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Scale 3117: "TIJian"

Scale 3117: TIJian, Ian Ring Music Theory

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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).

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

 {0,2,3,5,10,11}

Forte Number

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

 6-Z11

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: 1671

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.

 1 (uncohemitonic)

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 includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.

 6

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

 no
prime: 183

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.

 [2, 1, 2, 5, 1, 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, 3, 2, 3, 1>

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.

<0.6, 0.5, 0.6, 0, 0.6, 0.333>

Interval Spectrum

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

 p3m2n3s3d3t

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

Spectra Variation

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

 3.667

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.

5.485

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.

(27, 11, 59)

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.

0.415

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.

0.213

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 TriadsA♯{10,2,5}110.5
Diminished Triadsb°{11,2,5}110.5

The following pitch classes are not present in any of the common triads: {0,3}

Parsimonious Voice Leading Between Common Triads of Scale 3117. Created by Ian Ring ©2019 A# A# b° b° A#->b°

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 3117 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1803		Scale 1803: LAPian, Ian Ring Music TheoryLAPian
3rd mode:
Scale 2949		Scale 2949: SIKian, Ian Ring Music TheorySIKian
4th mode:
Scale 1761		Scale 1761: KUQian, Ian Ring Music TheoryKUQian
5th mode:
Scale 183		Scale 183: BEBian, Ian Ring Music TheoryBEBianThis is the prime mode
6th mode:
Scale 2139		Scale 2139: NAMian, Ian Ring Music TheoryNAMian

Prime

The prime form of this scale is Scale 183

Scale 183Scale 183: BEBian, Ian Ring Music TheoryBEBian

Complement

The hexatonic modal family [3117, 1803, 2949, 1761, 183, 2139] (Forte: 6-Z11) is the complement of the hexatonic modal family [303, 753, 1929, 2199, 3147, 3621] (Forte: 6-Z40)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3117 is 1671

Scale 1671Scale 1671: KEMian, Ian Ring Music TheoryKEMian

Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

kHierarchizabilityBreakdown PatternDiagram
111011010000113117k = 1h = 1
211011010000113117k = 2h = 1
311011010000113117k = 3h = 1
411011010000113117k = 4h = 1
511011010000113117k = 5h = 1

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3117 is chiral, and its enantiomorph is scale 1671

Scale 1671Scale 1671: KEMian, Ian Ring Music TheoryKEMian

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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 3117       T0I <11,0> 1671
T1 <1,1> 2139      T1I <11,1> 3342
T2 <1,2> 183      T2I <11,2> 2589
T3 <1,3> 366      T3I <11,3> 1083
T4 <1,4> 732      T4I <11,4> 2166
T5 <1,5> 1464      T5I <11,5> 237
T6 <1,6> 2928      T6I <11,6> 474
T7 <1,7> 1761      T7I <11,7> 948
T8 <1,8> 3522      T8I <11,8> 1896
T9 <1,9> 2949      T9I <11,9> 3792
T10 <1,10> 1803      T10I <11,10> 3489
T11 <1,11> 3606      T11I <11,11> 2883
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1167      T0MI <7,0> 3621
T1M <5,1> 2334      T1MI <7,1> 3147
T2M <5,2> 573      T2MI <7,2> 2199
T3M <5,3> 1146      T3MI <7,3> 303
T4M <5,4> 2292      T4MI <7,4> 606
T5M <5,5> 489      T5MI <7,5> 1212
T6M <5,6> 978      T6MI <7,6> 2424
T7M <5,7> 1956      T7MI <7,7> 753
T8M <5,8> 3912      T8MI <7,8> 1506
T9M <5,9> 3729      T9MI <7,9> 3012
T10M <5,10> 3363      T10MI <7,10> 1929
T11M <5,11> 2631      T11MI <7,11> 3858

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 3119Scale 3119: TIKian, Ian Ring Music TheoryTIKian
Scale 3113Scale 3113: TIGian, Ian Ring Music TheoryTIGian
Scale 3115Scale 3115: TIHian, Ian Ring Music TheoryTIHian
Scale 3109Scale 3109: TIDian, Ian Ring Music TheoryTIDian
Scale 3125Scale 3125: TONian, Ian Ring Music TheoryTONian
Scale 3133Scale 3133: TOSian, Ian Ring Music TheoryTOSian
Scale 3085Scale 3085: TEPian, Ian Ring Music TheoryTEPian
Scale 3101Scale 3101: TIYian, Ian Ring Music TheoryTIYian
Scale 3149Scale 3149: Phrycrimic, Ian Ring Music TheoryPhrycrimic
Scale 3181Scale 3181: Rolian, Ian Ring Music TheoryRolian
Scale 3245Scale 3245: Mela Varunapriya, Ian Ring Music TheoryMela Varunapriya
Scale 3373Scale 3373: Lodian, Ian Ring Music TheoryLodian
Scale 3629Scale 3629: Boptian, Ian Ring Music TheoryBoptian
Scale 2093Scale 2093: MULian, Ian Ring Music TheoryMULian
Scale 2605Scale 2605: Rylimic, Ian Ring Music TheoryRylimic
Scale 1069Scale 1069: GOQian, Ian Ring Music TheoryGOQian

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.