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Scale 2965: "Darian"

Scale 2965: Darian, 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
Darian

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

{0,2,4,7,8,9,11}

Forte Number

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

7-27

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: 1339

Hemitonia

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

3 (trihemitonic)

Cohemitonia

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

1 (uncohemitonic)

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.

2

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: 695

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.

[2, 2, 3, 1, 1, 2, 1] 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.

<3, 4, 4, 4, 5, 1>

Interval Spectrum

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

p5m4n4s4d3t

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

Spectra Variation

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

2.286

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.

2.549

Polygon Perimeter

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

5.967

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 TriadsC{0,4,7}231.57
E{4,8,11}321.29
G{7,11,2}241.86
Minor Triadsem{4,7,11}331.43
am{9,0,4}142.14
Augmented TriadsC+{0,4,8}331.43
Diminished Triadsg♯°{8,11,2}231.71
Parsimonious Voice Leading Between Common Triads of Scale 2965. Created by Ian Ring ©2019 C C C+ C+ C->C+ em em C->em E E C+->E am am C+->am em->E Parsimonious Voice Leading Between Common Triads of Scale 2965. Created by Ian Ring ©2019 G em->G g#° g#° E->g#° G->g#°

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.

Diameter4
Radius2
Self-Centeredno
Central VerticesE
Peripheral VerticesG, am

Modes

Modes are the rotational transformation of this scale. Scale 2965 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1765
Scale 1765: Lonian, Ian Ring Music TheoryLonian
3rd mode:
Scale 1465
Scale 1465: Mela Ragavardhani, Ian Ring Music TheoryMela Ragavardhani
4th mode:
Scale 695
Scale 695: Sarian, Ian Ring Music TheorySarianThis is the prime mode
5th mode:
Scale 2395
Scale 2395: Zoptian, Ian Ring Music TheoryZoptian
6th mode:
Scale 3245
Scale 3245: Mela Varunapriya, Ian Ring Music TheoryMela Varunapriya
7th mode:
Scale 1835
Scale 1835: Byptian, Ian Ring Music TheoryByptian

Prime

The prime form of this scale is Scale 695

Scale 695Scale 695: Sarian, Ian Ring Music TheorySarian

Complement

The heptatonic modal family [2965, 1765, 1465, 695, 2395, 3245, 1835] (Forte: 7-27) is the complement of the pentatonic modal family [299, 689, 1417, 1573, 2197] (Forte: 5-27)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2965 is 1339

Scale 1339Scale 1339: Kycrian, Ian Ring Music TheoryKycrian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2965 is chiral, and its enantiomorph is scale 1339

Scale 1339Scale 1339: Kycrian, Ian Ring Music TheoryKycrian

Transformations:

T0 2965  T0I 1339
T1 1835  T1I 2678
T2 3670  T2I 1261
T3 3245  T3I 2522
T4 2395  T4I 949
T5 695  T5I 1898
T6 1390  T6I 3796
T7 2780  T7I 3497
T8 1465  T8I 2899
T9 2930  T9I 1703
T10 1765  T10I 3406
T11 3530  T11I 2717

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 2967Scale 2967: Madyllic, Ian Ring Music TheoryMadyllic
Scale 2961Scale 2961: Bygimic, Ian Ring Music TheoryBygimic
Scale 2963Scale 2963: Bygian, Ian Ring Music TheoryBygian
Scale 2969Scale 2969: Tholian, Ian Ring Music TheoryTholian
Scale 2973Scale 2973: Panyllic, Ian Ring Music TheoryPanyllic
Scale 2949Scale 2949, Ian Ring Music Theory
Scale 2957Scale 2957: Thygian, Ian Ring Music TheoryThygian
Scale 2981Scale 2981: Ionolian, Ian Ring Music TheoryIonolian
Scale 2997Scale 2997: Major Bebop, Ian Ring Music TheoryMajor Bebop
Scale 3029Scale 3029: Ionocryllic, Ian Ring Music TheoryIonocryllic
Scale 2837Scale 2837: Aelothimic, Ian Ring Music TheoryAelothimic
Scale 2901Scale 2901: Lydian Augmented, Ian Ring Music TheoryLydian Augmented
Scale 2709Scale 2709: Raga Kumud, Ian Ring Music TheoryRaga Kumud
Scale 2453Scale 2453: Raga Latika, Ian Ring Music TheoryRaga Latika
Scale 3477Scale 3477: Kyptian, Ian Ring Music TheoryKyptian
Scale 3989Scale 3989: Sythyllic, Ian Ring Music TheorySythyllic
Scale 917Scale 917: Dygimic, Ian Ring Music TheoryDygimic
Scale 1941Scale 1941: Aeranian, Ian Ring Music TheoryAeranian

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.