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Scale 3143: "Polimic"

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: Polimic

Dozenal: Toyian

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	{0,1,2,6,10,11}
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-Z37
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.	[0]
Palindromicity A palindromic scale has the same pattern of intervals both ascending and descending.	yes
Chirality A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.	no
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.	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.	4
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.	5
Prime Form Describes if this scale is in prime form, using the Rahn/Ring formula.	no prime: 287
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.	[1, 1, 4, 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.	<4, 3, 2, 3, 2, 1>
Interval Spectrum The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.	p²m³n²s³d⁴t
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> = {1,4} <2> = {2,5,8} <3> = {3,6,9} <4> = {4,7,10} <5> = {8,11}
Spectra Variation Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.	4
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.	5.535
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.	[0]
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.	(41, 4, 50)

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 Type	Triad^*	Pitch Classes	Degree	Eccentricity	Closeness Centrality
Major Triads	F♯	{6,10,1}	1	2	1
Minor Triads	bm	{11,2,6}	1	2	1
Augmented Triads	D+	{2,6,10}	2	1	0.67

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

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.

Diameter	2
Radius	1
Self-Centered	no
Central Vertices	D+
Peripheral Vertices	F♯, bm

Modes

Modes are the rotational transformation of this scale. Scale 3143 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode: Scale 3619	Thanimic
3rd mode: Scale 3857	Ponimic
4th mode: Scale 497	Kadimic
5th mode: Scale 287	Gynimic	This is the prime mode
6th mode: Scale 2191	Thydimic

Prime

The prime form of this scale is Scale 287

Scale 287

Gynimic

Complement

The hexatonic modal family [3143, 3619, 3857, 497, 287, 2191] (Forte: 6-Z37) is the complement of the hexatonic modal family [119, 1799, 2107, 2947, 3101, 3521] (Forte: 6-Z4)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3143 is itself, because it is a palindromic scale!

Scale 3143

Polimic

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>	3143	T₀I	<11,0>	3143
T₁	<1,1>	2191	T₁I	<11,1>	2191
T₂	<1,2>	287	T₂I	<11,2>	287
T₃	<1,3>	574	T₃I	<11,3>	574
T₄	<1,4>	1148	T₄I	<11,4>	1148
T₅	<1,5>	2296	T₅I	<11,5>	2296
T₆	<1,6>	497	T₆I	<11,6>	497
T₇	<1,7>	994	T₇I	<11,7>	994
T₈	<1,8>	1988	T₈I	<11,8>	1988
T₉	<1,9>	3976	T₉I	<11,9>	3976
T₁₀	<1,10>	3857	T₁₀I	<11,10>	3857
T₁₁	<1,11>	3619	T₁₁I	<11,11>	3619
Abbrev	Operation	Result	Abbrev	Operation	Result
T₀M	<5,0>	1253	T₀MI	<7,0>	1253
T₁M	<5,1>	2506	T₁MI	<7,1>	2506
T₂M	<5,2>	917	T₂MI	<7,2>	917
T₃M	<5,3>	1834	T₃MI	<7,3>	1834
T₄M	<5,4>	3668	T₄MI	<7,4>	3668
T₅M	<5,5>	3241	T₅MI	<7,5>	3241
T₆M	<5,6>	2387	T₆MI	<7,6>	2387
T₇M	<5,7>	679	T₇MI	<7,7>	679
T₈M	<5,8>	1358	T₈MI	<7,8>	1358
T₉M	<5,9>	2716	T₉MI	<7,9>	2716
T₁₀M	<5,10>	1337	T₁₀MI	<7,10>	1337
T₁₁M	<5,11>	2674	T₁₁MI	<7,11>	2674

The transformations that map this set to itself are: T₀, T₀I

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 3141				Kanitonic
Scale 3139				Towian
Scale 3147				Ryrimic
Scale 3151				Pacrian
Scale 3159				Stocrian
Scale 3175				Eponian
Scale 3079				Pentatonic Chromatic 3
Scale 3111				Tifian
Scale 3207				Ucoian
Scale 3271				Mela Raghupriya
Scale 3399				Zonian
Scale 3655				Mathian
Scale 2119				Mubian
Scale 2631				Macrimic
Scale 1095				Phrythitonic

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.