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'''Mintaka''' is a [[temperament]] in the 3.7.11 [[subgroup]] where [[~]][[11/7]] is a [[generator]], and the comma [[1331/1323]] is [[tempering out|tempered out]], so a stack of two generators represents [[27/11]] in addition to 121/49, and a stack of three generators, [[3/1|tritave]]-reduced, represents [[9/7]]. As 11/7 as a generator against the tritave produces a [[5L 2s (3/1-equivalent)|5L 2s]] (macrodiatonic) scale, with the generator here occupying the role of a [[4/3|perfect fourth]], it is possible to use an analogue of the [[chain-of-fifths notation]] that is standardly used for [[diatonic]] scales, with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭). [[22edt|9\22]] is a very good tuning for the generator, and 22edt overall excels in the 3.7.11 subgroup, but other tunings such as [[17edt|7\17]] and [[39edt|16\39]] are also useful.  
{{Infobox regtemp
| Subgroups = 3.7.11
| Comma basis = [[1331/1323]]
| Edo join 1 = b5 | Edo join 2 = b17
| Mapping = 1; -3 -2
| Generators = 11/7 | Generators tuning = 778.7 | Optimization method = CWE
| MOS scales = [[2L 3s (3/1-equivalent)|2L 3s]], [[5L 2s (3/1-equivalent)|5L 2s]], [[5L 7s (3/1-equivalent)|5L 7s]], [[5L 12s (3/1-equivalent)|5L 12s]]
| Ploidacot = alpha-trigem
| Odd limit 1 = 3.7.11 11 | Mistuning 1 = 3.48 | Complexity 1 = 7
}}
'''Mintaka''' is a [[non-octave]] [[temperament]] in the 3.7.11 [[subgroup]] where [[~]][[11/7]] is a [[generator]], and the comma [[1331/1323]] is [[tempering out|tempered out]], so a stack of two generators represents [[27/11]] in addition to 121/49, and a stack of three generators, [[3/1|tritave]]-reduced, represents [[9/7]]. As 11/7 as a generator against the tritave produces a [[5L 2s (3/1-equivalent)|5L 2s]] (macrodiatonic) scale, with the generator here occupying the role of a [[4/3|perfect fourth]], it is possible to use an analogue of the [[chain-of-fifths notation]] that is standardly used for [[diatonic]] scales, with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) and that all intervals are extremely stretched, though the [[5L 7s (3/1-equivalent)|5L 7s]] macrochromatic scale is suggested for musical use due to the hardness of the macrodiatonic and the increased breadth of the tritave. [[22edt|9\22]]edt is a very good tuning for the generator, and 22edt overall excels in the 3.7.11 subgroup, but other tunings such as [[17edt|7\17]]edt and [[39edt|16\39]]edt are also useful, especially for extensions involving primes 5 and 13 (see below).
 
As perhaps the simplest temperament of this subgroup delivering decent accuracy—and, in particular, the simplest supported by tunings such as 17edt and 22edt—Mintaka can be considered the 3.7.11 analog of 3.5.7 [[Bohlen–Pierce–Stearns]] or 2.3.5 [[meantone]], using [[7:9:11]] as its fundamental consonant chord in the place of [[3:5:7]] or of [[4:5:6]].
 
[[Mos scale]]s of reasonable tunings have cardinalities of 5 (2L 3s), 7 (5L 2s), 12 (5L 7s), or 17 (5L 12s).


As perhaps the simplest temperament of this subgroup delivering decent accuracy - and, in particular, the simplest supported by tunings such as 17edt and 22edt - Mintaka can be considered the 3.7.11 analog of 3.5.7 [[Bohlen-Pierce-Stearns|BPS]] or 2.3.5 [[meantone]], using 7:9:11 as its fundamental consonant chord in the place of 3:5:7 or of 4:5:6.
{{Tdlink|No-twos subgroup temperaments #Mintaka}}


[[Mos scale]]s of reasonable tunings have cardinalities of 5 (2L 3s), 7 (5L 2s), 12 (5L 7s), or 17 (5L 12s, 12L 5s).
 
