User:Contribution/17-EDO resources: Difference between revisions
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!Cents | !Cents | ||
!Approximate Ratios | !Approximate Ratios | ||
!Weight | |||
!Approximate Ratios | !Approximate Ratios | ||
!Cents | !Cents | ||
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=== What these columns mean === | |||
* Degree: Number of steps within the octave period. | |||
* Name: Abbreviated spoken name of the interval, used in staff notation: | |||
** Interval quality: | |||
*** ''dim'' = diminished | |||
*** ''min'' = minor | |||
*** ''neu'' = neutral | |||
*** ''Maj'' = major | |||
*** ''Aug'' = augmented | |||
*** Unspecified quality (a bare ''4'' or ''5'') means a perfect interval. | |||
** Interval heptatonic degree: | |||
*** ''2, 3, 4, 5, 6, 7'' | |||
* Cents: Size of the interval. | |||
* Approximate Ratios: Small ratios close to the interval, given as possible just-intonation interpretations; other choices are possible. | |||
* Weight: Arbitrary weight values assigned to each interval, used to rank chord types in the '''Some chords''' section below. This is only one possible way to order or classify the intervals. | |||
* Layout: The left half of the table lists ascending intervals from the root; the right half lists their complements within the 17-step octave (degree 17 − ''n''), with the same weights. | |||
= Some chords = | = Some chords = | ||
=== What these chord tables show === | |||
For each set cardinality, the tables list the first members of the chord classes ranked by the interval weight values defined in the previous section. | |||
The ranking is obtained with a simple generative model on the 17-note circle: | |||
* Starting from degree 0, the algorithm successively extends the set by applying one of the 17 possible intervals above every degree that has already been generated. | |||
* Each interval carries a weight: simpler intervals receive larger weights than more remote ones. | |||
* All pitch-class sets obtained this way, modulo the 17-step equave, are grouped into chord classes (up to inversion and transposition). | |||
* Each chord class receives a score, obtained by adding the weights of the intervals along all the chains of intervals that generate it. | |||
* The classes are then sorted from highest to lowest score; only the best-scoring ones for each cardinality are shown in the tables. | |||
Each small inner table groups the inversions (rotations) of a single chord class. In every row, the left cell names the chord by listing its intervals above the root note of that inversion, and the right cell gives the same information as a set of degrees. | |||
The first row of each inner table is a distinguished representative: among all inversions of the chord, it is the one whose pitch classes, when ordered along the 17-EDO circle of fifths (0–10–3–13–6–16–9–2–12–5–15–8–1–11–4–14–7), form the lexicographically smallest sequence. The remaining rows list the other inversions of the same chord, in the order obtained by successive rotations. | |||
== Dyads == | == Dyads == | ||
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{| class="wikitable" | {| class="wikitable" | ||
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{| class="wikitable" | {| class="wikitable" | ||
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|0, 3, 7, 13 | |0, 3, 7, 13 | ||
