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Scale 71: "ALOian"

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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.	4 (tetratonic)
Pitch Class Set The tones in this scale, expressed as numbers from 0 to 11	{0,1,2,6}
Forte Number A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.	4-5
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: 3137
Hemitonia A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.	2 (dihemitonic)
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.	4
Prime Form Describes if this scale is in prime form, using the Starr/Rahn algorithm.	yes
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, 1, 4, 6]
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.	<2, 1, 0, 1, 1, 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.	<0.667, 0.333, 0, 0.333, 0.333, 0.5>
Interval Spectrum The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.	pmsd²t
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> = {1,4,6} <2> = {2,5,7,10} <3> = {6,8,11}
Spectra Variation Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.	4.5
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.	0.933
Polygon Perimeter Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.	4.767
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.	(6, 1, 16)
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
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.	0.111

Generator

This scale has no generator.

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.

Modes

Modes are the rotational transformation of this scale. Scale 71 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode: Scale 2083				MOFian
3rd mode: Scale 3089				TIRian
4th mode: Scale 449				CUJian

Prime

This is the prime form of this scale.

Complement

The tetratonic modal family [71, 2083, 3089, 449] (Forte: 4-5) is the complement of the octatonic modal family [479, 1991, 2287, 3043, 3191, 3569, 3643, 3869] (Forte: 8-5)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 71 is 3137

Scale 3137

TOVian

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.

k	Hierarchizability	Breakdown Pattern
1	1	`111000100000`
2	1	`111000100000`
3	1	`111000100000`
4	3	`11(1[0][0][0])(1[0][0][0])00`
5	3	`11(1[0][0][0])(1[0][0][0])00`

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 71 is chiral, and its enantiomorph is scale 3137

Scale 3137

TOVian

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>	71	T₀I	<11,0>	3137
T₁	<1,1>	142	T₁I	<11,1>	2179
T₂	<1,2>	284	T₂I	<11,2>	263
T₃	<1,3>	568	T₃I	<11,3>	526
T₄	<1,4>	1136	T₄I	<11,4>	1052
T₅	<1,5>	2272	T₅I	<11,5>	2104
T₆	<1,6>	449	T₆I	<11,6>	113
T₇	<1,7>	898	T₇I	<11,7>	226
T₈	<1,8>	1796	T₈I	<11,8>	452
T₉	<1,9>	3592	T₉I	<11,9>	904
T₁₀	<1,10>	3089	T₁₀I	<11,10>	1808
T₁₁	<1,11>	2083	T₁₁I	<11,11>	3616
Abbrev	Operation	Result	Abbrev	Operation	Result
T₀M	<5,0>	1121	T₀MI	<7,0>	197
T₁M	<5,1>	2242	T₁MI	<7,1>	394
T₂M	<5,2>	389	T₂MI	<7,2>	788
T₃M	<5,3>	778	T₃MI	<7,3>	1576
T₄M	<5,4>	1556	T₄MI	<7,4>	3152
T₅M	<5,5>	3112	T₅MI	<7,5>	2209
T₆M	<5,6>	2129	T₆MI	<7,6>	323
T₇M	<5,7>	163	T₇MI	<7,7>	646
T₈M	<5,8>	326	T₈MI	<7,8>	1292
T₉M	<5,9>	652	T₉MI	<7,9>	2584
T₁₀M	<5,10>	1304	T₁₀MI	<7,10>	1073
T₁₁M	<5,11>	2608	T₁₁MI	<7,11>	2146

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 69				DEZian
Scale 67				Viennese Trichord
Scale 75				ILOian
Scale 79				APPian
Scale 87				ASRian
Scale 103				APUian
Scale 7				Tritonic Chromatic
Scale 39				AFUian
Scale 135				ATTian
Scale 199				Raga Nabhomani
Scale 327				Syptitonic
Scale 583				Aeritonic
Scale 1095				Phrythitonic
Scale 2119				MUBian

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.

Scale 71: "ALOian"

Bracelet Diagram

Tonnetz Diagram

Analysis

Cardinality

Pitch Class Set

Proportional Saturation Vector

Polygon Perimeter

Heteromorphic Profile

Coherence Quotient

Sameness Quotient