[[File:Mintaka_scale_guide.png|alt=Mintaka_scale_guide.png|960x320px]]
 
A pictorial representation of Mintaka[7], two modes of Mintaka[12], and Mintaka[17], and how they vary across the tuning spectrum, with representative tunings of the two main 13-limit extensions depicted in more detail including approximate JI ratios of each interval.


== Extensions of Mintaka ==
== Extensions of Mintaka ==
Several extensions of this temperament are possible to incorporate additional harmonics.
Several extensions of this temperament are possible to incorporate additional harmonics.


=== Add 20 and 23/4 ===
=== Add 23/4 & 20 ===
Off the bat, given that 1331/1323 is a [[Square superparticular#Sk2_.2A_S.28k_.2B_1.29_and_S.28k_-_1.29_.2A_Sk2_.28lopsided_commas.29|lopsided comma]] with S-expression S22<sup>2</sup> * S23, one can reliably choose to temper both S22 = [[484/483]] and S23 = [[529/528]] in the 3.7.11.23/4 subgroup, which equates the 11/7 generator to [[36/23]], and the interval [[11/9]] to [[28/23]]. Furthermore, the tiny comma S161 = [[25921/25920]] can be tempered to add harmonic 20 to the subgroup, finding it 8 generators down. More neatly, this can be expressed as the temperament that tempers out the commas [[253/252]], 484/483, and [[540/539]] in the 3.7.11.20.23/4 subgroup.
Off the bat, given that 1331/1323 is a [[Square superparticular#Sk2_.2A_S.28k_.2B_1.29_and_S.28k_-_1.29_.2A_Sk2_.28lopsided_commas.29|lopsided comma]] with S-expression {{nowrap|S22<sup>2</sup> × S23}}, one can reliably choose to temper both {{nowrap|S22 {{=}} [[484/483]]}} and {{nowrap|S23 {{=}} [[529/528]]}} in the 3.7.11.23/4 subgroup, which equates the 11/7 generator to [[36/23]], and the interval [[11/9]] to [[28/23]]. Furthermore, the tiny comma {{nowrap|S161 {{=}} [[25921/25920]]}} can be tempered to add harmonic 20 to the subgroup, finding it 8 generators down. More neatly, this can be expressed as the temperament that tempers out the commas [[253/252]], 484/483, and [[540/539]] in the 3.7.11.20.23/4 subgroup.


=== Add 4 and 5 ===
=== Add 19 ===
For tunings of the generator that possess a sharp 9/7 (sharper than 1/3 comma), it is reasonable to combine this temperament with BPS, and additionally temper out [[245/243]], thereby equating [[5/3]] to 81/49 at 6 generators up. With the inclusion of 20 in the subgroup above, [[4/3]] would therefore also appear, at the position of (20/9)/(5/3), 14 generators down.
Noting that the interval [[21/19]] is fairly close to the square of [[81/77]], in tunings near 2/5-comma it is reasonable to temper out [[41553/41503]] or equivalently [[16929/16807]] to place 21/19 ten generators up, and the 19th harmonic itself 13 generators down.


=== Add 19 ===
=== Add 13 ===
There are two reasonable ways to incorporate prime 19 into the subgroup. For tunings of the generator ''flatter'' than 9\22edt, it is the most accurate to find [[19/9]] at (9/7)^3, 9 generators up, tempering out the comma [[6561/6517]]; for tunings ''sharper'' than 9\22edt, the step 81/77 approaches or exceeds 20/19 in quality, and therefore can be identified with 20/19 by tempering out S20 = [[400/399]], equating 19/9 to 1540/729 = (77/81)(20/9), 13 generators down. The two representations meet at 22edt.
There are two reasonable ways to incorporate prime 13 into the subgroup. For tunings of the generator ''sharper'' than 9\22edt, the step 81/77 approaches or exceeds 260/243 in quality, and therefore can be identified with 260/243 by tempering out [[20020/19683]], equating 27/13 to (77/81)(20/9), 13 generators down (or alternatively, if one does not include the even number 20 into the subgroup, by tempering out [[218491/216513]]); this is the extension listed as "tridecimal Mintaka". The alternative extension to include prime 13, known as '''Minalzidar''', works better for tunings ''flatter'' than 9\22edt, where it is the most accurate to find [[13/9]] at 3(9/7)<sup>–3</sup>, 9 generators down, tempering out the comma [[351/343]]. The two representations meet at 22edt.
 
