Mintaka: Difference between revisions
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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. | '''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]]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. | ||
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. | 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. | ||
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| | | 49/33 | ||
| 779.883 | | 779.883 | ||
| 1/4-comma | | 1/4-comma | ||
Revision as of 19:22, 21 August 2024
Mintaka is a temperament in the 3.7.11 subgroup where ~11/7 is a generator, and the comma 1331/1323 is tempered out, so a stack of two generators represents 27/11 in addition to 121/49, and a stack of three generators, tritave-reduced, represents 9/7. As 11/7 as a generator against the tritave produces a 5L 2s (macrodiatonic) scale, with the generator here occupying the role of a 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♭). 9\22edt is a very good tuning for the generator, and 22edt overall excels in the 3.7.11 subgroup, but other tunings such as 7\17edt and 16\39edt are also useful.
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 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.
Mos scales of reasonable tunings have cardinalities of 5 (2L 3s), 7 (5L 2s), 12 (5L 7s), or 17 (5L 12s, 12L 5s).
Extensions of Mintaka
Several extensions of this temperament are possible to incorporate additional harmonics.
Add 20 and 23/4
Off the bat, given that 1331/1323 is a lopsided comma with S-expression S222 * 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.
Add 4 and 5
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.
Add 19
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.
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.
Interval chains
| # | Cents* | Approximate Ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 778.5 | 11/7, 36/23 |
| 2 | 1556.9 | 27/11, 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 |
Tuning spectrum
| Edt Generator |
Eigenmonzo (Unchanged-interval) |
Generator (¢) |
Comments |
|---|---|---|---|
| 7\17 | 783.158 | ||
| 11/7 | 782.492 | 0-comma | |
| 23\56 | 781.160 | ||
| 363/343 | 780.405 | 1/5-comma | |
| 16\39 | 780.289 | ||
| 49/33 | 779.883 | 1/4-comma | |
| 41\100 | 779.802 | ||
| 25\61 | 779.490 | ||
| 59\144 | 779.273 | ||
| 34\83 | 779.114 | ||
| 9/7 | 779.013 | 1/3-comma | |
| 43\105 | 778.896 | ||
| 52\127 | 778.753 | ||
| 81/77 | 778.317 | 2/5-comma | |
| 778.124 | DR 7:9:11, close to 18/43-comma | ||
| 9\22 | 778.073 | ||
| 11/9 | 777.274 | 1/2-comma | |
| 29\71 | 776.855 | ||
| 20\49 | 776.308 | ||
| 11\27 | 774.871 | ||
| 121/63 | 772.055 | Full comma |
Other tunings
- DKW (3.7.11): ~3 = 1\1, ~11/7 = 778.466