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Scale 823: "Stodian"

Scale 823: Stodian, 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
Stodian
Dozenal
Favian

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

{0,1,2,4,5,8,9}

Forte Number

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

7-21

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: 3481

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.

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.

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.

yes

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.

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

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, 2, 4, 6, 4, 1>

Interval Spectrum

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

p4m6n4s2d4t

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}
<3> = {4,5,6,7}
<4> = {5,6,7,8}
<5> = {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.

2.433

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

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.

(7, 33, 90)

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}431.5
F{5,9,0}331.7
A{9,1,4}331.7
Minor Triadsc♯m{1,4,8}331.7
dm{2,5,9}242.1
fm{5,8,0}331.7
am{9,0,4}341.9
Augmented TriadsC+{0,4,8}341.9
C♯+{1,5,9}431.5
Diminished Triads{2,5,8}242.1
Parsimonious Voice Leading Between Common Triads of Scale 823. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m fm fm C+->fm am am C+->am C# C# c#m->C# A A c#m->A C#+ C#+ C#->C#+ C#->d° C#->fm dm dm C#+->dm F F C#+->F C#+->A d°->dm fm->F F->am am->A

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
Radius3
Self-Centeredno
Central Verticesc♯m, C♯, C♯+, fm, F, A
Peripheral VerticesC+, d°, dm, am

Modes

Modes are the rotational transformation of this scale. Scale 823 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2459
Scale 2459: Ionocrian, Ian Ring Music TheoryIonocrian
3rd mode:
Scale 3277
Scale 3277: Mela Nitimati, Ian Ring Music TheoryMela Nitimati
4th mode:
Scale 1843
Scale 1843: Ionygian, Ian Ring Music TheoryIonygian
5th mode:
Scale 2969
Scale 2969: Tholian, Ian Ring Music TheoryTholian
6th mode:
Scale 883
Scale 883: Ralian, Ian Ring Music TheoryRalian
7th mode:
Scale 2489
Scale 2489: Mela Gangeyabhusani, Ian Ring Music TheoryMela Gangeyabhusani

Prime

This is the prime form of this scale.

Complement

The heptatonic modal family [823, 2459, 3277, 1843, 2969, 883, 2489] (Forte: 7-21) is the complement of the pentatonic modal family [307, 787, 817, 2201, 2441] (Forte: 5-21)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 823 is 3481

Scale 3481Scale 3481: Katathian, Ian Ring Music TheoryKatathian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 823 is chiral, and its enantiomorph is scale 3481

Scale 3481Scale 3481: Katathian, Ian Ring Music TheoryKatathian

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> 823       T0I <11,0> 3481
T1 <1,1> 1646      T1I <11,1> 2867
T2 <1,2> 3292      T2I <11,2> 1639
T3 <1,3> 2489      T3I <11,3> 3278
T4 <1,4> 883      T4I <11,4> 2461
T5 <1,5> 1766      T5I <11,5> 827
T6 <1,6> 3532      T6I <11,6> 1654
T7 <1,7> 2969      T7I <11,7> 3308
T8 <1,8> 1843      T8I <11,8> 2521
T9 <1,9> 3686      T9I <11,9> 947
T10 <1,10> 3277      T10I <11,10> 1894
T11 <1,11> 2459      T11I <11,11> 3788
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1843      T0MI <7,0> 2461
T1M <5,1> 3686      T1MI <7,1> 827
T2M <5,2> 3277      T2MI <7,2> 1654
T3M <5,3> 2459      T3MI <7,3> 3308
T4M <5,4> 823       T4MI <7,4> 2521
T5M <5,5> 1646      T5MI <7,5> 947
T6M <5,6> 3292      T6MI <7,6> 1894
T7M <5,7> 2489      T7MI <7,7> 3788
T8M <5,8> 883      T8MI <7,8> 3481
T9M <5,9> 1766      T9MI <7,9> 2867
T10M <5,10> 3532      T10MI <7,10> 1639
T11M <5,11> 2969      T11MI <7,11> 3278

The transformations that map this set to itself are: T0, T4M

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 821Scale 821: Aeranimic, Ian Ring Music TheoryAeranimic
Scale 819Scale 819: Augmented Inverse, Ian Ring Music TheoryAugmented Inverse
Scale 827Scale 827: Mixolocrian, Ian Ring Music TheoryMixolocrian
Scale 831Scale 831: Rodyllic, Ian Ring Music TheoryRodyllic
Scale 807Scale 807: Raga Suddha Mukhari, Ian Ring Music TheoryRaga Suddha Mukhari
Scale 815Scale 815: Bolian, Ian Ring Music TheoryBolian
Scale 791Scale 791: Aeoloptimic, Ian Ring Music TheoryAeoloptimic
Scale 855Scale 855: Porian, Ian Ring Music TheoryPorian
Scale 887Scale 887: Sathyllic, Ian Ring Music TheorySathyllic
Scale 951Scale 951: Thogyllic, Ian Ring Music TheoryThogyllic
Scale 567Scale 567: Aeoladimic, Ian Ring Music TheoryAeoladimic
Scale 695Scale 695: Sarian, Ian Ring Music TheorySarian
Scale 311Scale 311: Stagimic, Ian Ring Music TheoryStagimic
Scale 1335Scale 1335: Elephant Scale, Ian Ring Music TheoryElephant Scale
Scale 1847Scale 1847: Thacryllic, Ian Ring Music TheoryThacryllic
Scale 2871Scale 2871: Stanyllic, Ian Ring Music TheoryStanyllic

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.