=== Add 5 ===
==== Mintra ====
{{Infobox regtemp
| Title = Mintra
| Subgroups = 3.5.7.11, 3.5.7.11.13
| Comma basis = [[245/243]], [[1331/1323]] (3.5.7.11);<br>[[245/243]], [[275/273]], [[1331/1323]] (3.5.7.11.13)
| Edo join 1 = b17 | Edo join 2 = b22
| Mapping = 1; 6 -3 -2 13
| Generators = 11/7 | Generators tuning = 780.4 | Optimization method = CWE
| MOS scales = [[2L 3s (3/1-equivalent)|2L 3s]], [[5L 2s (3/1-equivalent)|5L 2s]], [[5L 7s (3/1-equivalent)|5L 7s]], [[5L 12s (3/1-equivalent)|5L 12s]], [[17L 5s (3/1-equivalent)|17L 5s]]
| Odd limit 1 = 11 | Mistuning 1 = 6.16 | Complexity 1 = 12
| Odd limit 2 = 3.5.7.11.13 25 | Mistuning 2 = 8.77 | Complexity 2 = 17
}}
For tunings of the generator that possess a sharp 9/7 (sharper than {{frac|1|3}}-comma, or effectively between [[17edt]] and [[22edt]] tuning), it is reasonable to combine this temperament with [[BPS]] (as well as [[Deneb]] in the 3.5.11 subgroup), and additionally temper out [[245/243]], thereby equating [[5/3]] to 81/49 at 6 generators up. This is ''Mintra'' temperament, which splits the BPS generator in three. A good tuning for this temperament is [[39edt]], the triple BP equalized scale, though others such as [[95edt]] are possible.
 
In this range, the "canonical" extension to prime 13 makes sense, though it is worth noting that the Minalzidar extension corresponds directly to the 3.5.7.13 extension of BPS. This extension then is equivalent to tempering out [[275/273]] and equating [[13/11]] to [[25/21]]. Furthermore, 13/11 appears 15 generators up, and has a cube root in the temperament: 35/33. Therefore, as {{nowrap|13/11 {{=}} ([[35/33]])([[37/35]])([[39/37]])}}, it is "free" to equate 35/33 additionally to 37/35 and 39/37 (which amounts to tempering out [[407/405]]), placing the 37th harmonic 8 generators up.
 
With the inclusion of 20 in the subgroup above, [[4/3]] would therefore also appear, at the position of (20/9)/(5/3), 14 generators down; though the more interesting case with regard to harmonic 20 is documented below.
 
==== Nekkar and Eshurizel ====
In the ''flatter'' generator range (supported by the Minalzidar extension), the optimal representation of 5 is instead that obtained by tempering out [[120285/117649]], which equates 5 with (529/243)<sup>2</sup>, placing it 16 generators down; this leads to the 3.5.7.11.13 subgroup version of ''Nekkar'' temperament.
 
Nekkar, as soon as harmonic 20 is inserted, this also equates 5 with (20/9)<sup>2</sup>, tempering [[81/80]] in the 3.4.5 subgroup. ''Furthermore'', this then equates 4/3 to 27/20, 8 generators up, therefore creating a square root of 4 at 4 generators up and making this an [[insane]] restriction of [[meantone]] that must be fixed by including a mapping for 2, which turns out to equate it to the false octave of 243/121 or 99/49. Therefore, as soon as prime 5 is incorporated, this temperament folds into ''Eshurizel'', an elaborate add-19 add-23 extension of 11-limit [[squares]] (with commas 81/80, [[99/98]], and [[243/242]]).


