The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 2621: "Ionogian"

Scale 2621: Ionogian, 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
Ionogian
Dozenal
Qenian

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

{0,2,3,4,5,9,11}

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-Z36

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: 1931

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.

3

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: 367

Generator

Indicates if the scale can be constructed using a generator, and an origin.

none

Deep Scale

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

no

Interval Structure

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

[2, 1, 1, 1, 4, 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.

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

Interval Spectrum

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

p4m3n4s4d4t2

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

Spectra Variation

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

3.143

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.

2.299

Polygon Perimeter

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

5.803

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

Heteromorphic Profile

Defined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where c is the number of contradictions, a is the number of ambiguities, and d is the number of differences. When c is zero, the scale is Proper. When a is also zero, the scale is Strictly Proper.

(38, 34, 100)

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 TriadsF{5,9,0}221.2
Minor Triadsdm{2,5,9}231.4
am{9,0,4}231.4
Diminished Triads{9,0,3}142
{11,2,5}142
Parsimonious Voice Leading Between Common Triads of Scale 2621. Created by Ian Ring ©2019 dm dm F F dm->F dm->b° am am F->am a°->am

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 VerticesF
Peripheral Verticesa°, b°

Modes

Modes are the rotational transformation of this scale. Scale 2621 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1679
Scale 1679: Kydian, Ian Ring Music TheoryKydian
3rd mode:
Scale 2887
Scale 2887: Gaptian, Ian Ring Music TheoryGaptian
4th mode:
Scale 3491
Scale 3491: Tharian, Ian Ring Music TheoryTharian
5th mode:
Scale 3793
Scale 3793: Aeopian, Ian Ring Music TheoryAeopian
6th mode:
Scale 493
Scale 493: Rygian, Ian Ring Music TheoryRygian
7th mode:
Scale 1147
Scale 1147: Epynian, Ian Ring Music TheoryEpynian

Prime

The prime form of this scale is Scale 367

Scale 367Scale 367: Aerodian, Ian Ring Music TheoryAerodian

Complement

The heptatonic modal family [2621, 1679, 2887, 3491, 3793, 493, 1147] (Forte: 7-Z36) is the complement of the pentatonic modal family [151, 737, 1801, 2123, 3109] (Forte: 5-Z36)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2621 is 1931

Scale 1931Scale 1931: Stogian, Ian Ring Music TheoryStogian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2621 is chiral, and its enantiomorph is scale 1931

Scale 1931Scale 1931: Stogian, Ian Ring Music TheoryStogian

Transformations:

In the abbreviation, the subscript number after "T" is the number of semitones of tranposition, "M" means the pitch class is multiplied by 5, and "I" means the result is inverted. Operation is an identical way to express the same thing; the syntax is <a,b> where each tone of the set x is transformed by the equation y = ax + b

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 2621       T0I <11,0> 1931
T1 <1,1> 1147      T1I <11,1> 3862
T2 <1,2> 2294      T2I <11,2> 3629
T3 <1,3> 493      T3I <11,3> 3163
T4 <1,4> 986      T4I <11,4> 2231
T5 <1,5> 1972      T5I <11,5> 367
T6 <1,6> 3944      T6I <11,6> 734
T7 <1,7> 3793      T7I <11,7> 1468
T8 <1,8> 3491      T8I <11,8> 2936
T9 <1,9> 2887      T9I <11,9> 1777
T10 <1,10> 1679      T10I <11,10> 3554
T11 <1,11> 3358      T11I <11,11> 3013
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1931      T0MI <7,0> 2621
T1M <5,1> 3862      T1MI <7,1> 1147
T2M <5,2> 3629      T2MI <7,2> 2294
T3M <5,3> 3163      T3MI <7,3> 493
T4M <5,4> 2231      T4MI <7,4> 986
T5M <5,5> 367      T5MI <7,5> 1972
T6M <5,6> 734      T6MI <7,6> 3944
T7M <5,7> 1468      T7MI <7,7> 3793
T8M <5,8> 2936      T8MI <7,8> 3491
T9M <5,9> 1777      T9MI <7,9> 2887
T10M <5,10> 3554      T10MI <7,10> 1679
T11M <5,11> 3013      T11MI <7,11> 3358

The transformations that map this set to itself are: T0, T0MI

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 2623Scale 2623: Aerylyllic, Ian Ring Music TheoryAerylyllic
Scale 2617Scale 2617: Pylimic, Ian Ring Music TheoryPylimic
Scale 2619Scale 2619: Ionyrian, Ian Ring Music TheoryIonyrian
Scale 2613Scale 2613: Raga Hamsa Vinodini, Ian Ring Music TheoryRaga Hamsa Vinodini
Scale 2605Scale 2605: Rylimic, Ian Ring Music TheoryRylimic
Scale 2589Scale 2589: Puvian, Ian Ring Music TheoryPuvian
Scale 2653Scale 2653: Sygian, Ian Ring Music TheorySygian
Scale 2685Scale 2685: Ionoryllic, Ian Ring Music TheoryIonoryllic
Scale 2749Scale 2749: Katagyllic, Ian Ring Music TheoryKatagyllic
Scale 2877Scale 2877: Phrylyllic, Ian Ring Music TheoryPhrylyllic
Scale 2109Scale 2109: Muvian, Ian Ring Music TheoryMuvian
Scale 2365Scale 2365: Sythian, Ian Ring Music TheorySythian
Scale 3133Scale 3133: Tosian, Ian Ring Music TheoryTosian
Scale 3645Scale 3645: Zycryllic, Ian Ring Music TheoryZycryllic
Scale 573Scale 573: Saptimic, Ian Ring Music TheorySaptimic
Scale 1597Scale 1597: Aeolodian, Ian Ring Music TheoryAeolodian

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. All other diagrams and visualizations are © Ian Ring. 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, and George Howlett for assistance with the Carnatic ragas.