Tutonic hexad: Difference between revisions

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The '''tutonic hexad''' is an [[essentially tempered chord]] of the [[tutone]] temperament, which is every other member of the [[generator]] chain of [[11-limit]] [[meantone]] ([[Huygens]]). This means it is 9/8-9/8-9/8-9/8-9/8-8/7 tempered by {[[81/80]], [[99/98]], [[126/125]]}, or in other words the tempering of 9/8-5/4-7/5-11/7-7/4-2.
A '''tutonic hexad''' is an [[essentially tempered chord]] of the [[tutone]] temperament, which is every other member of the [[generator]] chain of [[11-limit]] [[meantone]] (also called as [[Huygens]], tempered by [[81/80]], [[99/98]] and [[126/125]]).


[[Category:11-odd-limit]]
There are eight hexads as [[11-odd-limit]] essentially tempered chords of the 11-limit meantone, all of them are available in the tutone temperament; in other words, all of the essentially tempered hexads of the 11-limit meantone are tutonic hexads.
[[Category:Hexad]]
 
The palindromic hexads are
* 1–9/8–5/4–7/5–14/9–7/4 with steps of 9/8, 9/8, 9/8, 9/8, 9/8, 8/7 ([[Didymic chords|erato hexad]], tempered by 81/80 and 126/125);
* 1–9/8–9/7–10/7–11/7–7/4 with steps of 9/8, 8/7, 9/8, 11/10, 9/8, 8/7 (euterpe hexad, tempered by 81/80 and 99/98);
* 1–9/8–9/7–7/5–8/5–7/4 with steps of 9/8, 8/7, 11/10, 8/7, 11/10, 8/7 (minerva hexad, tempered by 99/98 and 176/175);
* 1–9/8–5/4–7/5–8/5–7/4 with steps of 9/8, 9/8, 9/8, 8/7, 11/10, 8/7 (minerva hexad, tempered by 99/98 and 176/175).
 
The inversely related pairs of hexads are
* 1–9/8–5/4–10/7–11/7–7/4 with steps of 9/8, 9/8, 8/7, 11/10, 9/8, 8/7, and its inverse
* 1–9/8–11/9–7/5–11/7–7/4 with steps of 9/8, 11/10, 8/7, 9/8, 9/8, 8/7;
* 1–9/8–9/7–7/5–11/7–7/4 with steps of 9/8, 8/7, 11/10, 9/8, 9/8, 8/7, and its inverse
* 1–9/8–5/4–11/8–11/7–7/4 with steps of 9/8, 9/8, 11/10, 8/7, 9/8, 8/7.
 
[[Category:9-odd-limit chords]]
[[Category:11-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Essentially tempered chords]]
[[Category:Hexads]]
[[Category:Tutone]]
[[Category:Tutone]]

Latest revision as of 13:47, 1 January 2025

A tutonic hexad is an essentially tempered chord of the tutone temperament, which is every other member of the generator chain of 11-limit meantone (also called as Huygens, tempered by 81/80, 99/98 and 126/125).

There are eight hexads as 11-odd-limit essentially tempered chords of the 11-limit meantone, all of them are available in the tutone temperament; in other words, all of the essentially tempered hexads of the 11-limit meantone are tutonic hexads.

The palindromic hexads are

  • 1–9/8–5/4–7/5–14/9–7/4 with steps of 9/8, 9/8, 9/8, 9/8, 9/8, 8/7 (erato hexad, tempered by 81/80 and 126/125);
  • 1–9/8–9/7–10/7–11/7–7/4 with steps of 9/8, 8/7, 9/8, 11/10, 9/8, 8/7 (euterpe hexad, tempered by 81/80 and 99/98);
  • 1–9/8–9/7–7/5–8/5–7/4 with steps of 9/8, 8/7, 11/10, 8/7, 11/10, 8/7 (minerva hexad, tempered by 99/98 and 176/175);
  • 1–9/8–5/4–7/5–8/5–7/4 with steps of 9/8, 9/8, 9/8, 8/7, 11/10, 8/7 (minerva hexad, tempered by 99/98 and 176/175).

The inversely related pairs of hexads are

  • 1–9/8–5/4–10/7–11/7–7/4 with steps of 9/8, 9/8, 8/7, 11/10, 9/8, 8/7, and its inverse
  • 1–9/8–11/9–7/5–11/7–7/4 with steps of 9/8, 11/10, 8/7, 9/8, 9/8, 8/7;
  • 1–9/8–9/7–7/5–11/7–7/4 with steps of 9/8, 8/7, 11/10, 9/8, 9/8, 8/7, and its inverse
  • 1–9/8–5/4–11/8–11/7–7/4 with steps of 9/8, 9/8, 11/10, 8/7, 9/8, 8/7.
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