Breedsmic temperaments: Difference between revisions

This page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[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.

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

* ''[[Newt]]'' (+33554432/33480783) → [[Garischismic clan #Beatles|Garischismic clan]]

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

* ''[[Sesquiquartififths]]'' (+32805/32768) → [[Schismatic family #Sesquiquartififths|Schismatic family]]

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

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

* [[Harry]] (+19683/19600) → [[Gravity family #Harry|Gravity family]]

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

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

* [[Ennealimmal]] (+4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]

* ''[[Maviloid]]'' (+1224440064/1220703125) → [[Parakleismic family #Maviloid|Parakleismic family]]

* ''[[Hemicountercomp]]'' (+52613349376/52301766015) → [[Countercomp family #Hemicountercomp|Countercomp family]]

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

{{See also| Tertiaseptal extensions }}

 

 

[[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.

==== 2.3.5.7.13 subgroup ====

Subgroup: 2.3.5.7.13

 

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

* WE: ~2 = 1199.9931{{c}}, ~224/117 = 1122.8203{{c}} (~117/112 = 77.1729{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.8267{{c}} (~117/112 = 77.1733{{c}})

Badness (Sintel): 0.682

Subgroup: 2.3.5.7.13.17

Comma list: 625/624, 833/832, 1225/1224, 4096/4095

{{Mapping|legend=2| 1 -19 7 0 43 49 | 0 22 -5 3 -42 -48 }}

{{Mapping|legend=4| 1 -19 7 0 0 43 49 | 0 22 -5 3 0 -42 -48 }}

* WE: ~2 = 1200.0139{{c}}, ~65/34 = 1122.8361{{c}} (~68/65 = 77.1779{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8230{{c}} (~68/65 = 77.1770{{c}})

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

Badness (Sintel): 0.574

Subgroup: 2.3.5.7.11

{{Mapping|legend=0| 1 -19 7 0 -19 | 0 22 -5 3 24 }}

{{Mapping|legend=0| 1 -19 7 0 -19 43 | 0 22 -5 3 24 -42 }}

{{Mapping|legend=0| 1 -19 7 0 -19 43 49 | 0 22 -5 3 24 -42 -48 }}

=== Septia ===

Comma list: 243/242, 441/440, 65625/65536

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

* 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

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 }}

* 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

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 }}

* 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 ===

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

Comma list: 2401/2400, 6250/6237, 65625/65536

{{Mapping|legend=0| 1 -19 7 0 112 | 0 22 -5 3 -116 }}

* WE: ~2 = 1200.0053{{c}}, ~245/128 = 1122.8357{{c}} (~256/245 = 77.1696{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8308{{c}} (~256/245 = 77.1692{{c}})

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

Badness (Sintel): 1.88

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400

{{Mapping|legend=0| 1 -19 7 0 112 43 | 0 22 -5 3 -116 -42 }}

* WE: ~2 = 1199.9823{{c}}, ~224/117 = 1122.8150{{c}} (~117/112 = 77.1673{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.8316{{c}} (~117/112 = 77.1684{{c}})

{{Optimal ET sequence|legend=0| 31e, 140, 171, 311, 1073 }}

Badness (Sintel): 1.14

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

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

{{Mapping|legend=0| 1 -19 7 0 112 43 49 | 0 22 -5 3 -116 -42 -48 }}

* WE: ~2 = 1200.0092{{c}}, ~65/34 = 1122.8392{{c}} (~68/65 = 77.1700{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8305{{c}} (~68/65 = 77.1695{{c}})

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

Badness (Sintel): 0.956

==== 2.3.5.7.11.13.17.23 subgroup ====

Subgroup: 2.3.5.7.11.13.17.23

Comma list: 595/594, 625/624, 833/832, 1105/1104, 1156/1155, 2200/2197

{{Mapping|legend=0| 1 -19 7 0 112 43 49 114 | 0 22 -5 3 -116 -42 -48 -117 }}

* WE: ~2 = 1200.0047{{c}}, ~44/23 = 1122.8363{{c}} (~23/22 = 77.1684{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8319{{c}} (~23/22 = 77.1681{{c}})

