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Scale 1871: "Aeolyllic"

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: Aeolyllic

Dozenal: LIFian

Analysis

Cardinality Cardinality is the count of how many pitches are in the scale.	8 (octatonic)
Pitch Class Set The tones in this scale, expressed as numbers from 0 to 11	{0,1,2,3,6,8,9,10}
Forte Number A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.	8-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: 3677
Hemitonia A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.	5 (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.	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.	8
Prime Form Describes if this scale is in prime form, using the Starr/Rahn algorithm.	no prime: 751
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, 1, 3, 2, 1, 1, 2]
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.	<5, 5, 5, 5, 5, 3>
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.	p⁵m⁵n⁵s⁵d⁵t³
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} <2> = {2,3,4,5} <3> = {3,4,5,6} <4> = {5,6,7} <5> = {6,7,8,9} <6> = {7,8,9,10} <7> = {9,10,11}
Spectra Variation Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.	2.25
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.	2.616
Polygon Perimeter Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.	6.002
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.	0.176776
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.	(30, 60, 141)
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.	0.281

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	D	{2,6,9}	3	4	1.9
	F♯	{6,10,1}	2	4	2.1
	G♯	{8,0,3}	2	4	2.3
Minor Triads	d♯m	{3,6,10}	3	4	1.9
Minor Triads	f♯m	{6,9,1}	3	4	1.9
Augmented Triads	D+	{2,6,10}	3	4	1.9
Diminished Triads	c°	{0,3,6}	2	4	2.1
	d♯°	{3,6,9}	2	4	2.1
	f♯°	{6,9,0}	2	4	2.1
	a°	{9,0,3}	2	4	2.3

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.

Diameter	4
Radius	4
Self-Centered	yes

Modes

Modes are the rotational transformation of this scale. Scale 1871 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode: Scale 2983				Zythyllic
3rd mode: Scale 3539				Aeoryllic
4th mode: Scale 3817				Zoryllic
5th mode: Scale 989				Phrolyllic
6th mode: Scale 1271				Kolyllic
7th mode: Scale 2683				Thodyllic
8th mode: Scale 3389				Socryllic

Prime

The prime form of this scale is Scale 751

Scale 751

Epacryllic

Complement

The octatonic modal family [1871, 2983, 3539, 3817, 989, 1271, 2683, 3389] (Forte: 8-Z29) is the complement of the tetratonic modal family [139, 353, 1553, 2117] (Forte: 4-Z29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1871 is 3677

Scale 3677

Katylyllic

Interval Matrix

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

Contradictions (30)

Ambiguities(60)

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	`111100101110`
2	1	`111100101110`
3	1	`111100101110`
4	3	`1([1][1][1]0)010([1][1][1]0)`
5	3	`1([1][1][1]0)010([1][1][1]0)`

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1871 is chiral, and its enantiomorph is scale 3677

Scale 3677

Katylyllic

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	(-0.045753, -0.170753)
Distance from Center	0.176776
Angle in degrees measured clockwise starting from the root.	345
Angle in cents 100 cents = 1 semitone.	1150

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>	1871	T₀I	<11,0>	3677
T₁	<1,1>	3742	T₁I	<11,1>	3259
T₂	<1,2>	3389	T₂I	<11,2>	2423
T₃	<1,3>	2683	T₃I	<11,3>	751
T₄	<1,4>	1271	T₄I	<11,4>	1502
T₅	<1,5>	2542	T₅I	<11,5>	3004
T₆	<1,6>	989	T₆I	<11,6>	1913
T₇	<1,7>	1978	T₇I	<11,7>	3826
T₈	<1,8>	3956	T₈I	<11,8>	3557
T₉	<1,9>	3817	T₉I	<11,9>	3019
T₁₀	<1,10>	3539	T₁₀I	<11,10>	1943
T₁₁	<1,11>	2983	T₁₁I	<11,11>	3886
Abbrev	Operation	Result	Abbrev	Operation	Result
T₀M	<5,0>	1661	T₀MI	<7,0>	1997
T₁M	<5,1>	3322	T₁MI	<7,1>	3994
T₂M	<5,2>	2549	T₂MI	<7,2>	3893
T₃M	<5,3>	1003	T₃MI	<7,3>	3691
T₄M	<5,4>	2006	T₄MI	<7,4>	3287
T₅M	<5,5>	4012	T₅MI	<7,5>	2479
T₆M	<5,6>	3929	T₆MI	<7,6>	863
T₇M	<5,7>	3763	T₇MI	<7,7>	1726
T₈M	<5,8>	3431	T₈MI	<7,8>	3452
T₉M	<5,9>	2767	T₉MI	<7,9>	2809
T₁₀M	<5,10>	1439	T₁₀MI	<7,10>	1523
T₁₁M	<5,11>	2878	T₁₁MI	<7,11>	3046

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 1869				Katyrian
Scale 1867				Solian
Scale 1863				Pycrian
Scale 1879				Mixoryllic
Scale 1887				Aerocrygic
Scale 1903				Diminishing Nonamode Basic
Scale 1807				LARian
Scale 1839				Zogyllic
Scale 1935				Mycryllic
Scale 1999				Zacrygic
Scale 1615				Sydian
Scale 1743				Epigyllic
Scale 1359				Aerygian
Scale 847				Ganian
Scale 2895				Aeragyllic
Scale 3919				Lynygic

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. (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.