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Scale 2755: "Rivian"

Scale 2755: Rivian, 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




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


Forte Number

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


Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.


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.



A palindromic scale has the same pattern of intervals both ascending and descending.



A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

enantiomorph: 2155


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

3 (trihemitonic)


A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

1 (uncohemitonic)


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.



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.


Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

prime: 215


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


Deep Scale

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


Interval Structure

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

[1, 5, 1, 2, 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.

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

Interval Spectrum

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


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

Spectra Variation

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


Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.


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.


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.


Polygon Perimeter

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


Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.



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.


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.



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".


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.

(22, 14, 61)

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
Minor Triadsf♯m{6,9,1}110.5
Diminished Triadsf♯°{6,9,0}110.5

The following pitch classes are not present in any of the common triads: {7,11}

Parsimonious Voice Leading Between Common Triads of Scale 2755. Created by Ian Ring ©2019 f#° f#° f#m f#m f#°->f#m

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.



Modes are the rotational transformation of this scale. Scale 2755 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3425
Scale 3425: Vihian, Ian Ring Music TheoryVihian
3rd mode:
Scale 235
Scale 235: Bihian, Ian Ring Music TheoryBihian
4th mode:
Scale 2165
Scale 2165: Necian, Ian Ring Music TheoryNecian
5th mode:
Scale 1565
Scale 1565: Jozian, Ian Ring Music TheoryJozian
6th mode:
Scale 1415
Scale 1415: Impian, Ian Ring Music TheoryImpian


The prime form of this scale is Scale 215

Scale 215Scale 215: Bivian, Ian Ring Music TheoryBivian


The hexatonic modal family [2755, 3425, 235, 2165, 1565, 1415] (Forte: 6-Z12) is the complement of the hexatonic modal family [335, 965, 1265, 2215, 3155, 3625] (Forte: 6-Z41)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2755 is 2155

Scale 2155Scale 2155: Newian, Ian Ring Music TheoryNewian


Only scales that are chiral will have an enantiomorph. Scale 2755 is chiral, and its enantiomorph is scale 2155

Scale 2155Scale 2155: Newian, Ian Ring Music TheoryNewian


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> 2755       T0I <11,0> 2155
T1 <1,1> 1415      T1I <11,1> 215
T2 <1,2> 2830      T2I <11,2> 430
T3 <1,3> 1565      T3I <11,3> 860
T4 <1,4> 3130      T4I <11,4> 1720
T5 <1,5> 2165      T5I <11,5> 3440
T6 <1,6> 235      T6I <11,6> 2785
T7 <1,7> 470      T7I <11,7> 1475
T8 <1,8> 940      T8I <11,8> 2950
T9 <1,9> 1880      T9I <11,9> 1805
T10 <1,10> 3760      T10I <11,10> 3610
T11 <1,11> 3425      T11I <11,11> 3125
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2785      T0MI <7,0> 235
T1M <5,1> 1475      T1MI <7,1> 470
T2M <5,2> 2950      T2MI <7,2> 940
T3M <5,3> 1805      T3MI <7,3> 1880
T4M <5,4> 3610      T4MI <7,4> 3760
T5M <5,5> 3125      T5MI <7,5> 3425
T6M <5,6> 2155      T6MI <7,6> 2755
T7M <5,7> 215      T7MI <7,7> 1415
T8M <5,8> 430      T8MI <7,8> 2830
T9M <5,9> 860      T9MI <7,9> 1565
T10M <5,10> 1720      T10MI <7,10> 3130
T11M <5,11> 3440      T11MI <7,11> 2165

The transformations that map this set to itself are: T0, T6MI

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 2753Scale 2753: Ritian, Ian Ring Music TheoryRitian
Scale 2757Scale 2757: Raga Nishadi, Ian Ring Music TheoryRaga Nishadi
Scale 2759Scale 2759: Mela Pavani, Ian Ring Music TheoryMela Pavani
Scale 2763Scale 2763: Mela Suvarnangi, Ian Ring Music TheoryMela Suvarnangi
Scale 2771Scale 2771: Marva That, Ian Ring Music TheoryMarva That
Scale 2787Scale 2787: Zyrian, Ian Ring Music TheoryZyrian
Scale 2691Scale 2691: Rahian, Ian Ring Music TheoryRahian
Scale 2723Scale 2723: Raga Jivantika, Ian Ring Music TheoryRaga Jivantika
Scale 2627Scale 2627: Qerian, Ian Ring Music TheoryQerian
Scale 2883Scale 2883: Savian, Ian Ring Music TheorySavian
Scale 3011Scale 3011, Ian Ring Music Theory
Scale 2243Scale 2243: Noyian, Ian Ring Music TheoryNoyian
Scale 2499Scale 2499: Pirian, Ian Ring Music TheoryPirian
Scale 3267Scale 3267: Urfian, Ian Ring Music TheoryUrfian
Scale 3779Scale 3779, Ian Ring Music Theory
Scale 707Scale 707: Ehoian, Ian Ring Music TheoryEhoian
Scale 1731Scale 1731: Koxian, Ian Ring Music TheoryKoxian

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 ( 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.