{{Optimal ET sequence|legend=0| 31ei, 140, 171, 311 }}

Badness (Sintel): 0.944

==== 2.3.5.7.11.13.17.23.29 subgroup ====

Subgroup: 2.3.5.7.11.13.17.23.29

Comma list: 595/594, 625/624, 784/783, 833/832, 1015/1014, 1105/1104, 1156/1155

{{Mapping|legend=0| 1 -19 7 0 112 43 49 114 61 | 0 22 -5 3 -116 -42 -48 -117 -60 }}

* WE: ~2 = 1199.9945{{c}}, ~44/23 = 1122.8270{{c}} (~23/22 = 77.1675{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8322{{c}} (~23/22 = 77.1678{{c}})

{{Optimal ET sequence|legend=0| 31ei, 140, 311, 762g }}

Badness (Sintel): 0.858

{{Mapping|legend=0| 1 -41 12 -3 -73 | 0 44 -10 6 79 }}

{{Mapping|legend=0| 1 -41 12 -3 -73 85 | 0 44 -10 6 79 -84 }}

{{Mapping|legend=0| 1 -41 12 -3 -73 85 97| 0 44 -10 6 79 -84 -96 }}

{{Mapping|legend=0| 2 -16 9 3 47 | 0 22 -5 3 -46 }}

Emmthird tempers out the [[scheme comma]] and may be described as the {{nowrap| 58 & 171 }} temperament. The generator for emmthird is flatter than [[81/64]] by a lee comma, [[177147/175616]], and sharper than [[5/4]] by the hemimage comma, [[10976/10935]]. The [[ploidacot]] for this temperament is delta-14-cot.

 

The [[11-limit]] version, which tempers out [[243/242]] and [[441/440]], has much lower accuracy and is [[support]]ed by much fewer equal temperaments.  

{{Mapping|legend=0| 1 -3 -17 -8 -8 | 0 14 59 33 35 }}

{{Mapping|legend=0| 1 -3 -17 -8 -8 -13 | 0 14 59 33 35 51 }}

{{Mapping|legend=0| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}

{{Mapping|legend=0| 1 1 -5 -1 2 | 0 2 25 13 5 }}

{{Mapping|legend=0| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}

{{Mapping|legend=0| 2 0 -35 -15 -47 | 0 2 25 13 34 }}

{{Mapping|legend=0| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}

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

{{Mapping|legend=0| 1 1 -5 -1 8 | 0 4 50 26 -31 }}

{{Mapping|legend=0| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}

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 {{nowrap| 31 & 270 }} temperament, and its [[ploidacot]] is eta-38-cot (or omega-triseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure). 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.

{{Mapping|legend=0| 1 -7 3 1 -11 | 0 38 -3 8 64 }}

{{Mapping|legend=0| 1 -7 3 1 -11 22 | 0 38 -3 8 64 -81 }}

Quinmite may be described as the {{nowrap| 99 & 103 }} temperament. The generator for quinmite is the quasi-tempered minor third [[25/21]], sharper than [[32/27]] by the marvel comma, [[225/224]]. It is also generated by 1/5 of the 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>. Its [[ploidacot]] is eta-34-cot.  

Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit, and may be described as the {{nowrap| 10 & 171 }} temperament. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]), with a [[ploidacot]] of beta-26-cot. It is an excellent temperament for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.

: mapping 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]].

{{Mapping|legend=0| 1 -1 6 4 -3 | 0 26 -37 -12 65 }}

{{Mapping|legend=0| 1 -1 6 4 -3 4 | 0 26 -37 -12 65 -3 }}

{{Mapping|legend=0| 1 -1 6 4 -3 4 2 | 0 26 -37 -12 65 -3 21 }}

Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[~]][[77/60]] from the 11-limit onwards, and 74 generator steps give the interval class of [[3/1|3]]; its [[ploidacot]] is 26-sheared 74-cot. 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 has a [[temperament merging|join]] 103 & 270, hence the name. The name was proposed in 2022 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 [[41/1|41st]] [[harmonic]] such that the generator is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.

