Porcupine/Chords: Difference between revisions
m Clarify that the scope of this article is currently the 11-odd-limit. Correct a ratio. Style |
Correction: 15-odd-limit. Fill in the tables |
||
| Line 1: | Line 1: | ||
Below are listed the [[ | Below are listed the [[15-odd-limit]] [[dyadic chord]]s of [[11-limit]] [[porcupine|porcupine temperament]] that do not have generator steps 7 or 13 as [[dyad]]s. Typing the chords requires consideration of the fact that porcupine conflates [[10/9]], [[11/10]] and [[12/11]] and also [[16/9]] and [[7/4]]. If a [[transversal]] can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. If a chord is essentially tempered, the chord is analyzed in terms of the transversals 11/10, 6/5, 16/11 and 7/4. Chords that require only [[64/63]] tempering are marked [[archytas chords|archytas]], by [[100/99]] [[ptolemismic chords|ptolemismic]], by [[121/120]] [[biyatismic chords|biyatismic]], by [[176/175]] [[valinorsmic chords|valinorsmic]], and by [[385/384]] [[keenanismic chords|keenanismic]]. Chords that require 64/63 and 176/175 tempering are marked ares, 100/99 and 385/384 tempered chords are [[supermagic chords|supermagic]], and 176/175 and 385/384 tempered chords are marked zeus. Chords that receive tempering by three independent commas above are labeled porcupine. | ||
The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in | The transversal is in generator order. This is useful because it tells how common the chords are: For instance, a chord that appears on the sixth generation will appear exactly once in Porcupine[7], twice in porcupine[8], and nine times in Porcupine[15]. | ||
The "As generated" column takes the intervals that were generated and places them in size order. The 1st and 2nd inversion (and so on) columns show the inversions of those generated tones. Note that this gives different results than you might be used to: the major chord (1/1–5/4–3/2, or 4:5:6) is the second inversion of the generated 0–2–5 chord. | The "As generated" column takes the intervals that were generated and places them in size order. The 1st and 2nd inversion (and so on) columns show the inversions of those generated tones. Note that this gives different results than you might be used to: the major chord (1/1–5/4–3/2, or 4:5:6) is the second inversion of the generated 0–2–5 chord. | ||
| Line 671: | Line 671: | ||
| 0-2-4-5 | | 0-2-4-5 | ||
| 1-6/5-16/11-8/5 | | 1-6/5-16/11-8/5 | ||
| | | ptolemismic | ||
| 1/1-6/5-16/11-8/5 | | 1/1-6/5-16/11-8/5 | ||
| | | 1/1-6/5-4/3-5/3 | ||
| | | 1/1-11/10-11/8-5/3 | ||
| '''1/1-5/4-3/2-9/5''' | | '''1/1-5/4-3/2-9/5''' | ||
| Cv^7 | | Cv^7 | ||
| Line 716: | Line 716: | ||
== Pentads == | == Pentads == | ||
{{ | {{Todo|inline=1| research |comment=investigate what's missing there }} | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 724: | Line 724: | ||
! Type | ! Type | ||
! Name | ! Name | ||
|- | |- | ||
| 0-1-2-3-6 | | 0-1-2-3-6 | ||
| Line 736: | Line 731: | ||
|- | |- | ||
| 0-2-3-4-6 | | 0-2-3-4-6 | ||
| | | 1-6/5-4/3-16/11-16/9 | ||
| | | supermagic | ||
| C^m,7,11(v5) ''or''<br>C4^7v9 ''or'' C^4v6,9 | | C^m,7,11(v5) ''or''<br>C4^7v9 ''or'' C^4v6,9 | ||
|- | |- | ||
| 0-3-4-5-6 | | 0-3-4-5-6 | ||
| | | 1-4/3-16/11-8/5-16/9 | ||
| | | utonal | ||
| C^mv9,11 | | C^mv9,11 | ||
|- | |- | ||