|} | |} | ||
|- | |- | ||
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|0, 5, 8, 15 | |0, 5, 8, 15 | ||
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{| class="wikitable" | {| class="wikitable" | ||
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|0, 2, 7, 12 | |0, 2, 7, 12 | ||
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{| class="wikitable" | {| class="wikitable" | ||
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|0, 1, 7, 11 | |0, 1, 7, 11 | ||
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|0, 7, 10, 16 | |0, 7, 10, 16 | ||
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|0, 7, 9, 10 | |0, 7, 9, 10 | ||
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{| class="wikitable" | {| class="wikitable" | ||
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|0, 6, 7, 16 | |0, 6, 7, 16 | ||
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|0, 3, 9, 13 | |0, 3, 9, 13 | ||
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|0, 1, 11, 14 | |0, 1, 11, 14 | ||
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|0, 3, 13, 14 | |0, 3, 13, 14 | ||
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{| class="wikitable" | {| class="wikitable" | ||
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|0, 2, 12, 15 | |0, 2, 12, 15 | ||
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{| class="wikitable" | {| class="wikitable" | ||
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|0, 7, 11, 15 | |0, 7, 11, 15 | ||
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{| class="wikitable" | {| class="wikitable" | ||
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|0, 5, 11, 15 | |0, 5, 11, 15 | ||
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|- | |||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 993: | Line 1,037: | ||
|0, 7, 12, 15 | |0, 7, 12, 15 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 1,009: | Line 1,052: | ||
|0, 7, 9, 12 | |0, 7, 9, 12 | ||
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|- | |||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 1,130: | Line 1,174: | ||
|0, 5, 13, 15 | |0, 5, 13, 15 | ||
|} | |} | ||
|- | |||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 1,145: | Line 1,190: | ||
|0, 7, 13, 16 | |0, 7, 13, 16 | ||
|} | |} | ||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 1,191: | Line 1,235: | ||
|0, 4, 14, 15 | |0, 4, 14, 15 | ||
|} | |} | ||
|- | |||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 1,221: | Line 1,266: | ||
|0, 3, 12, 13 | |0, 3, 12, 13 | ||
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| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 1,252: | Line 1,296: | ||
|0, 2, 12, 16 | |0, 2, 12, 16 | ||
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|- | |||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 1,297: | Line 1,342: | ||
|0, 7, 9, 15 | |0, 7, 9, 15 | ||
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|} | |} | ||
| Line 1,744: | Line 1,790: | ||
== Hexads == | == Hexads == | ||
{| class="wikitable" | {| class="wikitable" | ||
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| Line 2,108: | Line 2,153: | ||
|0, 2, 5, 7, 12, 14 | |0, 2, 5, 7, 12, 14 | ||
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|} | |} | ||
| Line 2,161: | Line 2,207: | ||
|0, 1, 3, 4, 7, 11, 14 | |0, 1, 3, 4, 7, 11, 14 | ||