{{Mapping|legend=0| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}

{{Mapping|legend=0| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}

{{Mapping|legend=0| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}

Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[temperament merging|join]] of two tuning systems well-known for their high accuracy. It is generated by the interval of [[14/11]] (<u>un</u>decimal major <u>third</u>, hence the name) tuned less than a cent flat, 42 of which [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]. Its [[ploidacot]] is 14-sheared 42-cot. The 23-note [[mos]] from the generator serves as a well temperament of, of all things, [[23edo]]. The 49-note mos is needed to access the [[3/1|3rd]], [[5/1|5th]], [[7/1|7th]], and [[11/1|11th]] [[harmonic]]s.

The commas it tempers out in the 11-limit include the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).

{{Mapping|legend=0| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}

{{Mapping|legend=0| 1 -13 -14 -9 -8 -47 | 0 42 47 34 33 146 }}

Neominor tempers out [[177147/175616]] and may be described as the {{nowrap| 72 & 89 }} temperament. The generator is a neogothic minor third, which represents [[13/11]][[~]][[20/17]], or its [[octave complement]], which represents [[17/10]]~[[22/13]]. The latter stacked six times [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]], and the temperament has a [[ploidacot]] of delta-hexacot. [[72edo]] and [[89edo]] can be used as tunings.  

{{Mapping|legend=0| 1 -3 -29 -14 -8 | 0 6 41 22 15 }}

{{Mapping|legend=0| 1 -3 -29 -14 -8 -7 | 0 6 41 22 15 14 }}

{{Mapping|legend=0| 1 -3 -29 -14 -8 -7 -28 | 0 6 41 22 15 14 42 }}

{{Mapping|legend=0| 1 -13 -4 -4 2 | 0 30 13 14 3 }}

{{Mapping|legend=0| 1 -13 -4 -4 2 -7 | 0 30 13 14 3 22 }}

{{Mapping|legend=0| 1 -27 -7 -9 -4 | 0 46 15 19 12 }}

{{Mapping|legend=0| 1 -27 -7 -9 -4 -5 | 0 46 15 19 12 14 }}

{{Mapping|legend=0| 1 1 9 6 2 | 0 2 -23 -11 5 }}

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 may be described as the {{nowrap| 58 & 103 }} temperament. It has a generator of [[~]][[10/9]], tuned to around [[49/44]]. Note that in the data below, the generator is its [[octave complement]], ~[[9/5]], so that 22 of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]. Its [[ploidacot]] is 18-sheared 22-cot. As one might expect, 25\161 makes for an excellent tuning choice.

{{Mapping|legend=0| 1 -17 -34 -20 -43 | 0 22 43 27 55 }}

{{Mapping|legend=0| 1 -17 -34 -20 -43 -36 | 0 22 43 27 55 47 }}

{{Mapping|legend=0| 1 -17 -34 -20 -43 -36 10 | 0 22 43 27 55 47 -7 }}

{{Mapping|legend=0| 1 -13 8 2 14 | 0 18 -7 1 -13 }}

{{Mapping|legend=0| 1 -13 8 2 14 11 | 0 18 -7 1 -13 -9 }}

Hemigoldis may be described as the {{nowrap| 68 & 89 }} temperament. Though fairly complex in the [[7-limit]], it does a lot better in badness metrics than pure [[5-limit]] [[goldis]], and yet again has many possible extensions to higher 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 can be described as the {{nowrap| 311 & 431 }} temperament, starting with the 7-limit to the 19-limit. Its [[ploidacot]] is 28-sheared 70-cot. It was named by [[Eliora]] in 2022 for the generator fifth, 675/448 being 225/224 (marvel comma) sharp of 3/2.  

{{Mapping|legend=0| 1 -27 55 22 -19 | 0 70 -129 -47 55 }}

{{Mapping|legend=0| 1 -27 55 22 -19 -11 | 0 70 -129 -47 55 36 }}

{{Mapping|legend=0| 1 -27 55 22 -19 -11 78 | 0 70 -129 -47 55 36 -181 }}

{{Mapping|legend=0| 1 -27 55 22 -19 -11 78 41 | 0 70 -129 -47 55 36 -181 -90 }}

[[Category:Catalogs of rank-2 temperaments]]