Valinorsmic chords: Difference between revisions
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A '''valinorsmic chord''' is a [[valinorsma]] (176/175) tempered [[essentially tempered chord]] in the 2.5.7.11 subgroup | A '''valinorsmic chord''' is a [[valinorsma]] (176/175) tempered [[essentially tempered chord]]. | ||
Valinorsmic chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 1a]] in the 2.5.7.11 [[subgroup]] [[11-odd-limit]], meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures. | |||
There are three valinorsmic triads: the valinorsmic version of the augmented triad, | |||
* 1-5/4-8/5 with steps 5/4-14/11-5/4, | * 1-5/4-8/5 with steps 5/4-14/11-5/4, | ||
and an inversely related pair, | and an inversely related pair, | ||
* 1-8/5-7/4 with steps 8/5-11/10-8/7 and 1-8/7-5/4 with steps 8/7-11/10-8/5. | * 1-8/5-7/4 with steps 8/5-11/10-8/7, and its inverse | ||
* 1-8/7-5/4 with steps 8/7-11/10-8/5. | |||
Valinorsmic tetrads are six in number, consisting of two palindromic chords and a pair of pairs of inversely related chords. The palindromes are | Valinorsmic tetrads are six in number, consisting of two palindromic chords and a pair of pairs of inversely related chords. The palindromes are | ||
* 1-11/10-5/4-7/4 with steps 11/10-8/7-7/5-8/7 and | * 1-11/10-5/4-7/4 with steps 11/10-8/7-7/5-8/7, and its inverse | ||
* 1-11/10-5/4-11/8 with steps 11/10-8/7-11/10-16/11. | * 1-11/10-5/4-11/8 with steps 11/10-8/7-11/10-16/11. | ||
The rest are | The rest are | ||
* 1-5/4-8/5-7/4 with steps 5/4-14/11-11/10-8/7 and 1-5/4-10/7-11/7 with steps 5/4-8/7-11/10-14/11 | * 1-5/4-8/5-7/4 with steps 5/4-14/11-11/10-8/7, and its inverse | ||
* 1-5/4-11/8-11/7 with steps 5/4-11/10-8/7-14/11 and 1-11/10-11/8-7/4 with steps 11/10-5/4-14/11-8/7. | * 1-5/4-10/7-11/7 with steps 5/4-8/7-11/10-14/11; | ||
* 1-5/4-11/8-11/7 with steps 5/4-11/10-8/7-14/11, and its inverse | |||
1-11/10-11/8-7/4 with steps 11/10-5/4-14/11-8/7. | |||
Finally, there are the two inversely related pentads, | Finally, there are the two inversely related pentads, | ||
* 1-11/10-5/4-8/5-7/4 with steps 11/10-8/7-14/11-11/10-8/7 and 1-11/10-5/4-11/8-7/4 with steps 11/10-8/7-11/10-14/11-8/7 | * 1-11/10-5/4-8/5-7/4 with steps 11/10-8/7-14/11-11/10-8/7, and its inverse | ||
* 1-11/10-5/4-11/8-7/4 with steps 11/10-8/7-11/10-14/11-8/7. | |||
If we are willing to go to the [[15-odd-limit]], we get four alternative, valinorsmic, temperings of the four [[keenanismic chords|keenanismic tetrads]] which have steps which are permutations of the otonal/utonal 7-limit tetrads. The reason for the similarity between valinorsmic and keenanismic tetrads is that they both temper 48/35, valinorsmic to 15/11 and keenanismic to 11/8. Another valinorsmic tetrad has steps 11/9-6/5-8/7-6/5, leading to 1-11/9-22/15-5/3 chords. | If we are willing to go to the [[15-odd-limit]], we get four alternative, valinorsmic, temperings of the four [[keenanismic chords|keenanismic tetrads]] which have steps which are permutations of the otonal/utonal 7-limit tetrads. The reason for the similarity between valinorsmic and keenanismic tetrads is that they both temper 48/35, valinorsmic to 15/11 and keenanismic to 11/8. Another valinorsmic tetrad has steps 11/9-6/5-8/7-6/5, leading to 1-11/9-22/15-5/3 chords. | ||
Equal temperaments with valinorsmic chords include {{EDOs| 22, 31, 43, 46, 53, 58, 68, 80, 89, 111 }} with 111edo giving the optimal patent val. | Equal temperaments with valinorsmic chords include {{EDOs| 22, 31, 43, 46, 53, 58, 68, 80, 89, 111 }} with 111edo giving the [[optimal patent val]]. | ||
[[Category:11-odd-limit]] | [[Category:11-odd-limit]] | ||
Revision as of 12:44, 8 August 2023
A valinorsmic chord is a valinorsma (176/175) tempered essentially tempered chord.
Valinorsmic chords are of pattern 1a in the 2.5.7.11 subgroup 11-odd-limit, meaning that there are 3 triads, 6 tetrads and 2 pentads, for a total of 11 distinct chord structures.
There are three valinorsmic triads: the valinorsmic version of the augmented triad,
- 1-5/4-8/5 with steps 5/4-14/11-5/4,
and an inversely related pair,
- 1-8/5-7/4 with steps 8/5-11/10-8/7, and its inverse
- 1-8/7-5/4 with steps 8/7-11/10-8/5.
Valinorsmic tetrads are six in number, consisting of two palindromic chords and a pair of pairs of inversely related chords. The palindromes are
- 1-11/10-5/4-7/4 with steps 11/10-8/7-7/5-8/7, and its inverse
- 1-11/10-5/4-11/8 with steps 11/10-8/7-11/10-16/11.
The rest are
- 1-5/4-8/5-7/4 with steps 5/4-14/11-11/10-8/7, and its inverse
- 1-5/4-10/7-11/7 with steps 5/4-8/7-11/10-14/11;
- 1-5/4-11/8-11/7 with steps 5/4-11/10-8/7-14/11, and its inverse
1-11/10-11/8-7/4 with steps 11/10-5/4-14/11-8/7.
Finally, there are the two inversely related pentads,
- 1-11/10-5/4-8/5-7/4 with steps 11/10-8/7-14/11-11/10-8/7, and its inverse
- 1-11/10-5/4-11/8-7/4 with steps 11/10-8/7-11/10-14/11-8/7.
If we are willing to go to the 15-odd-limit, we get four alternative, valinorsmic, temperings of the four keenanismic tetrads which have steps which are permutations of the otonal/utonal 7-limit tetrads. The reason for the similarity between valinorsmic and keenanismic tetrads is that they both temper 48/35, valinorsmic to 15/11 and keenanismic to 11/8. Another valinorsmic tetrad has steps 11/9-6/5-8/7-6/5, leading to 1-11/9-22/15-5/3 chords.
Equal temperaments with valinorsmic chords include 22, 31, 43, 46, 53, 58, 68, 80, 89, 111 with 111edo giving the optimal patent val.