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|- | |||
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{| class="wikitable" | {| class="wikitable" | ||
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|0, 1, 4, 7, 11, 14, 15 | |0, 1, 4, 7, 11, 14, 15 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 2,210: | Line 2,256: | ||
|0, 4, 6, 7, 10, 13, 14 | |0, 4, 6, 7, 10, 13, 14 | ||
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|- | |||
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|0, 1, 4, 7, 11, 13, 14 | |0, 1, 4, 7, 11, 13, 14 | ||
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|0, 1, 4, 5, 7, 11, 14 | |0, 1, 4, 5, 7, 11, 14 | ||
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|0, 2, 5, 7, 10, 12, 15 | |0, 2, 5, 7, 10, 12, 15 | ||
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|0, 4, 5, 7, 10, 12, 14 | |0, 4, 5, 7, 10, 12, 14 | ||
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|0, 1, 4, 7, 11, 14, 16 | |0, 1, 4, 7, 11, 14, 16 | ||
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|0, 1, 2, 4, 7, 11, 14 | |0, 1, 2, 4, 7, 11, 14 | ||
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{| class="wikitable" | {| class="wikitable" | ||
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|0, 3, 6, 7, 13, 14, 16 | |0, 3, 6, 7, 13, 14, 16 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 2,623: | Line 2,673: | ||
|0, 2, 3, 5, 10, 12, 15 | |0, 2, 3, 5, 10, 12, 15 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 2,648: | Line 2,697: | ||
|0, 2, 5, 7, 12, 14, 15 | |0, 2, 5, 7, 12, 14, 15 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 2,745: | Line 2,795: | ||
|0, 4, 5, 7, 12, 14, 15 | |0, 4, 5, 7, 12, 14, 15 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 2,769: | Line 2,820: | ||
|0, 4, 5, 7, 8, 14, 15 | |0, 4, 5, 7, 8, 14, 15 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 2,794: | Line 2,844: | ||
|0, 4, 6, 7, 13, 14, 16 | |0, 4, 6, 7, 13, 14, 16 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 2,818: | Line 2,869: | ||
|0, 3, 6, 9, 12, 13, 16 | |0, 3, 6, 9, 12, 13, 16 | ||
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|} | |} | ||
== Octads == | == Octads == | ||
{| class="wikitable" | {| class="wikitable" | ||
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| Line 2,877: | Line 2,928: | ||
|0, 1, 4, 7, 10, 11, 14, 15 | |0, 1, 4, 7, 10, 11, 14, 15 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 2,904: | Line 2,956: | ||
|0, 1, 4, 7, 10, 11, 13, 14 | |0, 1, 4, 7, 10, 11, 13, 14 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 2,932: | Line 2,983: | ||
|0, 1, 3, 4, 7, 11, 13, 14 | |0, 1, 3, 4, 7, 11, 13, 14 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 3,041: | Line 3,093: | ||
|0, 4, 6, 7, 10, 13, 14, 16 | |0, 4, 6, 7, 10, 13, 14, 16 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 3,068: | Line 3,121: | ||
|0, 3, 6, 7, 9, 13, 14, 16 | |0, 3, 6, 7, 9, 13, 14, 16 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 3,096: | Line 3,148: | ||
|0, 2, 3, 5, 8, 10, 12, 15 | |0, 2, 3, 5, 8, 10, 12, 15 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 3,205: | Line 3,258: | ||
|0, 1, 4, 7, 10, 11, 12, 14 | |0, 1, 4, 7, 10, 11, 12, 14 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 3,232: | Line 3,286: | ||
|0, 1, 4, 6, 7, 11, 13, 14 | |0, 1, 4, 6, 7, 11, 13, 14 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 3,260: | Line 3,313: | ||
