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Scale 3373: "Lodian"

Scale 3373: Lodian, 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
Lodian
Dozenal
Vebian

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

{0,2,3,5,8,10,11}

Forte Number

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

7-25

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: 1687

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.

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: 733

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, 2, 3, 2, 1, 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, 5, 3, 4, 2>

Interval Spectrum

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

p4m3n5s4d3t2

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

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.

(19, 28, 92)

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 TriadsG♯{8,0,3}231.75
A♯{10,2,5}231.88
Minor Triadsfm{5,8,0}331.63
g♯m{8,11,3}331.63
Diminished Triads{2,5,8}231.75
{5,8,11}231.75
g♯°{8,11,2}231.75
{11,2,5}231.88
Parsimonious Voice Leading Between Common Triads of Scale 3373. Created by Ian Ring ©2019 fm fm d°->fm A# A# d°->A# f°->fm g#m g#m f°->g#m G# G# fm->G# g#° g#° g#°->g#m g#°->b° g#m->G# A#->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 3373 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1867
Scale 1867: Solian, Ian Ring Music TheorySolian
3rd mode:
Scale 2981
Scale 2981: Ionolian, Ian Ring Music TheoryIonolian
4th mode:
Scale 1769
Scale 1769: Blues Heptatonic II, Ian Ring Music TheoryBlues Heptatonic II
5th mode:
Scale 733
Scale 733: Donian, Ian Ring Music TheoryDonianThis is the prime mode
6th mode:
Scale 1207
Scale 1207: Aeoloptian, Ian Ring Music TheoryAeoloptian
7th mode:
Scale 2651
Scale 2651: Panian, Ian Ring Music TheoryPanian

Prime

The prime form of this scale is Scale 733

Scale 733Scale 733: Donian, Ian Ring Music TheoryDonian

Complement

The heptatonic modal family [3373, 1867, 2981, 1769, 733, 1207, 2651] (Forte: 7-25) is the complement of the pentatonic modal family [301, 721, 1099, 1673, 2597] (Forte: 5-25)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3373 is 1687

Scale 1687Scale 1687: Phralian, Ian Ring Music TheoryPhralian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3373 is chiral, and its enantiomorph is scale 1687

Scale 1687Scale 1687: Phralian, Ian Ring Music TheoryPhralian

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> 3373       T0I <11,0> 1687
T1 <1,1> 2651      T1I <11,1> 3374
T2 <1,2> 1207      T2I <11,2> 2653
T3 <1,3> 2414      T3I <11,3> 1211
T4 <1,4> 733      T4I <11,4> 2422
T5 <1,5> 1466      T5I <11,5> 749
T6 <1,6> 2932      T6I <11,6> 1498
T7 <1,7> 1769      T7I <11,7> 2996
T8 <1,8> 3538      T8I <11,8> 1897
T9 <1,9> 2981      T9I <11,9> 3794
T10 <1,10> 1867      T10I <11,10> 3493
T11 <1,11> 3734      T11I <11,11> 2891
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1183      T0MI <7,0> 3877
T1M <5,1> 2366      T1MI <7,1> 3659
T2M <5,2> 637      T2MI <7,2> 3223
T3M <5,3> 1274      T3MI <7,3> 2351
T4M <5,4> 2548      T4MI <7,4> 607
T5M <5,5> 1001      T5MI <7,5> 1214
T6M <5,6> 2002      T6MI <7,6> 2428
T7M <5,7> 4004      T7MI <7,7> 761
T8M <5,8> 3913      T8MI <7,8> 1522
T9M <5,9> 3731      T9MI <7,9> 3044
T10M <5,10> 3367      T10MI <7,10> 1993
T11M <5,11> 2639      T11MI <7,11> 3986

The transformations that map this set to itself are: T0

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 3375Scale 3375: Vecian, Ian Ring Music TheoryVecian
Scale 3369Scale 3369: Mixolimic, Ian Ring Music TheoryMixolimic
Scale 3371Scale 3371: Aeolylian, Ian Ring Music TheoryAeolylian
Scale 3365Scale 3365: Katolimic, Ian Ring Music TheoryKatolimic
Scale 3381Scale 3381: Katanian, Ian Ring Music TheoryKatanian
Scale 3389Scale 3389: Socryllic, Ian Ring Music TheorySocryllic
Scale 3341Scale 3341: Vahian, Ian Ring Music TheoryVahian
Scale 3357Scale 3357: Phrodian, Ian Ring Music TheoryPhrodian
Scale 3405Scale 3405: Stynian, Ian Ring Music TheoryStynian
Scale 3437Scale 3437: Vopian, Ian Ring Music TheoryVopian
Scale 3501Scale 3501: Maqam Nahawand, Ian Ring Music TheoryMaqam Nahawand
Scale 3117Scale 3117: Tijian, Ian Ring Music TheoryTijian
Scale 3245Scale 3245: Mela Varunapriya, Ian Ring Music TheoryMela Varunapriya
Scale 3629Scale 3629: Boptian, Ian Ring Music TheoryBoptian
Scale 3885Scale 3885: Styryllic, Ian Ring Music TheoryStyryllic
Scale 2349Scale 2349: Raga Ghantana, Ian Ring Music TheoryRaga Ghantana
Scale 2861Scale 2861: Katothian, Ian Ring Music TheoryKatothian
Scale 1325Scale 1325: Phradimic, Ian Ring Music TheoryPhradimic

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.