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Scale 553: "Rothic 2"

Scale 553: Rothic 2, 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
Rothic 2
Dozenal
Ditian

Analysis

Cardinality

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

4 (tetratonic)

Pitch Class Set

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

{0,3,5,9}

Forte Number

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

4-27

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: 649

Hemitonia

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

0 (anhemitonic)

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.

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.

3

Prime Form

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

no
prime: 293

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.

[3, 2, 4, 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.

<0, 1, 2, 1, 1, 1>

Interval Spectrum

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

pmn2st

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

Spectra Variation

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

1.5

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.

1.866

Polygon Perimeter

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

5.56

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".

Strictly Proper

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.

(0, 0, 15)

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 TriadsF{5,9,0}110.5
Diminished Triads{9,0,3}110.5
Parsimonious Voice Leading Between Common Triads of Scale 553. Created by Ian Ring ©2019 F F F->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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 553 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 581
Scale 581: Eporic 2, Ian Ring Music TheoryEporic 2
3rd mode:
Scale 1169
Scale 1169: Raga Mahathi, Ian Ring Music TheoryRaga Mahathi
4th mode:
Scale 329
Scale 329: Mynic 2, Ian Ring Music TheoryMynic 2

Prime

The prime form of this scale is Scale 293

Scale 293Scale 293: Raga Haripriya, Ian Ring Music TheoryRaga Haripriya

Complement

The tetratonic modal family [553, 581, 1169, 329] (Forte: 4-27) is the complement of the octatonic modal family [1463, 1757, 1771, 1883, 2779, 2933, 2989, 3437] (Forte: 8-27)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 553 is 649

Scale 649Scale 649: Byptic, Ian Ring Music TheoryByptic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 553 is chiral, and its enantiomorph is scale 649

Scale 649Scale 649: Byptic, Ian Ring Music TheoryByptic

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> 553       T0I <11,0> 649
T1 <1,1> 1106      T1I <11,1> 1298
T2 <1,2> 2212      T2I <11,2> 2596
T3 <1,3> 329      T3I <11,3> 1097
T4 <1,4> 658      T4I <11,4> 2194
T5 <1,5> 1316      T5I <11,5> 293
T6 <1,6> 2632      T6I <11,6> 586
T7 <1,7> 1169      T7I <11,7> 1172
T8 <1,8> 2338      T8I <11,8> 2344
T9 <1,9> 581      T9I <11,9> 593
T10 <1,10> 1162      T10I <11,10> 1186
T11 <1,11> 2324      T11I <11,11> 2372
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 523      T0MI <7,0> 2569
T1M <5,1> 1046      T1MI <7,1> 1043
T2M <5,2> 2092      T2MI <7,2> 2086
T3M <5,3> 89      T3MI <7,3> 77
T4M <5,4> 178      T4MI <7,4> 154
T5M <5,5> 356      T5MI <7,5> 308
T6M <5,6> 712      T6MI <7,6> 616
T7M <5,7> 1424      T7MI <7,7> 1232
T8M <5,8> 2848      T8MI <7,8> 2464
T9M <5,9> 1601      T9MI <7,9> 833
T10M <5,10> 3202      T10MI <7,10> 1666
T11M <5,11> 2309      T11MI <7,11> 3332

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 555Scale 555: Aeolycritonic, Ian Ring Music TheoryAeolycritonic
Scale 557Scale 557: Raga Abhogi, Ian Ring Music TheoryRaga Abhogi
Scale 545Scale 545: Dewian, Ian Ring Music TheoryDewian
Scale 549Scale 549: Raga Bhavani, Ian Ring Music TheoryRaga Bhavani
Scale 561Scale 561: Phratic, Ian Ring Music TheoryPhratic
Scale 569Scale 569: Mothitonic, Ian Ring Music TheoryMothitonic
Scale 521Scale 521: Astian, Ian Ring Music TheoryAstian
Scale 537Scale 537: Atuian, Ian Ring Music TheoryAtuian
Scale 585Scale 585: Diminished Seventh, Ian Ring Music TheoryDiminished Seventh
Scale 617Scale 617: Katycritonic, Ian Ring Music TheoryKatycritonic
Scale 681Scale 681: Kyemyonjo, Ian Ring Music TheoryKyemyonjo
Scale 809Scale 809: Dogitonic, Ian Ring Music TheoryDogitonic
Scale 41Scale 41: Vietnamese Tritonic, Ian Ring Music TheoryVietnamese Tritonic
Scale 297Scale 297: Mynic, Ian Ring Music TheoryMynic
Scale 1065Scale 1065: Gonian, Ian Ring Music TheoryGonian
Scale 1577Scale 1577: Raga Chandrakauns (Kafi), Ian Ring Music TheoryRaga Chandrakauns (Kafi)
Scale 2601Scale 2601: Raga Chandrakauns, Ian Ring Music TheoryRaga Chandrakauns

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.