|0, 1, 4, 5, 7, 11, 12, 14 | |0, 1, 4, 5, 7, 11, 12, 14 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 3,369: | Line 3,423: | ||
|0, 1, 2, 4, 7, 10, 11, 14 | |0, 1, 2, 4, 7, 10, 11, 14 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 3,396: | Line 3,451: | ||
|0, 4, 5, 7, 10, 12, 14, 15 | |0, 4, 5, 7, 10, 12, 14, 15 | ||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 3,424: | Line 3,478: | ||
|0, 2, 5, 7, 8, 12, 14, 15 | |0, 2, 5, 7, 8, 12, 14, 15 | ||
|} | |} | ||
|- | |||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 3,533: | Line 3,588: | ||
|0, 4, 5, 7, 8, 10, 14, 15 | |0, 4, 5, 7, 8, 10, 14, 15 | ||
|} | |} | ||
|- | |||
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{| class="wikitable" | {| class="wikitable" | ||
| Line 3,560: | Line 3,616: | ||
|0, 1, 4, 6, 7, 11, 14, 16 | |0, 1, 4, 6, 7, 11, 14, 16 | ||
|} | |} | ||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 3,588: | Line 3,643: | ||
|0, 1, 2, 4, 7, 11, 12, 14 | |0, 1, 2, 4, 7, 11, 12, 14 | ||
|} | |} | ||
|- | |||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 3,615: | Line 3,671: | ||
|0, 4, 6, 7, 9, 13, 14, 16 | |0, 4, 6, 7, 9, 13, 14, 16 | ||
|} | |} | ||
| | |||
|} | |} | ||
== Nonads == | == Nonads == | ||
{| class="wikitable" | {| class="wikitable" | ||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |||
|neu2 Maj2 Maj3 Aug4 5 neu6 Maj6 Maj7 | |||
|0, 2, 3, 6, 9, 10, 12, 13, 16 | |||
|- | |||
|min2 min3 4 neu4 5 min6 min7 neu7 | |||
|0, 1, 4, 7, 8, 10, 11, 14, 15 | |||
|- | |- | ||
|Maj2 Maj3 4 neu5 5 Maj6 min7 Maj7 | |Maj2 Maj3 4 neu5 5 Maj6 min7 Maj7 | ||
| Line 3,643: | Line 3,705: | ||
|min2 Maj2 min3 4 5 min6 Maj6 min7 | |min2 Maj2 min3 4 5 min6 Maj6 min7 | ||
|0, 1, 3, 4, 7, 10, 11, 13, 14 | |0, 1, 3, 4, 7, 10, 11, 13, 14 | ||
|} | |} | ||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |||
|neu2 Maj2 Maj3 Aug4 5 Maj6 min7 Maj7 | |||
|0, 2, 3, 6, 9, 10, 13, 14, 16 | |||
|- | |||
|min2 min3 4 dim5 min6 neu6 min7 neu7 | |||
|0, 1, 4, 7, 8, 11, 12, 14, 15 | |||
|- | |- | ||
|Maj2 Maj3 4 5 min6 Maj6 min7 Maj7 | |Maj2 Maj3 4 5 min6 Maj6 min7 Maj7 | ||
| Line 3,673: | Line 3,735: | ||
|min2 Maj2 min3 4 5 min6 min7 neu7 | |min2 Maj2 min3 4 5 min6 min7 neu7 | ||
|0, 1, 3, 4, 7, 10, 11, 14, 15 | |0, 1, 3, 4, 7, 10, 11, 14, 15 | ||
|} | |} | ||
|- | |- | ||
| Line 3,684: | Line 3,740: | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
|neu2 Maj2 neu3 Maj3 Aug4 5 Maj6 Maj7 | |||
|0, 2, 3, 5, 6, 9, 10, 13, 16 | |||
|neu2 Maj2 neu3 Maj3 Aug4 5 Maj6 Maj7 | |||
|0, 2, 3, 5, 6, 9, 10, 13, 16 | |||
|- | |- | ||
|min2 Maj2 min3 4 dim5 min6 min7 neu7 | |min2 Maj2 min3 4 dim5 min6 min7 neu7 | ||
| Line 3,710: | Line 3,757: | ||
|min2 min3 4 neu4 5 min6 Maj6 min7 | |min2 min3 4 neu4 5 min6 Maj6 min7 | ||
|0, 1, 4, 7, 8, 10, 11, 13, 14 | |0, 1, 4, 7, 8, 10, 11, 13, 14 | ||
|- | |||
|Maj2 Maj3 4 neu5 5 neu6 Maj6 Maj7 | |||
|0, 3, 6, 7, 9, 10, 12, 13, 16 | |||
|- | |||
|Maj2 min3 Maj3 4 neu5 5 Maj6 min7 | |||
|0, 3, 4, 6, 7, 9, 10, 13, 14 | |||
|- | |||
|min2 Maj2 min3 Maj3 4 5 min6 min7 | |||
|0, 1, 3, 4, 6, 7, 10, 11, 14 | |||
|} | |} | ||
| | | | ||
| Line 3,835: | Line 3,891: | ||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
|neu2 Maj2 min3 Maj3 Aug4 5 Maj6 Maj7 | |neu2 Maj2 min3 Maj3 Aug4 5 Maj6 Maj7 | ||
| Line 3,862: | Line 3,909: | ||
|min2 min3 4 neu4 5 min6 neu6 min7 | |min2 min3 4 neu4 5 min6 neu6 min7 | ||
