Miracle/Chords: Difference between revisions

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^s 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.  

* [[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

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

* [[Jove chords|jove]], for the tempering of any two of 243/242, 441/440 or 540/539

* [[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.

The chords are ordered by generator steps. This chord ordering is useful because you can tell how often it will occur in a scale of miracle. For instance, Chord 0-2-5, which is 1/1-8/7-7/5, occurs for the first time on the fifth generation, and will therefore occur five times in miracle[10]. Chord 0-3-9, which is 1/1-11/9-11/6, occurs for the first time on the ninth generation, and therefore appears only once in miracle[10].  

Each of the triads generated has three "inversions", i.e., can be written starting on any one of the three tones in the triad.

This ordering does create results that may seem surprising: For example, chord #21, 1/1-6/5-8/5, is the first inversion of 1/1-5/4-3/2; in this list, 1/1-5/4-3/2 is the second inversion of chord #21. Unlike in traditional music theory, there is nothing canonical about these orderings and inversions.

All three inversions for all 63 chords are notated in this [[Media:41-EDO-Miracle-scale-and-triads.pdf|PDF]] and can be heard using a [[Media:41 EDO Miracle - scale and triads - piano.mp3|piano]], a [[Media:41 EDO Miracle - scale and triads - steel gtr.mp3|steel-stringed guitar]], a [[Media:41 EDO Miracle - scale and triads - nylon guitar.mp3|nylon-stringed guitar]], and a [[Media:41 EDO Miracle - scale and triads - strings.mp3|string ensemble]]. (All linked files are in MP3 format. They were created in [http://www.mus2.com.tr/en/ Mus2] using 41 EDO sagittal notation ([[Media:41 EDO Miracle - scale and triads.mus2|get Mus2 file]]), exported to tuned MIDI, and rendered with Timidity++ using soundfonts.) Finally, here is a [[Media:41 EDO Miracle - scale and triads.mid|tuned MIDI file]]. The sequence is: chord as generated, first inversion, second inversion, quarter-note rest.

{| class="wikitable"

! Number

 

{| class="wikitable"

! Number

 

{| class="wikitable"

! Number

 

{| class="wikitable"

! Number