Magic chords: Difference between revisions
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A '''magic chord''' is an [[essentially tempered chord]] of the [[magic]] temperament; they are included among those listed as [[chords of magic]]. Aside from the marvel, sensamagic, ptolemismic, keenanismic, swetismic, pentacircle, apollo, octarod, | A '''magic chord''' is an [[essentially tempered chord]] of the [[magic]] temperament; they are included among those listed as [[chords of magic]]. Aside from the [[marvel chords|marvel]], [[sensamagic chords|sensamagic]], [[ptolemismic chords|ptolemismic]], [[keenanismic chords|keenanismic]], [[swetismic chords|swetismic]], [[pentacircle chords|pentacircle]], [[apollo chords|apollo]], [[octarod chords|octarod]], [[undecimal marvel chords|undecimal marvel]], [[undecimal sensamagic chords|undecimal sensamagic]] and [[keemic chords]] listed there, there are 23 further essentially magic chords in the [[11-odd-limit]]. | ||
The two magic tetrads are in inverse relationship: | The two magic tetrads are in inverse relationship: | ||
* | * 1–5/4–14/9–9/5 with steps 5/4, 5/4, 7/6, 10/9, and its inverse | ||
* 1–7/6–16/11–9/5 with steps 7/6, 5/4, 5/4, 10/9; | |||
For pentads, there are seven pairs of chords in inverse relationship: | For pentads, there are seven pairs of chords in inverse relationship: | ||
* | * 1–9/8–5/4–14/9–9/5 with steps 9/8, 10/9, 5/4, 7/6, 10/9, and its inverse | ||
* | * 1–9/8–5/4–16/11–9/5 with steps 9/8, 10/9, 7/6, 5/4, 10/9; | ||
* | * 1–5/4–7/5–14/9–9/5 with steps 5/4, 9/8, 10/9, 7/6, 10/9, and its inverse | ||
* | * 1–9/8–7/5–14/9–9/5 with steps 9/8, 5/4, 10/9, 7/6, 10/9; | ||
* | * 1–12/11–5/4–14/9–9/5 with steps 12/11, 8/7, 5/4, 7/6, 10/9, and its inverse | ||
* | * 1–12/11–6/5–7/5–7/4 with steps 12/11, 10/9, 7/6, 5/4, 8/7; | ||
* | * 1–12/11–6/5–14/9–7/4 with steps 12/11, 10/9, 9/7, 9/8, 8/7, and its inverse | ||
* 1–12/11–5/4–7/5–9/5 with steps 12/11, 8/7, 9/8, 9/7, 10/9; | |||
* 1–7/6–16/11–18/11–9/5 with steps 7/6, 5/4, 9/8, 10/9, 10/9, and its inverse | |||
* 1–9/8–7/5–18/11–9/5 with steps 9/8, 5/4, 7/6, 10/9, 10/9; | |||
* 1–7/6–14/11–16/11–9/5 with steps 7/6, 12/11, 8/7, 5/4, 10/9, and its inverse | |||
* 1–12/11–14/11–7/5–7/4 with steps 12/11, 7/6, 10/9, 5/4, 8/7; | |||
* 1–9/8–14/11–14/9–9/5 with steps 9/8, 9/8, 11/9, 7/6, 10/9, and its inverse | |||
* 1–9/8–14/11–7/5–18/11 with steps 9/8, 9/8, 10/9, 7/6, 11/9. | |||
For hexads, there are one palindromic chord and three pairs of chords in inverse relationship. The palindromic chord is | For hexads, there are one palindromic chord and three pairs of chords in inverse relationship. The palindromic chord is | ||
* | * 1–9/8–5/4–7/5–14/9–9/5 with steps 9/8, 10/9, 9/8, 10/9, 7/6, 10/9. | ||
The inversely related pairs of chords are | The inversely related pairs of chords are | ||
* | * 1–12/11–5/4–7/5–14/9–9/5 with steps 12/11, 8/7, 9/8, 10/9, 7/6, 10/9, and its inverse | ||
* | * 1–12/11–6/5–7/5–14/9–7/4 with steps 12/11, 10/9, 7/6, 10/9, 9/8, 8/7; | ||
* | * 1–9/8–14/11–7/5–14/9–9/5 with steps 9/8, 9/8, 10/9, 10/9, 7/6, 10/9, and its inverse | ||
* 1–9/8–14/11–7/5–18/11–9/5 with steps 9/8, 9/8, 10/9, 7/6, 10/9, 10/9; | |||
* 1–12/11–14/11–7/5–14/9–7/4 with steps 12/11, 7/6, 10/9, 10/9, 9/8, 8/7, and its inverse | |||
* 1–7/6–14/11–16/11–18/11–9/5 with steps 7/6, 12/11, 8/7, 9/8, 10/9, 10/9. | |||
The count of magic chords is therefore tetrads: 2, pentads: 14, and hexads: 7. | The count of magic chords is therefore tetrads: 2, pentads: 14, and hexads: 7. | ||
Equal | [[Equal temperament]]s with magic chords include {{Optimal ET sequence| 19, 22, 41, 60, 63, 85 and 104 }}. | ||
[[Category:11-odd-limit]] | [[Category:11-odd-limit chords]] | ||
[[Category:Essentially tempered chords]] | [[Category:Essentially tempered chords]] | ||
[[Category:Tetrads]] | [[Category:Tetrads]] | ||