/40]] intervals exceed [[3/2]], and by which two [[60/49]] intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system ([[12edo]], for example) which does not possess a neutral third cannot be tempering out the breedsma.

The breedsma is also the amount by which four stacked [[10/7]] intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, [[25/24]]. We might note also that (49/40)(10/7) = 7/4 and (49/40)(10/7)² = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.

* ''[[Decimal]]'' (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]

* ''[[Beatles]]'' (+64/63 or 686/675) → [[Archytas clan #Beatles|Archytas clan]]

* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]

* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Greenwoodmic temperaments #Greenwood|Greenwoodmic temperaments]]

* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]

* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]

* [[Ennealimmal]] (+4375/4374) → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]]

* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]

* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]

Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, newt, unthirds, septidiasemi, subneutral, maviloid, lockerbie, neominor, catafourth, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].  

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. [[171edo]] makes for an excellent tuning, although 171edo - [[31edo]] = [[140edo]] also makes sense, and in very high limits 140edo + 171edo = [[311edo]] 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.

* [[WE]]: ~2 = 1200.1004{{c}}, ~245/128 = 1122.9024{{c}} (~256/245 = 77.1979)

* [[CWE]]: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8101{{c}} (~256/245 = 77.1899)

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197

Mapping: {{mapping| 1 -19 7 0 112 43 49 -94 | 0 22 -5 3 -116 -42 -48 105 }}

* WE: ~2 = 1200.0187{{c}}, ~65/34 = 1122.8489{{c}} (~68/65 = 77.1698{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8313{{c}} (~68/65 = 77.1687{{c}})

{{Optimal ET sequence|legend=0| 140, 171, 311 }}

Badness (Sintel): 1.07

==== 23-limit ====

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215

Mapping: {{mapping| 1 -19 7 0 112 43 49 -94 114 | 0 22 -5 3 -116 -42 -48 105 -117 }}

* WE: ~2 = 1200.0101{{c}}, ~44/23 = 1122.8418{{c}} (~23/22 = 77.1683{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8323{{c}} (~23/22 = 77.1677{{c}})

{{Optimal ET sequence|legend=0| 140, 311, 762g }}

Badness (Sintel): 1.08

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

{{Optimal ET sequence|legend=0| 140, 171, 311 }}

 

Badness (Sintel): 1.20

The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.

This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four.  

In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 ({{monzo| 22 -1 -10 1 }}). It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31 & 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)^1/8, giving just 7's, or 384^1/38, giving pure fifths.

The generator for quinmite is quasi-tempered minor third [[25/21]], flatter than 6/5 by the starling comma, [[126/125]]. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth", given by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.

Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.

: mpping generators: ~2, ~15/14

The ''sedia'' temperament (10 & 161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.

Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[120/77]] or [[77/60]]. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.  

The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament is defined as 103 & 270, hence the name. The name is proposed by Eliora, who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.  

Lockerbie also has a unique extension that adds the 41st harmonic such that the generator below 600 cents is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.

=== 2.3.5.7.11.13.17.41 subgroup ===

Subgroup: 2.3.5.7.11.13.17.41

Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224

Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }}

* WE: ~2 = 1199.8693{{c}}, ~41/32 = 431.0650{{c}}

* CWE: ~2 = 1200.000{{c}}, ~41/32 = 431.1109{{c}}

{{Optimal ET sequence|legend=0| 103, 167, 270 }}

Badness (Sintel): 1.25

The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.

In addition to 2401/2400, mintone tempers out 177147/175000 ({{monzo| -3 11 -5 -1 }}) in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58 & 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.

: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''

 

Though fairly complex in the [[7-limit]], hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~[[21/19]] to add prime 19 or perhaps more accurately ~[[31/28]] to add prime 7, or even simply as ~[[32/29]] to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again [[89edo]] is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.

''Surmarvelpyth'' is named for the generator fifth, 675/448 being 225/224 (marvel comma) sharp of 3/2. It can be described as the 311 & 431 temperament, starting with the 7-limit to the 19-limit.