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Scale 2789: "Zolian"

Scale 2789: Zolian, 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

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
Zolian

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

{0,2,5,6,7,9,11}

Forte Number

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

7-29

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: 1259

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: 727

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, 3, 1, 1, 2, 2, 1]

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, 3, 5, 2>

Interval Spectrum

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

p5m3n4s4d3t2

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}
<4> = {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

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 TriadsD{2,6,9}321.29
F{5,9,0}241.86
G{7,11,2}142.14
Minor Triadsdm{2,5,9}331.43
bm{11,2,6}331.43
Diminished Triadsf♯°{6,9,0}231.71
{11,2,5}231.57
Parsimonious Voice Leading Between Common Triads of Scale 2789. Created by Ian Ring ©2019 dm dm D D dm->D F F dm->F dm->b° f#° f#° D->f#° bm bm D->bm F->f#° Parsimonious Voice Leading Between Common Triads of Scale 2789. Created by Ian Ring ©2019 G G->bm b°->bm

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 VerticesD
Peripheral VerticesF, G

Modes

Modes are the rotational transformation of this scale. Scale 2789 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1721
Scale 1721: Mela Vagadhisvari, Ian Ring Music TheoryMela Vagadhisvari
3rd mode:
Scale 727
Scale 727: Phradian, Ian Ring Music TheoryPhradianThis is the prime mode
4th mode:
Scale 2411
Scale 2411: Aeolorian, Ian Ring Music TheoryAeolorian
5th mode:
Scale 3253
Scale 3253: Mela Naganandini, Ian Ring Music TheoryMela Naganandini
6th mode:
Scale 1837
Scale 1837: Dalian, Ian Ring Music TheoryDalian
7th mode:
Scale 1483
Scale 1483: Mela Bhavapriya, Ian Ring Music TheoryMela Bhavapriya

Prime

The prime form of this scale is Scale 727

Scale 727Scale 727: Phradian, Ian Ring Music TheoryPhradian

Complement

The heptatonic modal family [2789, 1721, 727, 2411, 3253, 1837, 1483] (Forte: 7-29) is the complement of the pentatonic modal family [331, 709, 1201, 1577, 2213] (Forte: 5-29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2789 is 1259

Scale 1259Scale 1259: Stadian, Ian Ring Music TheoryStadian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2789 is chiral, and its enantiomorph is scale 1259

Scale 1259Scale 1259: Stadian, Ian Ring Music TheoryStadian

Transformations:

T0 2789  T0I 1259
T1 1483  T1I 2518
T2 2966  T2I 941
T3 1837  T3I 1882
T4 3674  T4I 3764
T5 3253  T5I 3433
T6 2411  T6I 2771
T7 727  T7I 1447
T8 1454  T8I 2894
T9 2908  T9I 1693
T10 1721  T10I 3386
T11 3442  T11I 2677

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 2791Scale 2791: Mixothyllic, Ian Ring Music TheoryMixothyllic
Scale 2785Scale 2785, Ian Ring Music Theory
Scale 2787Scale 2787: Zyrian, Ian Ring Music TheoryZyrian
Scale 2793Scale 2793: Eporian, Ian Ring Music TheoryEporian
Scale 2797Scale 2797: Stalyllic, Ian Ring Music TheoryStalyllic
Scale 2805Scale 2805: Ishikotsucho, Ian Ring Music TheoryIshikotsucho
Scale 2757Scale 2757: Raga Nishadi, Ian Ring Music TheoryRaga Nishadi
Scale 2773Scale 2773: Lydian, Ian Ring Music TheoryLydian
Scale 2725Scale 2725: Raga Nagagandhari, Ian Ring Music TheoryRaga Nagagandhari
Scale 2661Scale 2661: Stydimic, Ian Ring Music TheoryStydimic
Scale 2917Scale 2917: Nohkan Flute Scale, Ian Ring Music TheoryNohkan Flute Scale
Scale 3045Scale 3045: Raptyllic, Ian Ring Music TheoryRaptyllic
Scale 2277Scale 2277: Kagimic, Ian Ring Music TheoryKagimic
Scale 2533Scale 2533: Podian, Ian Ring Music TheoryPodian
Scale 3301Scale 3301: Chromatic Mixolydian Inverse, Ian Ring Music TheoryChromatic Mixolydian Inverse
Scale 3813Scale 3813: Aeologyllic, Ian Ring Music TheoryAeologyllic
Scale 741Scale 741: Gathimic, Ian Ring Music TheoryGathimic
Scale 1765Scale 1765: Lonian, Ian Ring Music TheoryLonian

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.