|0, 1, 4, 7, 8, 10, 11, 12, 14 | |0, 1, 4, 7, 8, 10, 11, 12, 14 | ||
|- | |||
|Maj2 Maj3 4 neu5 5 min6 Maj6 Maj7 | |||
|0, 3, 6, 7, 9, 10, 11, 13, 16 | |||
|- | |||
|Maj2 min3 Maj3 4 neu4 5 Maj6 min7 | |||
|0, 3, 4, 6, 7, 8, 10, 13, 14 | |||
|- | |||
|min2 Maj2 min3 neu3 4 5 min6 min7 | |||
|0, 1, 3, 4, 5, 7, 10, 11, 14 | |||
|} | |} | ||
|- | |- | ||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |||
|neu2 Maj2 Maj3 Aug4 5 Maj6 neu7 Maj7 | |||
|0, 2, 3, 6, 9, 10, 13, 15, 16 | |||
|- | |||
|min2 min3 4 dim5 min6 Maj6 min7 neu7 | |||
|0, 1, 4, 7, 8, 11, 13, 14, 15 | |||
|- | |- | ||
|Maj2 Maj3 4 5 neu6 Maj6 min7 Maj7 | |Maj2 Maj3 4 5 neu6 Maj6 min7 Maj7 | ||
| Line 3,887: | Line 3,949: | ||
|min2 Maj2 min3 4 5 min6 min7 Maj7 | |min2 Maj2 min3 4 5 min6 min7 Maj7 | ||
|0, 1, 3, 4, 7, 10, 11, 14, 16 | |0, 1, 3, 4, 7, 10, 11, 14, 16 | ||
|} | |||
| | |||
{| class="wikitable" | |||
|- | |- | ||
|neu2 Maj2 neu3 Maj3 Aug4 5 neu6 Maj6 | |||
|neu2 Maj2 neu3 Maj3 Aug4 5 neu6 Maj6 | |||
|0, 2, 3, 5, 6, 9, 10, 12, 13 | |0, 2, 3, 5, 6, 9, 10, 12, 13 | ||
|- | |- | ||
| Line 4,110: | Line 4,166: | ||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
|neu2 Maj2 Maj3 neu4 neu5 5 Maj6 Maj7 | |neu2 Maj2 Maj3 neu4 neu5 5 Maj6 Maj7 | ||
| Line 4,137: | Line 4,184: | ||
|min2 min3 4 neu4 5 min6 min7 Maj7 | |min2 min3 4 neu4 5 min6 min7 Maj7 | ||
|0, 1, 4, 7, 8, 10, 11, 14, 16 | |0, 1, 4, 7, 8, 10, 11, 14, 16 | ||
|- | |||
|Maj2 Maj3 4 neu5 5 Maj6 neu7 Maj7 | |||
|0, 3, 6, 7, 9, 10, 13, 15, 16 | |||
|- | |||
|Maj2 min3 Maj3 4 5 neu6 Maj6 min7 | |||
|0, 3, 4, 6, 7, 10, 12, 13, 14 | |||
|- | |||
|min2 Maj2 min3 4 neu5 5 min6 min7 | |||
|0, 1, 3, 4, 7, 9, 10, 11, 14 | |||
|} | |} | ||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |||
|neu2 Maj2 Maj3 Aug4 5 min6 Maj6 Maj7 | |||
|0, 2, 3, 6, 9, 10, 11, 13, 16 | |||
|- | |||
|min2 min3 4 dim5 neu5 min6 min7 neu7 | |||
|0, 1, 4, 7, 8, 9, 11, 14, 15 | |||
|- | |- | ||
|Maj2 Maj3 4 neu4 5 Maj6 min7 Maj7 | |Maj2 Maj3 4 neu4 5 Maj6 min7 Maj7 | ||
| Line 4,161: | Line 4,223: | ||
|min2 Maj2 min3 4 5 min6 neu6 min7 | |min2 Maj2 min3 4 5 min6 neu6 min7 | ||
|0, 1, 3, 4, 7, 10, 11, 12, 14 | |0, 1, 3, 4, 7, 10, 11, 12, 14 | ||
|} | |} | ||
|- | |- | ||
| Line 4,446: | Line 4,502: | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
|Maj2 Maj3 | |min2 Maj2 Maj3 neu4 neu5 5 Maj6 Maj7 | ||
|0, 3, 6, | |0, 1, 3, 6, 8, 9, 10, 13, 16 | ||
|- | |- | ||
|neu2 neu3 4 dim5 neu5 neu6 neu7 Maj7 | |||
|0, 2, 5, 7, 8, 9, 12, 15, 16 | |||
|neu2 neu3 4 dim5 neu5 neu6 neu7 Maj7 | |||
|0, 2, 5, 7, 8, 9, 12, 15, 16 | |||
|- | |- | ||
|Maj2 neu3 Maj3 4 5 Maj6 min7 neu7 | |Maj2 neu3 Maj3 4 5 Maj6 min7 neu7 | ||
| Line 4,472: | Line 4,519: | ||
|min2 min3 4 dim5 neu5 min6 min7 Maj7 | |min2 min3 4 dim5 neu5 min6 min7 Maj7 | ||
|0, 1, 4, 7, 8, 9, 11, 14, 16 | |0, 1, 4, 7, 8, 9, 11, 14, 16 | ||
|- | |||
|Maj2 Maj3 4 neu4 5 Maj6 neu7 Maj7 | |||
|0, 3, 6, 7, 8, 10, 13, 15, 16 | |||
|- | |||
|Maj2 min3 neu3 4 5 neu6 Maj6 min7 | |||
|0, 3, 4, 5, 7, 10, 12, 13, 14 | |||
|- | |||
|min2 neu2 min3 4 neu5 5 min6 min7 | |||
|0, 1, 2, 4, 7, 9, 10, 11, 14 | |||
|} | |} | ||
|- | |- | ||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |||
|min2 neu2 Maj2 Maj3 Aug4 5 Maj6 Maj7 | |||
|0, 1, 2, 3, 6, 9, 10, 13, 16 | |||
|- | |||
|min2 neu2 neu3 neu4 neu5 neu6 neu7 Maj7 | |||