If we combine all of the above, we find the complete 3.4.5.7.11.19.23 temperament with commas [[100/99]], [[133/132]], 253/252, 484/483, and 540/539.
Even without the mappings for other primes, this method can be used to introduce octaves into Mintaka in a manner alike to [[sensi]] and [[hedgehog]] being produced as extensions of BPS. Equating the false octave ({{nowrap|243/121 ~ 99/49}}) to 2/1 provides 2.3.7.11 [[skwares]] temperament, to which the aforementioned Eshurizel is but an extension.


== Interval chains ==
== Interval chains ==
One important feature of subgroups involving 3 and 11 is the quasi-octave at the interval designated 243/121; in this temperament, it is equated to 99/49 and placed four generators up. In flatter tunings of the generator, this is closer to a true octave. This interval is meriting of special treatment in terms of consonance and dissonance.
Tritave-reduced harmonics below 243 are marked in '''bold'''.
<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Mintaka
|-
! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
|-
! 3.7.11 subgroup !! 3.7.11.20.23/4 extension
|-
| −4 || 690.0 || 49/33, '''121/81''' || 161/108, 180/121
|-
| −3 || 1468.5 || '''7/3''' || 180/77
|-
| −2 || 345.0 || '''11/9''', 147/121 || 28/23, 60/49
|-
| −1 || 1123.5 || 21/11, 121/63 || 23/12, 44/23
|-
| 0 || 0.0 || '''1/1''' || 253/252, 484/483, 540/539
|-
| 1 || 778.5 || 11/7, 189/121 || 36/23, 69/44
|-
| 2 || 1556.9 || 27/11, 121/49 || 69/28, 49/20
|-
| 3 || 433.4 || 9/7 || 77/60
|-
| 4 || 1211.9 || 99/49, 243/121 || 324/161, 121/60
|-
| 5 || 88.4 || 81/77, 363/343 || 207/196, 21/20
|-
| 6 || 866.9 || 81/49 || 33/20
|-
| 7 || 1645.4 || 891/343, 2187/847 || 207/80
|-
| 8 || 521.9 || 729/539 || 759/560, 27/20
|}
<nowiki />* In 3.7.11-targeted [[DKW theory|DKW]] tuning
</div>
=== Mintra ===
Good tunings of Mintra lie on the sharper side of the generator range, and include [[17edt]], [[39edt]], [[56edt]], and [[95edt]].


