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Scale 1997: "Raga Cintamani"

Scale 1997: Raga Cintamani, 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

Carnatic
Raga Cintamani
Dozenal
Median
Zeitler
Staryllic

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

{0,2,3,6,7,8,9,10}

Forte Number

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

8-Z15

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: 1661

Hemitonia

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

5 (multihemitonic)

Cohemitonia

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

3 (tricohemitonic)

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.

7

Prime Form

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

no
prime: 863

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, 3, 1, 1, 1, 1, 2]

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.

<5, 5, 5, 5, 5, 3>

Interval Spectrum

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

p5m5n5s5d5t3

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

Spectra Variation

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

2.25

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.

2.616

Polygon Perimeter

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

6.002

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.

(30, 59, 140)

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 TriadsD{2,6,9}342
D♯{3,7,10}341.91
G♯{8,0,3}242.27
Minor Triadscm{0,3,7}342
d♯m{3,6,10}441.82
gm{7,10,2}242.18
Augmented TriadsD+{2,6,10}341.91
Diminished Triads{0,3,6}242.09
d♯°{3,6,9}242.09
f♯°{6,9,0}242.27
{9,0,3}242.36
Parsimonious Voice Leading Between Common Triads of Scale 1997. Created by Ian Ring ©2019 cm cm c°->cm d#m d#m c°->d#m D# D# cm->D# G# G# cm->G# D D D+ D+ D->D+ d#° d#° D->d#° f#° f#° D->f#° D+->d#m gm gm D+->gm d#°->d#m d#m->D# D#->gm f#°->a° G#->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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1997 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1523
Scale 1523: Zothyllic, Ian Ring Music TheoryZothyllic
3rd mode:
Scale 2809
Scale 2809: Gythyllic, Ian Ring Music TheoryGythyllic
4th mode:
Scale 863
Scale 863: Pyryllic, Ian Ring Music TheoryPyryllicThis is the prime mode
5th mode:
Scale 2479
Scale 2479: Harmonic and Neapolitan Minor Mixed, Ian Ring Music TheoryHarmonic and Neapolitan Minor Mixed
6th mode:
Scale 3287
Scale 3287: Phrathyllic, Ian Ring Music TheoryPhrathyllic
7th mode:
Scale 3691
Scale 3691: Badyllic, Ian Ring Music TheoryBadyllic
8th mode:
Scale 3893
Scale 3893: Phrocryllic, Ian Ring Music TheoryPhrocryllic

Prime

The prime form of this scale is Scale 863

Scale 863Scale 863: Pyryllic, Ian Ring Music TheoryPyryllic

Complement

The octatonic modal family [1997, 1523, 2809, 863, 2479, 3287, 3691, 3893] (Forte: 8-Z15) is the complement of the tetratonic modal family [83, 773, 1217, 2089] (Forte: 4-Z15)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1997 is 1661

Scale 1661Scale 1661: Gonyllic, Ian Ring Music TheoryGonyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1997 is chiral, and its enantiomorph is scale 1661

Scale 1661Scale 1661: Gonyllic, Ian Ring Music TheoryGonyllic

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> 1997       T0I <11,0> 1661
T1 <1,1> 3994      T1I <11,1> 3322
T2 <1,2> 3893      T2I <11,2> 2549
T3 <1,3> 3691      T3I <11,3> 1003
T4 <1,4> 3287      T4I <11,4> 2006
T5 <1,5> 2479      T5I <11,5> 4012
T6 <1,6> 863      T6I <11,6> 3929
T7 <1,7> 1726      T7I <11,7> 3763
T8 <1,8> 3452      T8I <11,8> 3431
T9 <1,9> 2809      T9I <11,9> 2767
T10 <1,10> 1523      T10I <11,10> 1439
T11 <1,11> 3046      T11I <11,11> 2878
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3677      T0MI <7,0> 1871
T1M <5,1> 3259      T1MI <7,1> 3742
T2M <5,2> 2423      T2MI <7,2> 3389
T3M <5,3> 751      T3MI <7,3> 2683
T4M <5,4> 1502      T4MI <7,4> 1271
T5M <5,5> 3004      T5MI <7,5> 2542
T6M <5,6> 1913      T6MI <7,6> 989
T7M <5,7> 3826      T7MI <7,7> 1978
T8M <5,8> 3557      T8MI <7,8> 3956
T9M <5,9> 3019      T9MI <7,9> 3817
T10M <5,10> 1943      T10MI <7,10> 3539
T11M <5,11> 3886      T11MI <7,11> 2983

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 1999Scale 1999: Zacrygic, Ian Ring Music TheoryZacrygic
Scale 1993Scale 1993: Katoptian, Ian Ring Music TheoryKatoptian
Scale 1995Scale 1995: Sideways Scale, Ian Ring Music TheorySideways Scale
Scale 1989Scale 1989: Dydian, Ian Ring Music TheoryDydian
Scale 2005Scale 2005: Gygyllic, Ian Ring Music TheoryGygyllic
Scale 2013Scale 2013: Mocrygic, Ian Ring Music TheoryMocrygic
Scale 2029Scale 2029: Kiourdi, Ian Ring Music TheoryKiourdi
Scale 1933Scale 1933: Mocrian, Ian Ring Music TheoryMocrian
Scale 1965Scale 1965: Raga Mukhari, Ian Ring Music TheoryRaga Mukhari
Scale 1869Scale 1869: Katyrian, Ian Ring Music TheoryKatyrian
Scale 1741Scale 1741: Lydian Diminished, Ian Ring Music TheoryLydian Diminished
Scale 1485Scale 1485: Minor Romani, Ian Ring Music TheoryMinor Romani
Scale 973Scale 973: Mela Syamalangi, Ian Ring Music TheoryMela Syamalangi
Scale 3021Scale 3021: Stodyllic, Ian Ring Music TheoryStodyllic
Scale 4045Scale 4045: Gyptygic, Ian Ring Music TheoryGyptygic

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.