Miracle/Chords: Difference between revisions

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Below are listed the [[dyadic chord]]s of 11-limit [[Miracle|miracle temperament]]. They are listed in order of increasing [[Graham complexity]], with [[hash complexity]] used to break ties. For any linear temperament, if we chose a generator, we may put any octave-equivalent chord of the temperament in a canonical form by representing it as a set of nonnegative integers, starting with 0, which are the number of generator steps giving an octave-equivalent note class in the chord. If S is such a set, the sum Σ 2<sup>s</sup> for all s ∊ S provides a unique odd number encoding the chord, and the logarithm base two of this number is the hash complexity, listed under the heading "Hash" below. The floor of hash complexity, that is, the greatest integer less than the hash complexity, is equal to the Graham complexity.

Below are listed the [[dyadic chord]]s of 11-limit [[Miracle|miracle temperament]]. They are listed in order of increasing [[Graham complexity]], with [[hash complexity]] used to break ties.

For any linear temperament, if we chose a generator, we may put any octave-equivalent chord of the temperament in a canonical form by representing it as a set of nonnegative integers, starting with 0, which are the number of generator steps giving an octave-equivalent note class in the chord. If S is such a set, the sum Σ 2<sup>s</sup> for all s ∊ S provides a unique odd number encoding the chord, and the logarithm base two of this number is the hash complexity, listed under the heading "Hash" below. The floor of hash complexity, that is, the greatest integer less than the hash complexity, is equal to the Graham complexity.

The heading "transversal" shows a JI chord whose intervals temper to the marvel chord. Under "type" it says otonal if it is an essentially just chord best understood in terms of the overtone series, utonal if it is the inversion of an otonal chord, and ambitonal if it can equally well be seen as otonal or utonal.

The other names denote types of essentially tempered chord, giving the least amount of tempering needed to define the chord, ~~so that if~~ for ~~example it says "~~werckismic~~" only~~ 441/440 ~~needs to be tempered out. Also~~, ~~"marvel" denotes 9 odd limit (7-limit) marvel~~, ~~whereas "unimarvel" means 11-limit marvel.~~

The other names denote types of essentially tempered chord, giving the least amount of tempering needed to define the chord. These include:

* [[Marvel triad|marvel]], for the tempering of [[225/224]]

* [[Keenanismic chords|keenanismic]], for the tempering of [[385/384]]

* [[Swetismic chords|swetismic]], for the tempering of [[540/539]]

* [[Rastmic chords|rastmic]], for the tempering of [[243/242]]

* [[Werckismic chords|werckismic]], for the tempering of [[441/440]]

* [[undecimal marvel chords|unimarvel]], for the tempering of any two of 225/224, 385/384 or 540/539

* [[Jove chords|jove]], for the tempering of both 225/224 and 243/242

* [[Prodigy chords|prodigy]], for the tempering of both 225/224 and 441/440

* [[Miracle chords|miracle]], for the tempering of any three independent commas mentioned above

The generator used below is the standard choice for miracle temperament, the secor, 16/15~15/14. However, it should be noted that the choice of generator does not affect the list of chords, only how those chords are interpreted. Does 0-7-13 denote the major triad or the minor triad? It's on the list of chords either way, but with a secor generator it's the major triad.

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-Below are listed the [[dyadic chord]]s of 11-limit [[Miracle|miracle temperament]]. They are listed in order of increasing [[Graham complexity]], with [[hash complexity]] used to break ties. For any linear temperament, if we chose a generator, we may put any octave-equivalent chord of the temperament in a canonical form by representing it as a set of nonnegative integers, starting with 0, which are the number of generator steps giving an octave-equivalent note class in the chord. If S is such a set, the sum Σ 2<sup>s</sup> for all s ∊ S provides a unique odd number encoding the chord, and the logarithm base two of this number is the hash complexity, listed under the heading "Hash" below. The floor of hash complexity, that is, the greatest integer less than the hash complexity, is equal to the Graham complexity.
+Below are listed the [[dyadic chord]]s of 11-limit [[Miracle|miracle temperament]]. They are listed in order of increasing [[Graham complexity]], with [[hash complexity]] used to break ties.
+For any linear temperament, if we chose a generator, we may put any octave-equivalent chord of the temperament in a canonical form by representing it as a set of nonnegative integers, starting with 0, which are the number of generator steps giving an octave-equivalent note class in the chord. If S is such a set, the sum Σ 2<sup>s</sup> for all s ∊ S provides a unique odd number encoding the chord, and the logarithm base two of this number is the hash complexity, listed under the heading "Hash" below. The floor of hash complexity, that is, the greatest integer less than the hash complexity, is equal to the Graham complexity.
 The heading "transversal" shows a JI chord whose intervals temper to the marvel chord. Under "type" it says otonal if it is an essentially just chord best understood in terms of the overtone series, utonal if it is the inversion of an otonal chord, and ambitonal if it can equally well be seen as otonal or utonal.
-The other names denote types of essentially tempered chord, giving the least amount of tempering needed to define the chord, so that if for example it says "werckismic" only 441/440 needs to be tempered out. Also, "marvel" denotes 9 odd limit (7-limit) marvel, whereas "unimarvel" means 11-limit marvel.
+The other names denote types of essentially tempered chord, giving the least amount of tempering needed to define the chord. These include:
+* [[Marvel triad|marvel]], for the tempering of [[225/224]]
+* [[Keenanismic chords|keenanismic]], for the tempering of [[385/384]]
+* [[Swetismic chords|swetismic]], for the tempering of [[540/539]]
+* [[Rastmic chords|rastmic]], for the tempering of [[243/242]]
+* [[Werckismic chords|werckismic]], for the tempering of [[441/440]]
+* [[undecimal marvel chords|unimarvel]], for the tempering of any two of 225/224, 385/384 or 540/539
+* [[Jove chords|jove]], for the tempering of both 225/224 and 243/242
+* [[Prodigy chords|prodigy]], for the tempering of both 225/224 and 441/440
+* [[Miracle chords|miracle]], for the tempering of any three independent commas mentioned above
 The generator used below is the standard choice for miracle temperament, the secor, 16/15~15/14. However, it should be noted that the choice of generator does not affect the list of chords, only how those chords are interpreted. Does 0-7-13 denote the major triad or the minor triad? It's on the list of chords either way, but with a secor generator it's the major triad.