Tetracot: Difference between revisions
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| Title = Tetracot | | Title = Tetracot | ||
| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.13 | | Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.13 | ||
| Comma basis = [[20000/19683]] (2.3.5); <br>[[100/99]], [[243/242]] (2.3.5.11) <br>[[100/99]], [[144/143]], [[243/242]] (2.3.5.11.13) | | Comma basis = [[20000/19683]] (2.3.5);<br>[[100/99]], [[243/242]] (2.3.5.11)<br>[[100/99]], [[144/143]], [[243/242]] (2.3.5.11.13) | ||
| Edo join 1 = 7 | Edo join 2 = 27e | | Edo join 1 = 7 | Edo join 2 = 27e | ||
| Mapping = 1; 4 9 10 -2 | | Mapping = 1; 4 9 10 -2 | ||
| Line 24: | Line 24: | ||
'''Tetracot''', in this article, is the [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] in the 2.3.5.11.13 [[subgroup]] [[generator|generated]] by a submajor second of about 174–178{{cent}} which represents both [[10/9]] and [[11/10]]. It is so named because the generator is a quarter of fifth: four such generators make a perfect fifth which approximates [[3/2]], which cannot occur in [[12edo]], resulting in [[100/99]], [[144/143]], and [[243/242]] being [[tempering out|tempered out]]. This is in contrast to [[meantone]], where 10/9 is tuned sharper than or equal to just in order to be equated with [[9/8]]. | '''Tetracot''', in this article, is the [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] in the 2.3.5.11.13 [[subgroup]] [[generator|generated]] by a submajor second of about 174–178{{cent}} which represents both [[10/9]] and [[11/10]]. It is so named because the generator is a quarter of fifth: four such generators make a perfect fifth which approximates [[3/2]], which cannot occur in [[12edo]], resulting in [[100/99]], [[144/143]], and [[243/242]] being [[tempering out|tempered out]]. This is in contrast to [[meantone]], where 10/9 is tuned sharper than or equal to just in order to be equated with [[9/8]]. | ||
[[Equal temperament]]s that [[support]] tetracot include {{EDOs| 27, 34, and 41 }}. | |||
Tetracot has four strong [[extension]]s for the 7-, 11-, and 13-limit, which use the same methods of obtaining the [[11/1|11th]] and [[13/1|13th]] harmonics (10 generators up and 2 generators down, respectively) but differ in their methods of obtaining the [[7/1|7th harmonic]]: | |||
* [[Monkey]] (34 & 41) obtains the 7th harmonic at 15 generators down, tempering out [[875/864]] and thereby equating [[7/4]] with ([[6/5]])<sup>3</sup>; | |||
* [[Bunya]] (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out [[225/224]] and thereby equating [[7/2]] with ([[15/8]])<sup>2</sup>; | |||
* [[Modus]] (27e & 34d) obtains the 7th harmonic at 8 generators down, tempering out [[64/63]] and thereby equating 7/4 with [[16/9]]; | |||
* [[Wollemia]] (27e & 34) obtains the 7th harmonic at 19 generators up, tempering out [[126/125]] and thereby equating [[7/1]] with ([[5/3]])<sup>3</sup>([[3/2]]). | |||
== Interval chain == | See [[Tetracot family]] for technical data. | ||
== Intervals == | |||
=== Interval chain === | |||
In the following table, odd harmonics and subharmonics 1–15 are in '''bold'''. | |||
{| class="wikitable right-1 right-2" | {| class="wikitable right-1 right-2" | ||
| Line 101: | Line 108: | ||
| 15/13 | | 15/13 | ||
|} | |} | ||
<nowiki />* In 2.3.5.11.13 subgroup CTE tuning | <nowiki/>* In 2.3.5.11.13 subgroup CTE tuning | ||
