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Scale 209: "All-Interval Tetrachord 4"

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

Western Modern: All-Interval Tetrachord 4

Dozenal: BIRian

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	{0,4,6,7}
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-Z29
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: 353
Hemitonia A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.	1 (unhemitonic)
Cohemitonia A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.	0 (ancohemitonic)
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.	no prime: 139
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.	[4, 2, 1, 5]
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.	<1, 1, 1, 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.	<0.333, 0.333, 0.25, 0.333, 0.333, 0.5>
Interval Spectrum The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.	pmnsdt
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> = {1,2,4,5} <2> = {3,6,9} <3> = {7,8,10,11}
Spectra Variation Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.	3.5
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.	1.366
Polygon Perimeter Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.	5.182
Myhill Property A scale has Myhill Property if the Distribution Spectra have exactly two specific intervals for every generic interval.	no
Centre of Gravity Distance When tones of a scale are imagined as physical objects of equal weight arranged around a unit circle, this is the distance from the center of the circle to the center of gravity for all the tones. A perfectly balanced scale has a CoG distance of zero.	0.353553
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.	(4, 0, 17)
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.	0.429
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.	0.056

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 Type	Triad^*	Pitch Classes	Degree	Eccentricity	Closeness Centrality
Major Triads	C	{0,4,7}	0	0	0

The following pitch classes are not present in any of the common triads: {6}

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.

Modes

Modes are the rotational transformation of this scale. Scale 209 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode: Scale 269				BOCian
3rd mode: Scale 1091				PEDian
4th mode: Scale 2593				PUXian

Prime

The prime form of this scale is Scale 139

Scale 139

All-Interval Tetrachord 2

Complement

The tetratonic modal family [209, 269, 1091, 2593] (Forte: 4-Z29) is the complement of the octatonic modal family [751, 1913, 1943, 2423, 3019, 3259, 3557, 3677] (Forte: 8-Z29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 209 is 353

Scale 353

CEBian

Interval Matrix

Each row is a generic interval, cells contain the specific size of each generic. Useful for identifying contradictions and ambiguities.

Contradictions (4)

Ambiguities(0)

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	`100010110000`
2	1	`100010110000`
3	1	`100010110000`
4	3	`(1[0][0][0])101(1[0][0][0])0`
5	3	`(1[0][0][0])101(1[0][0][0])0`

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 209 is chiral, and its enantiomorph is scale 353

Scale 353

CEBian

Center of Gravity

If tones of the scale are imagined as identical physical objects spaced around a unit circle, the center of gravity is the point where the scale is balanced.

Position with origin in the center	(0.091506, 0.341506)
Distance from Center	0.353553
Angle in degrees measured clockwise starting from the root.	165
Angle in cents 100 cents = 1 semitone.	550

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>	209	T₀I	<11,0>	353
T₁	<1,1>	418	T₁I	<11,1>	706
T₂	<1,2>	836	T₂I	<11,2>	1412
T₃	<1,3>	1672	T₃I	<11,3>	2824
T₄	<1,4>	3344	T₄I	<11,4>	1553
T₅	<1,5>	2593	T₅I	<11,5>	3106
T₆	<1,6>	1091	T₆I	<11,6>	2117
T₇	<1,7>	2182	T₇I	<11,7>	139
T₈	<1,8>	269	T₈I	<11,8>	278
T₉	<1,9>	538	T₉I	<11,9>	556
T₁₀	<1,10>	1076	T₁₀I	<11,10>	1112
T₁₁	<1,11>	2152	T₁₁I	<11,11>	2224
Abbrev	Operation	Result	Abbrev	Operation	Result
T₀M	<5,0>	2369	T₀MI	<7,0>	83
T₁M	<5,1>	643	T₁MI	<7,1>	166
T₂M	<5,2>	1286	T₂MI	<7,2>	332
T₃M	<5,3>	2572	T₃MI	<7,3>	664
T₄M	<5,4>	1049	T₄MI	<7,4>	1328
T₅M	<5,5>	2098	T₅MI	<7,5>	2656
T₆M	<5,6>	101	T₆MI	<7,6>	1217
T₇M	<5,7>	202	T₇MI	<7,7>	2434
T₈M	<5,8>	404	T₈MI	<7,8>	773
T₉M	<5,9>	808	T₉MI	<7,9>	1546
T₁₀M	<5,10>	1616	T₁₀MI	<7,10>	3092
T₁₁M	<5,11>	3232	T₁₁MI	<7,11>	2089

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 211				BISian
Scale 213				BITian
Scale 217				BIWian
Scale 193				Raga Ongkari
Scale 201				BEMian
Scale 225				BIBian
Scale 241				BILian
Scale 145				Major Triad
Scale 177				BEXian
Scale 81				Karen 3 Tone Type 6
Scale 337				Kmhmu 4 Tone Type 2
Scale 465				Zoditonic
Scale 721				Raga Dhavalashri
Scale 1233				Ionoditonic
Scale 2257				Lydian Pentatonic

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, and ©2023 Robert Bedwell, used with permission.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (DOI, Patent owner: Dokuz Eylül University, Used with Permission.

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 naming the Carnatic ragas. Thanks to Niels Verosky for collaborating on the Hierarchizability diagrams. Thanks to u/howaboot for inventing the Center of Gravity metrics.