Tetracot: Difference between revisions

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| Title = Tetracot

| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.13

| Comma basis = [[20000/19683]] (2.3.5); [[100/99]], [[243/242]] (2.3.5.11) [[100/99]], [[144/143]], [[243/242]] (2.3.5.11.13)

| Comma basis = [[20000/19683]] (2.3.5); [[100/99]], [[243/242]] (2.3.5.11) [[100/99]], [[144/143]], [[243/242]] (2.3.5.11.13)

| Edo join 1 = 7 | Edo join 2 = 27e

| Mapping = 1; 4 9 10 -2

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'''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 has many~~ [[~~extension~~]]s ~~for the 7-, 11-, and 13-limit. See~~ [[~~Tetracot extensions~~]]~~. Equal temperaments that support~~ tetracot include {{EDOs| 27, 34, and 41 }}.

[[Equal temperament]]s that [[support]] tetracot include {{EDOs| 27, 34, and 41 }}.

See [[Tetracot family]] for ~~more~~ technical data.

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]])3;

* [[Bunya]] (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out [[225/224]] and thereby equating [[7/2]] with ([[15/8]])2;

* [[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]])3([[3/2]]).

See [[Tetracot family]] for technical data.

== Intervals ==

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[[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.

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 ==

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== 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 ===

{| class="wikitable center-all left-4"

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|

| 171.429

| Lower bound of 2.3.5.11 subgroup 11-odd-limit, 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone

| Lower bound of 2.3.5.11 subgroup 11-odd-limit, 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone

|-

|

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|

| 177.778

| ~~Upper~~ bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone

| 27e val, upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone

|-

|

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|

| 180.000

| ~~Upper~~ bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone

| 20ce val, upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone

|-

|

@@ Line 8: / Line 8: @@
 | Title = Tetracot
 | 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
 | 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 has many [[extension]]s for the 7-, 11-, and 13-limit. See [[Tetracot extensions]]. Equal temperaments that support tetracot include {{EDOs| 27, 34, and 41 }}.
+[[Equal temperament]]s that [[support]] tetracot include {{EDOs| 27, 34, and 41 }}.
-See [[Tetracot family]] for more technical data.
+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]]).
+See [[Tetracot family]] for technical data.
 == Intervals ==
@@ Line 107: / Line 113: @@
 [[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.
+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 ==
@@ Line 115: / Line 121: @@
 == 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 ===
 {| class="wikitable center-all left-4"
@@ Line 136: / Line 188: @@
 |
 | 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
 |-
 |
@@ Line 226: / Line 278: @@
 |
 | 177.778
-| Upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone
+| 27e val, upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone
 |-
 |
@@ Line 251: / Line 303: @@
 |
 | 180.000
-| Upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone
+| 20ce val, upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone
 |-
 |