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Scale 1615: "Sydian"

Scale 1615: Sydian, Ian Ring Music Theory

Bracelet Diagram

The bracelet shows tones that are in this scale, starting from the top (12 o'clock), going clockwise in ascending semitones. The "i" icon marks imperfect tones that do not have a tone a fifth above. Dotted lines indicate axes of symmetry.

Tonnetz Diagram

41161837294116105072918310504116183
Tonnetz diagrams are popular in Neo-Riemannian theory. Notes are arranged in a lattice where perfect 5th intervals are from left to right, major third are northeast, and major 6th intervals are northwest. Other directions are inverse of their opposite. This diagram helps to visualize common triads (they're triangles) and circle-of-fifth relationships (horizontal lines).

Common Names

Zeitler
Sydian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

7 (heptatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,1,2,3,6,9,10}

Forte Number

A code assigned by theorist Alan Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

7-16

Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

none

Reflection Axes

If a scale has an axis of reflective symmetry, then it can transform into itself by inversion. It also implies that the scale has Ridge Tones. Notably an axis of reflection can occur directly on a tone or half way between two tones.

none

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

no

Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

yes
enantiomorph: 3661

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

4 (multihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

2 (dicohemitonic)

Imperfections

An imperfection is a tone which does not have a perfect fifth above it in the scale. This value is the quantity of imperfections in this scale.

4

Modes

Modes are the rotational transformations of this scale. This number does not include the scale itself, so the number is usually one less than its cardinality; unless there are rotational symmetries then there are even fewer modes.

6

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 623

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 1, 1, 3, 3, 1, 2] 9

Interval Vector

Describes the intervallic content of the scale, read from left to right as the number of occurences of each interval size from semitone, up to six semitones.

<4, 3, 5, 4, 3, 2>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hansen.

p3m4n5s3d4t2

Distribution Spectra

Describes the specific interval sizes that exist for each generic interval size. Each generic <g> has a spectrum {n,...}. The Spectrum Width is the difference between the highest and lowest values in each spectrum.

<1> = {1,2,3}
<2> = {2,3,4,6}
<3> = {3,4,5,6,7}
<4> = {5,6,7,8,9}
<5> = {6,8,9,10}
<6> = {9,10,11}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

2.857

Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

no

Maximal Area Set

A scale is a maximal area set if a polygon described by vertices dodecimetrically placed around a circle produces the maximal interior area for scales of the same cardinality. All maximally even sets have maximal area, but not all maximal area sets are maximally even.

no

Interior Area

Area of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle, ie a circle with radius of 1.

2.433

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

5.899

Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

no

Balanced

A scale is balanced if the distribution of its tones would satisfy the "centrifuge problem", ie are placed such that it would balance on its centre point.

no

Ridge Tones

Ridge Tones are those that appear in all transpositions of a scale upon the members of that scale. Ridge Tones correspond directly with axes of reflective symmetry.

none

Propriety

Also known as Rothenberg Propriety, named after its inventor. Propriety describes whether every specific interval is uniquely mapped to a generic interval. A scale is either "Proper", "Strictly Proper", or "Improper".

Improper

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

Common Triads

These are the common triads (major, minor, augmented and diminished) that you can create from members of this scale.

* Pitches are shown with C as the root

Triad TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsD{2,6,9}331.67
F♯{6,10,1}231.89
Minor Triadsd♯m{3,6,10}331.67
f♯m{6,9,1}331.67
Augmented TriadsD+{2,6,10}331.67
Diminished Triads{0,3,6}231.89
d♯°{3,6,9}231.89
f♯°{6,9,0}231.89
{9,0,3}232
Parsimonious Voice Leading Between Common Triads of Scale 1615. Created by Ian Ring ©2019 d#m d#m c°->d#m c°->a° D D D+ D+ D->D+ d#° d#° D->d#° f#m f#m D->f#m D+->d#m F# F# D+->F# d#°->d#m f#° f#° f#°->f#m f#°->a° f#m->F#

view full size

Above is a graph showing opportunities for parsimonious voice leading between triads*. Each line connects two triads that have two common tones, while the third tone changes by one generic scale step.

