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Scale 1871: "Aeolyllic"

Scale 1871: Aeolyllic, 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
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

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.

(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 TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsD{2,6,9}341.9
F♯{6,10,1}242.1
G♯{8,0,3}242.3
Minor Triadsd♯m{3,6,10}341.9
f♯m{6,9,1}341.9
Augmented TriadsD+{2,6,10}341.9
Diminished Triadsc°{0,3,6}242.1
d♯°{3,6,9}242.1
f♯°{6,9,0}242.1
a°{9,0,3}242.3
Parsimonious Voice Leading Between Common Triads of Scale 1871. Created by Ian Ring ©2019 c° c° d#m d#m c°->d#m G# G# c°->G# D D D+ D+ D->D+ d#° d#° D->d#° f#m f#m D->f#m D+->d#m F# F# D+->F# d#°->d#m f#° f#° f#°->f#m a° a° f#°->a° f#m->F# G#->a°

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.

Diameter4
Radius4
Self-Centeredyes

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		Scale 2983: Zythyllic, Ian Ring Music TheoryZythyllic
3rd mode:
Scale 3539		Scale 3539: Aeoryllic, Ian Ring Music TheoryAeoryllic
4th mode:
Scale 3817		Scale 3817: Zoryllic, Ian Ring Music TheoryZoryllic
5th mode:
Scale 989		Scale 989: Phrolyllic, Ian Ring Music TheoryPhrolyllic
6th mode:
Scale 1271		Scale 1271: Kolyllic, Ian Ring Music TheoryKolyllic
7th mode:
Scale 2683		Scale 2683: Thodyllic, Ian Ring Music TheoryThodyllic
8th mode:
Scale 3389		Scale 3389: Socryllic, Ian Ring Music TheorySocryllic

Prime

The prime form of this scale is Scale 751

Scale 751Scale 751: Epacryllic, Ian Ring Music TheoryEpacryllic

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 3677Scale 3677: Katylyllic, Ian Ring Music TheoryKatylyllic

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
111111001011101871k = 1h = 1
211111001011101871k = 2h = 1
311111001011101871k = 3h = 1
431([1][1][1]0)010([1][1][1]0)1871k = 4h = 3
531([1][1][1]0)010([1][1][1]0)1871k = 5h = 3

Enantiomorph

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

Scale 3677Scale 3677: Katylyllic, Ian Ring Music TheoryKatylyllic

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 1869Scale 1869: Katyrian, Ian Ring Music TheoryKatyrian
Scale 1867Scale 1867: Solian, Ian Ring Music TheorySolian
Scale 1863Scale 1863: Pycrian, Ian Ring Music TheoryPycrian
Scale 1879Scale 1879: Mixoryllic, Ian Ring Music TheoryMixoryllic
Scale 1887Scale 1887: Aerocrygic, Ian Ring Music TheoryAerocrygic
Scale 1903Scale 1903: Diminishing Nonamode Basic, Ian Ring Music TheoryDiminishing Nonamode Basic
Scale 1807Scale 1807: LARian, Ian Ring Music TheoryLARian
Scale 1839Scale 1839: Zogyllic, Ian Ring Music TheoryZogyllic
Scale 1935Scale 1935: Mycryllic, Ian Ring Music TheoryMycryllic
Scale 1999Scale 1999: Zacrygic, Ian Ring Music TheoryZacrygic
Scale 1615Scale 1615: Sydian, Ian Ring Music TheorySydian
Scale 1743Scale 1743: Epigyllic, Ian Ring Music TheoryEpigyllic
Scale 1359Scale 1359: Aerygian, Ian Ring Music TheoryAerygian
Scale 847Scale 847: Ganian, Ian Ring Music TheoryGanian
Scale 2895Scale 2895: Aeragyllic, Ian Ring Music TheoryAeragyllic
Scale 3919Scale 3919: Lynygic, Ian Ring Music TheoryLynygic

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.