|0, 1, 2, 5, 8, 9, 12, 15, 16 | |||
|- | |||
|min2 min3 4 dim5 min6 min7 neu7 Maj7 | |||
|0, 1, 4, 7, 8, 11, 14, 15, 16 | |||
|- | |- | ||
|Maj2 Maj3 4 5 Maj6 min7 neu7 Maj7 | |Maj2 Maj3 4 5 Maj6 min7 neu7 Maj7 | ||
| Line 4,494: | Line 4,559: | ||
|min2 neu2 Maj2 min3 4 5 min6 min7 | |min2 neu2 Maj2 min3 4 5 min6 min7 | ||
|0, 1, 2, 3, 4, 7, 10, 11, 14 | |0, 1, 2, 3, 4, 7, 10, 11, 14 | ||
|} | |} | ||
| | |||
|} | |} | ||
| Line 4,540: | Line 4,597: | ||
|0, 2, 15 | |0, 2, 15 | ||
|} | |} | ||
|- | |||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 4,556: | Line 4,614: | ||
|0, 2, 13, 15 | |0, 2, 13, 15 | ||
|} | |} | ||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 4,576: | Line 4,633: | ||
|0, 2, 11, 13, 15 | |0, 2, 11, 13, 15 | ||
|} | |} | ||
|- | |||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 4,683: | Line 4,741: | ||
|0, 1, 3, 5, 7, 9, 11, 13, 15 | |0, 1, 3, 5, 7, 9, 11, 13, 15 | ||
|} | |} | ||
|} | |} | ||
| Line 5,108: | Line 5,165: | ||
|0, 1, 9 | |0, 1, 9 | ||
|} | |} | ||
|- | |||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 5,127: | Line 5,185: | ||
|0, 1, 8, 9, 16 | |0, 1, 8, 9, 16 | ||
|} | |} | ||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 5,153: | Line 5,210: | ||
|0, 1, 7, 8, 9, 15, 16 | |0, 1, 7, 8, 9, 15, 16 | ||
|} | |} | ||
|- | |||
| | | | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 5,183: | Line 5,241: | ||
|min2 Maj3 4 neu4 neu5 min7 neu7 Maj7 | |min2 Maj3 4 neu4 neu5 min7 neu7 Maj7 | ||
|0, 1, 6, 7, 8, 9, 14, 15, 16 | |0, 1, 6, 7, 8, 9, 14, 15, 16 | ||
|} | |||
| | |||
|} | |||
= Scales with 2 or 3 steps per degree = | |||
{| class="wikitable" | |||
|+ | |||
| | |||
{| class="wikitable" | |||
|+3 3 3 3 3 2 (MOS) | |||
|- | |||
|Maj2 Maj3 neu5 neu6 neu7 | |||
|0, 3, 6, 9, 12, 15 | |||
|- | |||
|Maj2 Maj3 neu5 neu6 min7 | |||
|0, 3, 6, 9, 12, 14 | |||
|- | |||
|Maj2 Maj3 neu5 min6 min7 | |||
|0, 3, 6, 9, 11, 14 | |||
|- | |||
|Maj2 Maj3 neu4 min6 min7 | |||
|0, 3, 6, 8, 11, 14 | |||
|- | |||
|Maj2 neu3 neu4 min6 min7 | |||
|0, 3, 5, 8, 11, 14 | |||
|- | |||
|neu2 neu3 neu4 min6 min7 | |||
|0, 2, 5, 8, 11, 14 | |||
|} | |||
| | |||
{| class="wikitable" | |||
|+3 2 3 2 3 2 2 (MOS) | |||
|- | |||
|Maj2 neu3 neu4 5 Maj6 neu7 | |||
|0, 3, 5, 8, 10, 13, 15 | |||
|- | |||
|neu2 neu3 4 5 neu6 min7 | |||
|0, 2, 5, 7, 10, 12, 14 | |||
|- | |||
|Maj2 neu3 neu4 5 neu6 neu7 | |||
|0, 3, 5, 8, 10, 12, 15 | |||
|- | |||
|neu2 neu3 4 neu5 neu6 min7 | |||
|0, 2, 5, 7, 9, 12, 14 | |||
|- | |||
|Maj2 neu3 4 5 neu6 neu7 | |||
|0, 3, 5, 7, 10, 12, 15 | |||
|- | |||
|neu2 min3 4 neu5 neu6 min7 | |||
|0, 2, 4, 7, 9, 12, 14 | |||
|- | |||
|neu2 neu3 4 5 neu6 neu7 | |||
|0, 2, 5, 7, 10, 12, 15 | |||
|} | |||
|- | |||
| | |||
{| class="wikitable" | |||
|+3 3 2 2 2 3 2 | |||
|- | |||
|Maj2 Maj3 neu4 5 neu6 neu7 | |||
|0, 3, 6, 8, 10, 12, 15 | |||
|- | |||
|Maj2 neu3 4 neu5 neu6 min7 | |||
|0, 3, 5, 7, 9, 12, 14 | |||
|- | |||
|neu2 min3 Maj3 neu5 min6 min7 | |||
|0, 2, 4, 6, 9, 11, 14 | |||
|- | |||
|neu2 min3 4 neu5 neu6 neu7 | |||
|0, 2, 4, 7, 9, 12, 15 | |||
|- | |||
|neu2 neu3 4 5 Maj6 neu7 | |||
|0, 2, 5, 7, 10, 13, 15 | |||
|- | |||
|Maj2 neu3 neu4 min6 Maj6 neu7 | |||
|0, 3, 5, 8, 11, 13, 15 | |||
|- | |||
|neu2 neu3 neu4 5 neu6 min7 | |||
|0, 2, 5, 8, 10, 12, 14 | |||
|} | |||
| | |||
{| class="wikitable" | |||
|+3 3 2 2 3 2 2 | |||
|- | |||
|Maj2 Maj3 neu4 5 Maj6 neu7 | |||