<div><div style="display: inline-grid; margin-right: 25px;">
<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Basic 3.7.11.20.23/4 temperament
|+ style="font-size: 105%;" | Mintra
|-
! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="3" | Approximate Ratios
|-
|-
! &#35; !! Cents&#42; !! Approximate Ratios
! 3.5.7.11 subgroup !! Tridecimal mintra !! Add-37 extension
|-
|-
| 0 || 0.0 || '''1/1'''
| −6 || 1021.3 || 9/5, '''49/27''' || 165/91 ||
|-
|-
| 1 || 778.5 || 11/7, 36/23
| −5 || 1802.1 || '''77/27''', 99/35 || 405/143 || 37/13, 105/37
|-
|-
| 2 || 1556.9 || 27/11, 69/28, 49/20
| −4 || 680.9 || 81/55, 49/33, '''121/81''' || 135/91, 175/117 || 37/25, 273/185
|-
|-
| 3 || 433.4 || 9/7, 77/60
| −3 || 1461.7 || '''7/3''', 81/35 || 275/117 || 333/143
|-
|-
| 4 || 1211.9 || 99/49, 243/121, 324/161, 121/60
| −2 || 340.5 || '''11/9''', 147/121, 297/245 || 91/75, 175/143 || 111/91, 315/259, 333/275
|-
|-
| 5 || 88.4 || 81/77, 363/343, 207/196, 21/20
| −1 || 1121.2 || 21/11, 121/63 || 25/13, 143/75 || 333/175, 351/185, 495/259
|-
|-
| 6 || 866.9 || 81/49, 33/20
| 0 || 0.0 || '''1/1''', 245/243 || 275/273 ||
|-
|-
| 7 || 1645.4 || 891/343, 2187/847, 207/80
| 1 || 780.8 || 11/7, 189/121 || 39/25 || 175/111, 185/117, 259/165
|-
|-
| 8 || 521.9 || 729/539, 759/560, 27/20
| 2 || 1561.5 || 27/11, 121/49, 245/99 || 225/91, 429/175 || 91/37, 259/105, 275/111
{{table notes|cols=3
|-
| In 3.7.11-subgroup [[DKW theory|DKW]] tuning
| 3 || 440.3 || 9/7, '''35/27''' || 351/275 || 143/111
}}
|-
| 4 || 1221.1 || '''55/27''', 99/49, 243/121 || 91/45, 351/175 || 75/37, 185/91
|-
| 5 || 99.9 || 35/33, 81/77 || 143/135 || 37/35, 39/37, 259/243
|-
| 6 || 880.6 || '''5/3''', 81/49 || 91/55 ||
|-
| 7 || 1661.4 || 55/21 || 13/5 || 259/99, 675/259
|-
| 8 || 540.2 || 15/11 || 143/105 || '''37/27''', 351/259
|-
| 9 || 1320.9 || 15/7, '''175/81''' || 117/55 || 259/121
|-
| 10 || 199.7 || 55/49, 135/121, 275/243 || 39/35, '''91/81''' || 37/33
|-
| 11 || 980.5 || 135/77, 175/99 || '''143/81''' || 37/21
|-
| 12 || 1761.2 || '''25/9''', 135/49 || 91/33 || 333/121
|-
| 13 || 640.0 || 175/121, 275/189 || '''13/9''', 351/245 || 111/77
|-
| 14 || 1420.8 || 25/11 || 143/63, 273/121 || 111/49, '''185/81'''
|-
| 15 || 299.5 || 25/21 || 13/11 ||
|}
|}
<nowiki />* In 3.5.7.11-subgroup [[CWE]] tuning
</div>
</div>


== Tuning spectrum ==
== Tuning spectrum ==
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! Edt<br>Generator
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Edt<br />Generator
! Generator<br>(¢)
! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]]
! Generator<br />(¢)
! Comments
! Comments
|-
|-
Line 66: Line 177:
| 782.492
| 782.492
| 0-comma
| 0-comma
|-
|
| 5/3
| 781.378
| in Mintra
|-
|-
| [[56edt|23\56]]
| [[56edt|23\56]]
Line 72: Line 188:
|  
|  
|-
|-
| [[95edt|39\95]]
|
| 780.803
|  
|  
| 363/343
|-
| 780.405
|
| 1/5-comma
| 28/23
| 780.702
|
|-
| [[134edt|55\134]]
|
| 780.653
|
|-
|
| 7/5
| 780.590
| in Mintra
|-
|
| 13/9
| 780.492
| in tridecimal Mintaka
|-
|
| 11/5
| 780.352
| in Mintra
|-
|-
| [[39edt|16\39]]
| [[39edt|16\39]]
Line 83: Line 224:
|-
|-
|  
|  
| 99/49
| 13/7
| 780.214
| in tridecimal Mintaka
|-
|
| 13/11
| 780.063
| in tridecimal Mintaka
|-
| [[139edt|57\139]]
|
| 779.938
|
|-
|
| 49/33
| 779.883
| 779.883
| 1/4-comma
| 1/4-comma
Line 91: Line 247:
| 779.802
| 779.802
|  
|  
|-
|
| 13/5
| 779.732
| in tridecimal Mintra
|-
|-
| [[61edt|25\61]]
| [[61edt|25\61]]
Line 108: Line 269:
|-
|-
|  
|  
| [[9/7]]
| 9/7
| 779.013
| 779.013
| 1/3-comma
| 1/3-comma
Line 120: Line 281:
|  
|  
| 778.753
| 778.753
|
|-
|
| 33/20
| 778.478
|  
|  
|-
|-
Line 126: Line 292:
| 778.317
| 778.317
| 2/5-comma
| 2/5-comma
|-
|
| 27/20
| 778.177
|
|-
|-
|  
|  
Line 138: Line 309:
|-
|-
|  
|  
| [[11/9]]
| 21/20
| 777.675
|
|-
|
| 11/9
| 777.274
| 777.274
| 1/2-comma
| 1/2-comma
|-
| [[93edt|38\93]]
|
| 777.143
|
|-
|-
| [[71edt|29\71]]
| [[71edt|29\71]]
Line 150: Line 331:
|  
|  
| 776.308
| 776.308
|
|-
| [[76edt|31\76]]
|
| 775.797
|
|-
|
| 49/20
| 775.669
|
|-
|
| 23/12
| 775.636
|  
|  
|-
|-
Line 164: Line 360:


