The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 3173: "Zarimic"

Scale 3173: Zarimic, 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
Zarimic
Dozenal
TURian

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

 {0,2,5,6,10,11}

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-Z17

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: 1223

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: 407

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, 3, 1, 4, 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, 2, 2, 3, 3, 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.

<0.6, 0.333, 0.4, 0.25, 0.6, 0.667>

Interval Spectrum

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

 p3m3n2s2d3t2

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

Spectra Variation

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

 2.667

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.

2.116

Polygon Perimeter

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

5.699

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.

(12, 10, 57)

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.

0.662

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.

0.24

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}221
Minor Triadsbm{11,2,6}221
Augmented TriadsD+{2,6,10}221
Diminished Triadsb°{11,2,5}221

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

Parsimonious Voice Leading Between Common Triads of Scale 3173. Created by Ian Ring ©2019 D+ D+ A# A# D+->A# bm bm D+->bm b° b° A#->b° b°->bm

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.

Diameter2
Radius2
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3173 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1817		Scale 1817: Phrythimic, Ian Ring Music TheoryPhrythimic
3rd mode:
Scale 739		Scale 739: Rorimic, Ian Ring Music TheoryRorimic
4th mode:
Scale 2417		Scale 2417: Kanimic, Ian Ring Music TheoryKanimic
5th mode:
Scale 407		Scale 407: All-Trichord Hexachord, Ian Ring Music TheoryAll-Trichord HexachordThis is the prime mode
6th mode:
Scale 2251		Scale 2251: Zodimic, Ian Ring Music TheoryZodimic

Prime

The prime form of this scale is Scale 407

Scale 407Scale 407: All-Trichord Hexachord, Ian Ring Music TheoryAll-Trichord Hexachord

Complement

The hexatonic modal family [3173, 1817, 739, 2417, 407, 2251] (Forte: 6-Z17) is the complement of the hexatonic modal family [359, 907, 1649, 2227, 2501, 3161] (Forte: 6-Z43)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3173 is 1223

Scale 1223Scale 1223: Phryptimic, Ian Ring Music TheoryPhryptimic

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
111010011000113173k = 1h = 1
211010011000113173k = 2h = 1
311010011000113173k = 3h = 1
42101(0011)0(0011)3173k = 4h = 2
52101(0011)0(0011)3173k = 5h = 2

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3173 is chiral, and its enantiomorph is scale 1223

Scale 1223Scale 1223: Phryptimic, Ian Ring Music TheoryPhryptimic

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> 3173       T0I <11,0> 1223
T1 <1,1> 2251      T1I <11,1> 2446
T2 <1,2> 407      T2I <11,2> 797
T3 <1,3> 814      T3I <11,3> 1594
T4 <1,4> 1628      T4I <11,4> 3188
T5 <1,5> 3256      T5I <11,5> 2281
T6 <1,6> 2417      T6I <11,6> 467
T7 <1,7> 739      T7I <11,7> 934
T8 <1,8> 1478      T8I <11,8> 1868
T9 <1,9> 2956      T9I <11,9> 3736
T10 <1,10> 1817      T10I <11,10> 3377
T11 <1,11> 3634      T11I <11,11> 2659
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1223      T0MI <7,0> 3173
T1M <5,1> 2446      T1MI <7,1> 2251
T2M <5,2> 797      T2MI <7,2> 407
T3M <5,3> 1594      T3MI <7,3> 814
T4M <5,4> 3188      T4MI <7,4> 1628
T5M <5,5> 2281      T5MI <7,5> 3256
T6M <5,6> 467      T6MI <7,6> 2417
T7M <5,7> 934      T7MI <7,7> 739
T8M <5,8> 1868      T8MI <7,8> 1478
T9M <5,9> 3736      T9MI <7,9> 2956
T10M <5,10> 3377      T10MI <7,10> 1817
T11M <5,11> 2659      T11MI <7,11> 3634

The transformations that map this set to itself are: T0, T0MI

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 3175Scale 3175: Eponian, Ian Ring Music TheoryEponian
Scale 3169Scale 3169: TUPian, Ian Ring Music TheoryTUPian
Scale 3171Scale 3171: Zythimic, Ian Ring Music TheoryZythimic
Scale 3177Scale 3177: Rothimic, Ian Ring Music TheoryRothimic
Scale 3181Scale 3181: Rolian, Ian Ring Music TheoryRolian
Scale 3189Scale 3189: Aeolonian, Ian Ring Music TheoryAeolonian
Scale 3141Scale 3141: Kanitonic, Ian Ring Music TheoryKanitonic
Scale 3157Scale 3157: Zyptimic, Ian Ring Music TheoryZyptimic
Scale 3109Scale 3109: TIDian, Ian Ring Music TheoryTIDian
Scale 3237Scale 3237: Raga Brindabani Sarang, Ian Ring Music TheoryRaga Brindabani Sarang
Scale 3301Scale 3301: Chromatic Mixolydian Inverse, Ian Ring Music TheoryChromatic Mixolydian Inverse
Scale 3429Scale 3429: Marian, Ian Ring Music TheoryMarian
Scale 3685Scale 3685: Kodian, Ian Ring Music TheoryKodian
Scale 2149Scale 2149: NASian, Ian Ring Music TheoryNASian
Scale 2661Scale 2661: Stydimic, Ian Ring Music TheoryStydimic
Scale 1125Scale 1125: Ionaritonic, Ian Ring Music TheoryIonaritonic

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.