The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 2359: "Gadian"

Scale 2359: Gadian, 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
Gadian

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

{0,1,2,4,5,8,11}

Forte Number

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

7-16

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: 3475

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, 2, 1, 3, 3, 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.

<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 TriadsC♯{1,5,8}331.67
E{4,8,11}331.67
Minor Triadsc♯m{1,4,8}231.89
fm{5,8,0}331.67
Augmented TriadsC+{0,4,8}331.67
Diminished Triads{2,5,8}231.89
{5,8,11}231.89
g♯°{8,11,2}231.89
{11,2,5}232
Parsimonious Voice Leading Between Common Triads of Scale 2359. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m E E C+->E fm fm C+->fm C# C# c#m->C# C#->d° C#->fm d°->b° E->f° g#° g#° E->g#° f°->fm g#°->b°

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 2359 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3227
Scale 3227: Aeolocrian, Ian Ring Music TheoryAeolocrian
3rd mode:
Scale 3661
Scale 3661: Mixodorian, Ian Ring Music TheoryMixodorian
4th mode:
Scale 1939
Scale 1939: Dathian, Ian Ring Music TheoryDathian
5th mode:
Scale 3017
Scale 3017: Gacrian, Ian Ring Music TheoryGacrian
6th mode:
Scale 889
Scale 889: Borian, Ian Ring Music TheoryBorian
7th mode:
Scale 623
Scale 623: Sycrian, Ian Ring Music TheorySycrianThis is the prime mode

Prime

The prime form of this scale is Scale 623

Scale 623Scale 623: Sycrian, Ian Ring Music TheorySycrian

Complement

The heptatonic modal family [2359, 3227, 3661, 1939, 3017, 889, 623] (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 2359 is 3475

Scale 3475Scale 3475: Kylian, Ian Ring Music TheoryKylian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2359 is chiral, and its enantiomorph is scale 3475

Scale 3475Scale 3475: Kylian, Ian Ring Music TheoryKylian

Transformations:

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

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 2357Scale 2357: Raga Sarasanana, Ian Ring Music TheoryRaga Sarasanana
Scale 2355Scale 2355: Raga Lalita, Ian Ring Music TheoryRaga Lalita
Scale 2363Scale 2363: Kataptian, Ian Ring Music TheoryKataptian
Scale 2367Scale 2367: Laryllic, Ian Ring Music TheoryLaryllic
Scale 2343Scale 2343: Tharimic, Ian Ring Music TheoryTharimic
Scale 2351Scale 2351: Gynian, Ian Ring Music TheoryGynian
Scale 2327Scale 2327: Epalimic, Ian Ring Music TheoryEpalimic
Scale 2391Scale 2391: Molian, Ian Ring Music TheoryMolian
Scale 2423Scale 2423, Ian Ring Music Theory
Scale 2487Scale 2487: Dothyllic, Ian Ring Music TheoryDothyllic
Scale 2103Scale 2103, Ian Ring Music Theory
Scale 2231Scale 2231: Macrian, Ian Ring Music TheoryMacrian
Scale 2615Scale 2615: Thoptian, Ian Ring Music TheoryThoptian
Scale 2871Scale 2871: Stanyllic, Ian Ring Music TheoryStanyllic
Scale 3383Scale 3383: Zoptyllic, Ian Ring Music TheoryZoptyllic
Scale 311Scale 311: Stagimic, Ian Ring Music TheoryStagimic
Scale 1335Scale 1335: Elephant Scale, Ian Ring Music TheoryElephant Scale

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.