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Scale 731: "Alternating Heptamode"

Scale 731: Alternating Heptamode, 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

Western Modern
Alternating Heptamode
Western
Hungarian Major 2nd Mode
Zeitler
Ionorian
Dozenal
EMOian

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

 {0,1,3,4,6,7,9}

Forte Number

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

 7-31

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: 2921

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.

 0 (ancohemitonic)

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 Starr/Rahn algorithm.

 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, an indicator of maximum hierarchization.

 no

Interval Structure

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

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

 <3, 3, 6, 3, 3, 3>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.25, 0.25, 1, 0, 0.25, 1>

Interval Spectrum

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

 p3m3n6s3d3t3

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

Spectra Variation

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

 1.714

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 Distribution Spectra have 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.

(4, 27, 84)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.779

Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.333

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.

Generator

This scale has no generator.

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{0,4,7}331.7
A{9,1,4}331.7
Minor Triadscm{0,3,7}331.8
f♯m{6,9,1}331.8
am{9,0,4}431.6
Diminished Triadsc°{0,3,6}232
c♯°{1,4,7}232
d♯°{3,6,9}232
f♯°{6,9,0}231.9
a°{9,0,3}231.9
Parsimonious Voice Leading Between Common Triads of Scale 731. Created by Ian Ring ©2019 c° c° cm cm c°->cm d#° d#° c°->d#° C C cm->C a° a° cm->a° c#° c#° C->c#° am am C->am A A c#°->A f#m f#m d#°->f#m f#° f#° f#°->f#m f#°->am f#m->A a°->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.

Diameter3
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 731 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2413		Scale 2413: Harmonic Minor Flat 5, Ian Ring Music TheoryHarmonic Minor Flat 5
3rd mode:
Scale 1627		Scale 1627: Hungarian Major 4th Mode, Ian Ring Music TheoryHungarian Major 4th Mode
4th mode:
Scale 2861		Scale 2861: Hungarian Major 5th Mode, Ian Ring Music TheoryHungarian Major 5th Mode
5th mode:
Scale 1739		Scale 1739: Mela Sadvidhamargini, Ian Ring Music TheoryMela Sadvidhamargini
6th mode:
Scale 2917		Scale 2917: Nohkan Flute Scale, Ian Ring Music TheoryNohkan Flute Scale
7th mode:
Scale 1753		Scale 1753: Hungarian Major, Ian Ring Music TheoryHungarian Major

Prime

This is the prime form of this scale.

Complement

The heptatonic modal family [731, 2413, 1627, 2861, 1739, 2917, 1753] (Forte: 7-31) is the complement of the pentatonic modal family [587, 601, 713, 1609, 2341] (Forte: 5-31)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 731 is 2921

Scale 2921Scale 2921: Pogian, Ian Ring Music TheoryPogian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 731 is chiral, and its enantiomorph is scale 2921

Scale 2921Scale 2921: Pogian, Ian Ring Music TheoryPogian

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> 731       T0I <11,0> 2921
T1 <1,1> 1462      T1I <11,1> 1747
T2 <1,2> 2924      T2I <11,2> 3494
T3 <1,3> 1753      T3I <11,3> 2893
T4 <1,4> 3506      T4I <11,4> 1691
T5 <1,5> 2917      T5I <11,5> 3382
T6 <1,6> 1739      T6I <11,6> 2669
T7 <1,7> 3478      T7I <11,7> 1243
T8 <1,8> 2861      T8I <11,8> 2486
T9 <1,9> 1627      T9I <11,9> 877
T10 <1,10> 3254      T10I <11,10> 1754
T11 <1,11> 2413      T11I <11,11> 3508
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2921      T0MI <7,0> 731
T1M <5,1> 1747      T1MI <7,1> 1462
T2M <5,2> 3494      T2MI <7,2> 2924
T3M <5,3> 2893      T3MI <7,3> 1753
T4M <5,4> 1691      T4MI <7,4> 3506
T5M <5,5> 3382      T5MI <7,5> 2917
T6M <5,6> 2669      T6MI <7,6> 1739
T7M <5,7> 1243      T7MI <7,7> 3478
T8M <5,8> 2486      T8MI <7,8> 2861
T9M <5,9> 877      T9MI <7,9> 1627
T10M <5,10> 1754      T10MI <7,10> 3254
T11M <5,11> 3508      T11MI <7,11> 2413

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 729Scale 729: Stygimic, Ian Ring Music TheoryStygimic
Scale 733Scale 733: Donian, Ian Ring Music TheoryDonian
Scale 735Scale 735: Sylyllic, Ian Ring Music TheorySylyllic
Scale 723Scale 723: Ionadimic, Ian Ring Music TheoryIonadimic
Scale 727Scale 727: Phradian, Ian Ring Music TheoryPhradian
Scale 715Scale 715: T4 Prime Mode, Ian Ring Music TheoryT4 Prime Mode
Scale 747Scale 747: Lynian, Ian Ring Music TheoryLynian
Scale 763Scale 763: Doryllic, Ian Ring Music TheoryDoryllic
Scale 667Scale 667: Rodimic, Ian Ring Music TheoryRodimic
Scale 699Scale 699: Aerothian, Ian Ring Music TheoryAerothian
Scale 603Scale 603: Aeolygimic, Ian Ring Music TheoryAeolygimic
Scale 859Scale 859: Ultralocrian, Ian Ring Music TheoryUltralocrian
Scale 987Scale 987: Aeraptyllic, Ian Ring Music TheoryAeraptyllic
Scale 219Scale 219: Istrian, Ian Ring Music TheoryIstrian
Scale 475Scale 475: Aeolygian, Ian Ring Music TheoryAeolygian
Scale 1243Scale 1243: Epylian, Ian Ring Music TheoryEpylian
Scale 1755Scale 1755: Octatonic, Ian Ring Music TheoryOctatonic
Scale 2779Scale 2779: Shostakovich, Ian Ring Music TheoryShostakovich

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.