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!Cents
!Cents
!Approximate Ratios
!Approximate Ratios
!Height
!Weight
!Approximate Ratios
!Approximate Ratios
!Cents
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* Degree: Number of steps within the octave period.
* Degree: Number of steps within the octave period.
* Name: Abbreviated spoken name of the interval:  
* Name: Abbreviated spoken name of the interval, used in staff notation:  
** min = minor
** Interval quality:
** neu = neutral
*** ''dim'' = diminished
** Maj = major
*** ''min'' = minor
** dim = diminished
*** ''neu'' = neutral
** Aug = augmented
*** ''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.
* Cents: Size of the interval.
* Approximate Ratios: Small ratios close to the interval, given as possible just-intonation interpretations; other choices are possible.
* Approximate Ratios: Small ratios close to the interval, given as possible just-intonation interpretations; other choices are possible.
* Height: Arbitrary “height” 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.
* 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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== Hexads ==
== Hexads ==
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== Octads ==
== Octads ==
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== Nonads ==
== Nonads ==
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= Scales with 2 or 3 steps per degree =
= Scales with 2 or 3 steps per degree =
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Retrieved from "https://en.xen.wiki/w/User:Contribution/17-EDO_resources"