=== Other tunings ===
=== Other tunings ===
* [[DKW theory|DKW]] (3.7.11): ~3 = 1\1, ~11/7 = 778.466
* [[DKW theory|DKW]] (3.7.11): ~3 = 1\1edt, ~11/7 = 778.466
* [[CEE]] (3.7.11): ~3 = 1\1edt, ~11/7 = 778.478 (5/13-comma)
 
== Scales ==
* [[Mintaka7]] – 7-note macrodiatonic scale (5L&nbsp;2s) for 3.7.11 Mintaka
* [[Mintaka12]] – 12-note macrochromatic scale (5L&nbsp;7s) for 3.7.11 Mintaka
* [[Mintaka17]] – 17-note macroënharmonic scale (5L&nbsp;12s) for 3.7.11 Mintaka
* [[Mintra7]] – 7-note macrodiatonic scale (5L&nbsp;2s) for tridecimal Mintra
* [[Mintra12]] – 12-note macrochromatic scale (5L&nbsp;7s) for tridecimal Mintra
* [[Mintra17]] – 17-note macroënharmonic scale (5L&nbsp;12s) for tridecimal Mintra
* [[Mintra22]] – 22-note scale (17L&nbsp;5s) for tridecimal Mintra
 
=== Properties and uses ===
The 7-note scale of Mintaka encompasses the entire (3,&nbsp;7,&nbsp;11) tonality diamond in its symmetric mode; as the most consonant chord in the subgroup is 7:9:11, it is useful to speak of modes that have this chord on the tonic, which are those that stack at least three 11/7s up: macro-Dorian (LsLLLsL), Aeolian (LsLLsLL), Phrygian (sLLLsLL), and Locrian (sLLsLLL); the macro-Dorian and Aeolian modes also include the inversion of this chord.
 
While the macrodiatonic might suffice for the 3.7.11 subgroup, larger scales are needed to represent the primes 5 and 13 in Mintra; the tetrads 5:7:9:11 (~7:9:11:15), 7:9:11:13, and 9:11:13:15 would be bases of harmony in this extended subgroup. In terms of amount of generators (11/7) up from the tonic, these become 0:−9:−6:−8 (~0:3:1:9); 0:3:1:16, and 0:-2:13:6; the 17-note scale therefore includes all tetrads, although not on the same tonic.
 
== Music ==
* [https://archive.org/details/TuneIn22Edt Tune in 22edt] – [[Peter Kosmorsky]] (2011) – uses the LssLssLsssLssLsss MOS (Mintaka[17])


[[Category:Temperaments]]
[[Category:Mintaka| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Non-octave temperaments]]
Retrieved from "https://en.xen.wiki/w/Mintaka"