|0, 3, 6, 8, 10, 13, 15 | |||
|- | |||
|Maj2 neu3 4 5 neu6 min7 | |||
|0, 3, 5, 7, 10, 12, 14 | |||
|- | |||
|neu2 min3 4 neu5 min6 min7 | |||
|0, 2, 4, 7, 9, 11, 14 | |||
|- | |||
|neu2 neu3 4 neu5 neu6 neu7 | |||
|0, 2, 5, 7, 9, 12, 15 | |||
|- | |||
|Maj2 neu3 4 5 Maj6 neu7 | |||
|0, 3, 5, 7, 10, 13, 15 | |||
|- | |||
|neu2 min3 4 5 neu6 min7 | |||
|0, 2, 4, 7, 10, 12, 14 | |||
|- | |||
|neu2 neu3 neu4 5 neu6 neu7 | |||
|0, 2, 5, 8, 10, 12, 15 | |||
|} | |||
|- | |||
| | |||
{| class="wikitable" | |||
|+3 3 2 3 2 2 2 | |||
|- | |||
|Maj2 Maj3 neu4 min6 Maj6 neu7 | |||
|0, 3, 6, 8, 11, 13, 15 | |||
|- | |||
|Maj2 neu3 neu4 5 neu6 min7 | |||
|0, 3, 5, 8, 10, 12, 14 | |||
|- | |||
|neu2 neu3 4 neu5 min6 min7 | |||
|0, 2, 5, 7, 9, 11, 14 | |||
|- | |||
|Maj2 neu3 4 neu5 neu6 neu7 | |||
|0, 3, 5, 7, 9, 12, 15 | |||
|- | |||
|neu2 min3 Maj3 neu5 neu6 min7 | |||
|0, 2, 4, 6, 9, 12, 14 | |||
|- | |||
|neu2 min3 4 5 neu6 neu7 | |||
|0, 2, 4, 7, 10, 12, 15 | |||
|- | |||
|neu2 neu3 neu4 5 Maj6 neu7 | |||
|0, 2, 5, 8, 10, 13, 15 | |||
|} | |||
| | |||
{| class="wikitable" | |||
|+3 3 3 2 2 2 2 | |||
|- | |||
|Maj2 Maj3 neu5 min6 Maj6 neu7 | |||
|0, 3, 6, 9, 11, 13, 15 | |||
|- | |||
|Maj2 Maj3 neu4 5 neu6 min7 | |||
|0, 3, 6, 8, 10, 12, 14 | |||
|- | |||
|Maj2 neu3 4 neu5 min6 min7 | |||
|0, 3, 5, 7, 9, 11, 14 | |||
|- | |||
|neu2 min3 Maj3 neu4 min6 min7 | |||
|0, 2, 4, 6, 8, 11, 14 | |||
|- | |||
|neu2 min3 Maj3 neu5 neu6 neu7 | |||
|0, 2, 4, 6, 9, 12, 15 | |||
|- | |||
|neu2 min3 4 5 Maj6 neu7 | |||
|0, 2, 4, 7, 10, 13, 15 | |||
|- | |||
|neu2 neu3 neu4 min6 Maj6 neu7 | |||
|0, 2, 5, 8, 11, 13, 15 | |||
|} | |||
|- | |||
| | |||
{| class="wikitable" | |||
|+3 2 2 2 2 2 2 2 (MOS) | |||
|- | |||
|Maj2 neu3 4 neu5 min6 Maj6 neu7 | |||
|0, 3, 5, 7, 9, 11, 13, 15 | |||
|- | |||
|neu2 min3 Maj3 neu4 5 neu6 min7 | |||
|0, 2, 4, 6, 8, 10, 12, 14 | |||
|- | |||
|neu2 min3 Maj3 neu4 5 neu6 neu7 | |||
|0, 2, 4, 6, 8, 10, 12, 15 | |||
|- | |||
|neu2 min3 Maj3 neu4 5 Maj6 neu7 | |||
|0, 2, 4, 6, 8, 10, 13, 15 | |||
|- | |||
|neu2 min3 Maj3 neu4 min6 Maj6 neu7 | |||
|0, 2, 4, 6, 8, 11, 13, 15 | |||
|- | |||
|neu2 min3 Maj3 neu5 min6 Maj6 neu7 | |||
|0, 2, 4, 6, 9, 11, 13, 15 | |||
|- | |||
|neu2 min3 4 neu5 min6 Maj6 neu7 | |||
|0, 2, 4, 7, 9, 11, 13, 15 | |||
|- | |||
|neu2 neu3 4 neu5 min6 Maj6 neu7 | |||
|0, 2, 5, 7, 9, 11, 13, 15 | |||
|} | |} | ||
| | | | ||
|} | |} | ||
Latest revision as of 23:14, 1 December 2025
Some mapping
sharpness = 2 ; L = 3 ; s = 1
| Degree | Name | Cents | Approximate Ratios | Weight | Approximate Ratios | Cents | Name | Degree |
|---|---|---|---|---|---|---|---|---|
| 0 | root | 0.0 | 1/1 | 2/1 | 1200.0 | octave | 17 | |
| 1 | min2 | 70.6 | 25/24 | 1/25 | 48/25 | 1129.4 | Maj7 | 16 |
| 2 | neu2 | 141.2 | 14/13, 13/12, 12/11 | 1/13 | 11/6, 24/13, 13/7 | 1058.8 | neu7 | 15 |
| 3 | Maj2 | 211.8 | 9/8, 8/7 | 1/9 | 7/4, 16/9 | 988.2 | min7 | 14 |
| 4 | min3 | 282.4 | 20/17, 13/11 | 1/10 | 22/13, 17/10 | 917.6 | Maj6 | 13 |
| 5 | neu3 | 352.9 | 11/9, 16/13 | 1/12 | 13/8, 18/11 | 847.1 | neu6 | 12 |
| 6 | Maj3 | 423.5 | 14/11, 23/18 | 1/11 | 36/23, 11/7 | 776.5 | min6 | 11 |
| 7 | 4 | 494.1 | 4/3 | 1/3 | 3/2 | 705.9 | 5 | 10 |
| 8 | neu4, dim5 | 564.7 | 18/13 | 1/13 | 13/9 | 635.3 | neu5, Aug4 | 9 |
What these columns mean
- Degree: Number of steps within the octave period.
- Name: Abbreviated spoken name of the interval, used in staff notation:
- Interval quality:
- dim = diminished
- min = minor
- neu = neutral
- Maj = major
- Aug = augmented
- Unspecified quality (a bare 4 or 5) means a perfect interval.
- Interval heptatonic degree:
- 2, 3, 4, 5, 6, 7
- Interval quality:
- Cents: Size of the interval.
- Approximate Ratios: Small ratios close to the interval, given as possible just-intonation interpretations; other choices are possible.
- Weight: Arbitrary weight values assigned to each interval, used to rank chord types in the Some chords section below. This is only one possible way to order or classify the intervals.
- Layout: The left half of the table lists ascending intervals from the root; the right half lists their complements within the 17-step octave (degree 17 − n), with the same weights.
Some chords
What these chord tables show
For each set cardinality, the tables list the first members of the chord classes ranked by the interval weight values defined in the previous section.
The ranking is obtained with a simple generative model on the 17-note circle:
- Starting from degree 0, the algorithm successively extends the set by applying one of the 17 possible intervals above every degree that has already been generated.
- Each interval carries a weight: simpler intervals receive larger weights than more remote ones.
- All pitch-class sets obtained this way, modulo the 17-step equave, are grouped into chord classes (up to inversion and transposition).
- Each chord class receives a score, obtained by adding the weights of the intervals along all the chains of intervals that generate it.
- The classes are then sorted from highest to lowest score; only the best-scoring ones for each cardinality are shown in the tables.
Each small inner table groups the inversions (rotations) of a single chord class. In every row, the left cell names the chord by listing its intervals above the root note of that inversion, and the right cell gives the same information as a set of degrees.
The first row of each inner table is a distinguished representative: among all inversions of the chord, it is the one whose pitch classes, when ordered along the 17-EDO circle of fifths (0–10–3–13–6–16–9–2–12–5–15–8–1–11–4–14–7), form the lexicographically smallest sequence. The remaining rows list the other inversions of the same chord, in the order obtained by successive rotations.
Dyads
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Triads
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Tetrads
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Pentads
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Hexads
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Heptads
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Octads
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Nonads
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Some MOS Regular Temperaments
MOS modes with 2-9 notes per octave (generator 1\17 excluded).
Period: 1\17, Generator: 1\17
excluded
Period: 17\17, Generator: 1\17
excluded
Period: 17\17, Generator: 2\17
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Period: 17\17, Generator: 3\17
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Period: 17\17, Generator: 4\17
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Period: 17\17, Generator: 5\17
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