=== As a detemperament of 7et === | |||
[[File: Tetracot 7et Detempering.png|thumb|Tetracot as a 34-tone 7et detempering]] | |||
Tetracot is considered as a [[cluster temperament]] with 7 clusters of notes in an octave, so it is naturally a [[detemperament]] of the [[7edo|7 equal temperament]]. The diagram on the right shows a 34-tone detempered scale, with a generator range of −16 to +17, which covers all the intervals in the no-7 13-odd-limit. Each category is divided into four or five qualities separated by 7 generator steps, which represent [[40/39]], [[45/44]], [[55/54]], [[65/64]], [[66/65]], [[81/80]], and [[121/120]] all at once. | |||
== Scales == | == Scales == | ||
| Line 109: | Line 121: | ||
== Tunings == | == Tunings == | ||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~10/9 = 176.0283{{c}} | |||
| CWE: ~10/9 = 176.0965{{c}} | |||
| POTE: ~10/9 = 176.1598{{c}} | |||
|} | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.11-subgroup norm-based tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~10/9 = 175.7765{{c}} | |||
| CWE: ~10/9 = 175.8847{{c}} | |||
| POTE: ~10/9 = 175.9849{{c}} | |||
|} | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.11.13-subgroup norm-based tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~10/9 = 175.8150{{c}} | |||
| CWE: ~10/9 = 176.0854{{c}} | |||
| POTE: ~10/9 = 176.1965{{c}} | |||
|} | |||
=== Tuning spectrum === | === Tuning spectrum === | ||
{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
| Line 122: | Line 180: | ||
| | | | ||
|- | |- | ||
| | | | ||
|243/200 | | 243/200 | ||
|168.574 | | 168.574 | ||
|1/2-comma | | 1/2-comma | ||
|- | |- | ||
| 1\7 | | 1\7 | ||
| | | | ||
| 171.429 | | 171.429 | ||
| Lower bound of 2.3.5.11 subgroup 11-odd-limit, <br />2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone | | Lower bound of 2.3.5.11 subgroup 11-odd-limit,<br />2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone | ||
|- | |- | ||
| | | | ||
|27/20 | | 27/20 | ||
|173.184 | | 173.184 | ||
|1/3-comma | | 1/3-comma | ||
|- | |- | ||
| | | | ||
| Line 142: | Line 200: | ||
| | | | ||
|- | |- | ||
| | | | ||
|81/80 | | 81/80 | ||
|174.501 | | 174.501 | ||
|2/7-comma | | 2/7-comma | ||
|- | |- | ||
| | | | ||
| Line 160: | Line 218: | ||
| 11/8 | | 11/8 | ||
| 175.132 | | 175.132 | ||
| 2.3.5.11 subgroup 11-odd-limit minimax | | 2.3.5.11-subgroup 11-odd-limit minimax | ||
|- | |- | ||
| | | | ||
| 3/2 | | 3/2 | ||
| 175.489 | | 175.489 | ||
|1/4-comma | | 1/4-comma | ||
|- | |- | ||
| 6\41 | | 6\41 | ||
| Line 175: | Line 233: | ||
| 13/11 | | 13/11 | ||
| 175.899 | | 175.899 | ||
| 2.3.5.11.13 subgroup 13- and 15-odd-limit minimax | | 2.3.5.11.13-subgroup 13- and 15-odd-limit minimax | ||
|- | |- | ||
| | | | ||
| Line 205: | Line 263: | ||
| 5/3 | | 5/3 | ||
| 176.872 | | 176.872 | ||
|1/5-comma | | 1/5-comma | ||
|- | |- | ||
| | | | ||
| Line 220: | Line 278: | ||
| | | | ||
| 177.778 | | 177.778 | ||
| | | 27e val, upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone | ||
|- | |- | ||
| | | | ||
|27/25 | | 27/25 | ||
|177.794 | | 177.794 | ||
|1/6-comma | | 1/6-comma | ||
|- | |- | ||
| | | | ||
|243/125 | | 243/125 | ||
|178.452 | | 178.452 | ||
|1/7-comma | | 1/7-comma | ||
|- | |- | ||
| | | | ||
| Line 245: | Line 303: | ||
| | | | ||
| 180.000 | | 180.000 | ||
| | | 20ce val, upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone | ||
|- | |- | ||
| | | | ||