Miracle/Chords: Difference between revisions
+links to each chord page |
m adjusted "overtone series" to "harmonic series" but kept references to "overtones" to explain the use of "otonal" |
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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. | 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 | 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 harmonic series (the overtones) and utonal if it is the inversion of an otonal chord (in other words, fitting the subharmonic series), 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. These include: | The other names denote types of essentially tempered chord, giving the least amount of tempering needed to define the chord. These include: | ||
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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. | 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 [[ | 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. | ||
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