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Scale 667: "Rodimic"

Scale 667: Rodimic, 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
Rodimic
Dozenal
Ecoian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

6 (hexatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

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

Forte Number

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

6-Z49

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.

[2]

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.

no

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

2 (dihemitonic)

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.

5

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, 2, 1, 3, 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.

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

Interval Spectrum

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

p2m3n4s2d2t2

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> = {4,6,8}
<4> = {7,8,9}
<5> = {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.

2.366

Polygon Perimeter

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

5.864

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.

[4]

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, 12, 57)

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}321.17
A{9,1,4}231.5
Minor Triadscm{0,3,7}231.5
am{9,0,4}321.17
Diminished Triadsc♯°{1,4,7}231.5
{9,0,3}231.5
Parsimonious Voice Leading Between Common Triads of Scale 667. Created by Ian Ring ©2019 cm cm C C cm->C cm->a° c#° c#° C->c#° am am C->am A A c#°->A a°->am am->A

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
Radius2
Self-Centeredno
Central VerticesC, am
Peripheral Verticescm, c♯°, a°, A

Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There are 2 ways that this hexatonic scale can be split into two common triads.


Minor: {0, 3, 7}
Major: {9, 1, 4}

Diminished: {1, 4, 7}
Diminished: {9, 0, 3}

Modes

Modes are the rotational transformation of this scale. Scale 667 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2381
Scale 2381: Takemitsu Linea Mode 1, Ian Ring Music TheoryTakemitsu Linea Mode 1
3rd mode:
Scale 1619
Scale 1619: Prometheus Neapolitan, Ian Ring Music TheoryPrometheus Neapolitan
4th mode:
Scale 2857
Scale 2857: Stythimic, Ian Ring Music TheoryStythimic
5th mode:
Scale 869
Scale 869: Kothimic, Ian Ring Music TheoryKothimic
6th mode:
Scale 1241
Scale 1241: Pygimic, Ian Ring Music TheoryPygimic

Prime

This is the prime form of this scale.

Complement

The hexatonic modal family [667, 2381, 1619, 2857, 869, 1241] (Forte: 6-Z49) is the complement of the hexatonic modal family [619, 857, 1427, 1613, 2357, 2761] (Forte: 6-Z28)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 667 is 2857

Scale 2857Scale 2857: Stythimic, Ian Ring Music TheoryStythimic

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> 667       T0I <11,0> 2857
T1 <1,1> 1334      T1I <11,1> 1619
T2 <1,2> 2668      T2I <11,2> 3238
T3 <1,3> 1241      T3I <11,3> 2381
T4 <1,4> 2482      T4I <11,4> 667
T5 <1,5> 869      T5I <11,5> 1334
T6 <1,6> 1738      T6I <11,6> 2668
T7 <1,7> 3476      T7I <11,7> 1241
T8 <1,8> 2857      T8I <11,8> 2482
T9 <1,9> 1619      T9I <11,9> 869
T10 <1,10> 3238      T10I <11,10> 1738
T11 <1,11> 2381      T11I <11,11> 3476
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2857      T0MI <7,0> 667
T1M <5,1> 1619      T1MI <7,1> 1334
T2M <5,2> 3238      T2MI <7,2> 2668
T3M <5,3> 2381      T3MI <7,3> 1241
T4M <5,4> 667       T4MI <7,4> 2482
T5M <5,5> 1334      T5MI <7,5> 869
T6M <5,6> 2668      T6MI <7,6> 1738
T7M <5,7> 1241      T7MI <7,7> 3476
T8M <5,8> 2482      T8MI <7,8> 2857
T9M <5,9> 869      T9MI <7,9> 1619
T10M <5,10> 1738      T10MI <7,10> 3238
T11M <5,11> 3476      T11MI <7,11> 2381

The transformations that map this set to itself are: T0, T4I, T4M, 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 665Scale 665: Raga Mohanangi, Ian Ring Music TheoryRaga Mohanangi
Scale 669Scale 669: Gycrimic, Ian Ring Music TheoryGycrimic
Scale 671Scale 671: Stycrian, Ian Ring Music TheoryStycrian
Scale 659Scale 659: Raga Rasika Ranjani, Ian Ring Music TheoryRaga Rasika Ranjani
Scale 663Scale 663: Phrynimic, Ian Ring Music TheoryPhrynimic
Scale 651Scale 651: Golitonic, Ian Ring Music TheoryGolitonic
Scale 683Scale 683: Stogimic, Ian Ring Music TheoryStogimic
Scale 699Scale 699: Aerothian, Ian Ring Music TheoryAerothian
Scale 731Scale 731: Alternating Heptamode, Ian Ring Music TheoryAlternating Heptamode
Scale 539Scale 539: Delian, Ian Ring Music TheoryDelian
Scale 603Scale 603: Aeolygimic, Ian Ring Music TheoryAeolygimic
Scale 795Scale 795: Aeologimic, Ian Ring Music TheoryAeologimic
Scale 923Scale 923: Ultraphrygian, Ian Ring Music TheoryUltraphrygian
Scale 155Scale 155: Bakian, Ian Ring Music TheoryBakian
Scale 411Scale 411: Lygimic, Ian Ring Music TheoryLygimic
Scale 1179Scale 1179: Sonimic, Ian Ring Music TheorySonimic
Scale 1691Scale 1691: Kathian, Ian Ring Music TheoryKathian
Scale 2715Scale 2715: Kynian, Ian Ring Music TheoryKynian

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.