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Scale 3427: "Zacrian"

Scale 3427: Zacrian, 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).

Common Names

Zeitler
Zacrian
Dozenal
VIJian

Analysis

Cardinality

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

 7 (heptatonic)

Pitch Class Set

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

 {0,1,5,6,8,10,11}

Forte Number

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

 7-14

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: 2263

Hemitonia

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

 4 (multihemitonic)

Cohemitonia

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

 2 (dicohemitonic)

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.

 2

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.

 7

Prime Form

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

 no
prime: 431

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.

 [1, 4, 1, 2, 2, 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.

 <4, 4, 3, 3, 5, 2>

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.

<0.5, 0.5, 0.25, 0, 0.75, 0.5>

Interval Spectrum

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

 p5m3n3s4d4t2

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

Spectra Variation

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

 2.857

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.

2.299

Polygon Perimeter

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

5.803

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.

(25, 38, 102)

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.

0.55

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.

0.19

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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 TriadsC♯{1,5,8}221.2
F♯{6,10,1}142
Minor Triadsfm{5,8,0}231.4
a♯m{10,1,5}231.4
Diminished Triadsf°{5,8,11}142
Parsimonious Voice Leading Between Common Triads of Scale 3427. Created by Ian Ring ©2019 C# C# fm fm C#->fm a#m a#m C#->a#m f° f° f°->fm F# F# F#->a#m

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
Radius2
Self-Centeredno
Central VerticesC♯
Peripheral Verticesf°, F♯

Modes

Modes are the rotational transformation of this scale. Scale 3427 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3761		Scale 3761: Raga Madhuri, Ian Ring Music TheoryRaga Madhuri
3rd mode:
Scale 491		Scale 491: Aeolyrian, Ian Ring Music TheoryAeolyrian
4th mode:
Scale 2293		Scale 2293: Gorian, Ian Ring Music TheoryGorian
5th mode:
Scale 1597		Scale 1597: Aeolodian, Ian Ring Music TheoryAeolodian
6th mode:
Scale 1423		Scale 1423: Doptian, Ian Ring Music TheoryDoptian
7th mode:
Scale 2759		Scale 2759: Mela Pavani, Ian Ring Music TheoryMela Pavani

Prime

The prime form of this scale is Scale 431

Scale 431Scale 431: Epyrian, Ian Ring Music TheoryEpyrian

Complement

The heptatonic modal family [3427, 3761, 491, 2293, 1597, 1423, 2759] (Forte: 7-14) is the complement of the pentatonic modal family [167, 901, 1249, 2131, 3113] (Forte: 5-14)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3427 is 2263

Scale 2263Scale 2263: Lycrian, Ian Ring Music TheoryLycrian

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
111100011010113427k = 1h = 1
211100011010113427k = 2h = 1
311100011010113427k = 3h = 1
411100011010113427k = 4h = 1
52(1)(1)000(1)(1)0(1)0(1)(1)3427k = 5h = 2

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3427 is chiral, and its enantiomorph is scale 2263

Scale 2263Scale 2263: Lycrian, Ian Ring Music TheoryLycrian

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> 3427       T0I <11,0> 2263
T1 <1,1> 2759      T1I <11,1> 431
T2 <1,2> 1423      T2I <11,2> 862
T3 <1,3> 2846      T3I <11,3> 1724
T4 <1,4> 1597      T4I <11,4> 3448
T5 <1,5> 3194      T5I <11,5> 2801
T6 <1,6> 2293      T6I <11,6> 1507
T7 <1,7> 491      T7I <11,7> 3014
T8 <1,8> 982      T8I <11,8> 1933
T9 <1,9> 1964      T9I <11,9> 3866
T10 <1,10> 3928      T10I <11,10> 3637
T11 <1,11> 3761      T11I <11,11> 3179
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 247      T0MI <7,0> 3553
T1M <5,1> 494      T1MI <7,1> 3011
T2M <5,2> 988      T2MI <7,2> 1927
T3M <5,3> 1976      T3MI <7,3> 3854
T4M <5,4> 3952      T4MI <7,4> 3613
T5M <5,5> 3809      T5MI <7,5> 3131
T6M <5,6> 3523      T6MI <7,6> 2167
T7M <5,7> 2951      T7MI <7,7> 239
T8M <5,8> 1807      T8MI <7,8> 478
T9M <5,9> 3614      T9MI <7,9> 956
T10M <5,10> 3133      T10MI <7,10> 1912
T11M <5,11> 2171      T11MI <7,11> 3824

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 3425Scale 3425: VIHian, Ian Ring Music TheoryVIHian
Scale 3429Scale 3429: Marian, Ian Ring Music TheoryMarian
Scale 3431Scale 3431: Zyptyllic, Ian Ring Music TheoryZyptyllic
Scale 3435Scale 3435: Prokofiev, Ian Ring Music TheoryProkofiev
Scale 3443Scale 3443: Verdi's Scala Enigmatica, Ian Ring Music TheoryVerdi's Scala Enigmatica
Scale 3395Scale 3395: VEPian, Ian Ring Music TheoryVEPian
Scale 3411Scale 3411: Enigmatic, Ian Ring Music TheoryEnigmatic
Scale 3363Scale 3363: Rogimic, Ian Ring Music TheoryRogimic
Scale 3491Scale 3491: Tharian, Ian Ring Music TheoryTharian
Scale 3555Scale 3555: Pylyllic, Ian Ring Music TheoryPylyllic
Scale 3171Scale 3171: Zythimic, Ian Ring Music TheoryZythimic
Scale 3299Scale 3299: Syptian, Ian Ring Music TheorySyptian
Scale 3683Scale 3683: Dycrian, Ian Ring Music TheoryDycrian
Scale 3939Scale 3939: Dogyllic, Ian Ring Music TheoryDogyllic
Scale 2403Scale 2403: Lycrimic, Ian Ring Music TheoryLycrimic
Scale 2915Scale 2915: Aeolydian, Ian Ring Music TheoryAeolydian
Scale 1379Scale 1379: Kycrimic, Ian Ring Music TheoryKycrimic

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.