Diameter3
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1615 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2855
Scale 2855: Epocrain, Ian Ring Music TheoryEpocrain
3rd mode:
Scale 3475
Scale 3475: Kylian, Ian Ring Music TheoryKylian
4th mode:
Scale 3785
Scale 3785: Epagian, Ian Ring Music TheoryEpagian
5th mode:
Scale 985
Scale 985: Mela Sucaritra, Ian Ring Music TheoryMela Sucaritra
6th mode:
Scale 635
Scale 635: Epolian, Ian Ring Music TheoryEpolian
7th mode:
Scale 2365
Scale 2365: Sythian, Ian Ring Music TheorySythian

Prime

The prime form of this scale is Scale 623

Scale 623Scale 623: Sycrian, Ian Ring Music TheorySycrian

Complement

The heptatonic modal family [1615, 2855, 3475, 3785, 985, 635, 2365] (Forte: 7-16) is the complement of the pentatonic modal family [155, 865, 1555, 2125, 2825] (Forte: 5-16)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1615 is 3661

Scale 3661Scale 3661: Mixodorian, Ian Ring Music TheoryMixodorian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1615 is chiral, and its enantiomorph is scale 3661

Scale 3661Scale 3661: Mixodorian, Ian Ring Music TheoryMixodorian

Transformations:

T0 1615  T0I 3661
T1 3230  T1I 3227
T2 2365  T2I 2359
T3 635  T3I 623
T4 1270  T4I 1246
T5 2540  T5I 2492
T6 985  T6I 889
T7 1970  T7I 1778
T8 3940  T8I 3556
T9 3785  T9I 3017
T10 3475  T10I 1939
T11 2855  T11I 3878

Nearby Scales:

These are other scales that are similar to this one, created by adding a tone, removing a tone, or moving one note up or down a semitone.

Scale 1613Scale 1613: Thylimic, Ian Ring Music TheoryThylimic
Scale 1611Scale 1611: Dacrimic, Ian Ring Music TheoryDacrimic
Scale 1607Scale 1607: Epytimic, Ian Ring Music TheoryEpytimic
Scale 1623Scale 1623: Lothian, Ian Ring Music TheoryLothian
Scale 1631Scale 1631: Rynyllic, Ian Ring Music TheoryRynyllic
Scale 1647Scale 1647: Polyllic, Ian Ring Music TheoryPolyllic
Scale 1551Scale 1551, Ian Ring Music Theory
Scale 1583Scale 1583: Salian, Ian Ring Music TheorySalian
Scale 1679Scale 1679: Kydian, Ian Ring Music TheoryKydian
Scale 1743Scale 1743: Epigyllic, Ian Ring Music TheoryEpigyllic
Scale 1871Scale 1871: Aeolyllic, Ian Ring Music TheoryAeolyllic
Scale 1103Scale 1103: Lynimic, Ian Ring Music TheoryLynimic
Scale 1359Scale 1359: Aerygian, Ian Ring Music TheoryAerygian
Scale 591Scale 591: Gaptimic, Ian Ring Music TheoryGaptimic
Scale 2639Scale 2639: Dothian, Ian Ring Music TheoryDothian
Scale 3663Scale 3663: Sonyllic, Ian Ring Music TheorySonyllic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. Some scale names used on this and other pages are ©2005 William Zeitler (http://allthescales.org) used with permission.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (An algorithm for spelling the pitches of any musical scale) Contact authors Patent owner: Dokuz Eylül University, Used with Permission. Contact TTO

Tons of background resources contributed to the production of this summary; for a list of these peruse this Bibliography. Special thanks to Richard Repp for helping with technical accuracy.