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Scale 1851: "Zacryllic"

Scale 1851: Zacryllic, 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

41161837294116105072918310504116183
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
Zacryllic

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

{0,1,3,4,5,8,9,10}

Forte Number

A code assigned by theorist Alan Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

8-20

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.

[0.5]

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.

no

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.

2 (dicohemitonic)

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.

2

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.

7

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 951

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 2, 1, 1, 3, 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, 4, 5, 6, 6, 2>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hansen.

p6m6n5s4d5t2

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}
<3> = {4,5}
<4> = {5,6,7}
<5> = {7,8}
<6> = {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.

1.5

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.

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 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.

[1]

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

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 TriadsC♯{1,5,8}342
F{5,9,0}331.83
G♯{8,0,3}252.5
A{9,1,4}431.67
Minor Triadsc♯m{1,4,8}331.83
fm{5,8,0}342
am{9,0,4}431.67
a♯m{10,1,5}252.5
Augmented TriadsC+{0,4,8}441.83
C♯+{1,5,9}441.83
Diminished Triads{9,0,3}242.33
a♯°{10,1,4}242.33
Parsimonious Voice Leading Between Common Triads of Scale 1851. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m fm fm C+->fm G# G# C+->G# am am C+->am C# C# c#m->C# A A c#m->A C#+ C#+ C#->C#+ C#->fm F F C#+->F C#+->A a#m a#m C#+->a#m fm->F F->am G#->a° a°->am am->A a#° a#° A->a#° a#°->a#m

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.

Diameter5
Radius3
Self-Centeredno
Central Verticesc♯m, F, am, A
Peripheral VerticesG♯, a♯m

Modes

Modes are the rotational transformation of this scale. Scale 1851 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2973
Scale 2973: Panyllic, Ian Ring Music TheoryPanyllic
3rd mode:
Scale 1767
Scale 1767: Dyryllic, Ian Ring Music TheoryDyryllic
4th mode:
Scale 2931
Scale 2931: Zathyllic, Ian Ring Music TheoryZathyllic
5th mode:
Scale 3513
Scale 3513: Dydyllic, Ian Ring Music TheoryDydyllic
6th mode:
Scale 951
Scale 951: Thogyllic, Ian Ring Music TheoryThogyllicThis is the prime mode
7th mode:
Scale 2523
Scale 2523: Mirage Scale, Ian Ring Music TheoryMirage Scale
8th mode:
Scale 3309
Scale 3309: Bycryllic, Ian Ring Music TheoryBycryllic

Prime

The prime form of this scale is Scale 951

Scale 951Scale 951: Thogyllic, Ian Ring Music TheoryThogyllic

Complement

The octatonic modal family [1851, 2973, 1767, 2931, 3513, 951, 2523, 3309] (Forte: 8-20) is the complement of the tetratonic modal family [291, 393, 561, 2193] (Forte: 4-20)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1851 is 2973

Scale 2973Scale 2973: Panyllic, Ian Ring Music TheoryPanyllic

Transformations:

T0 1851  T0I 2973
T1 3702  T1I 1851
T2 3309  T2I 3702
T3 2523  T3I 3309
T4 951  T4I 2523
T5 1902  T5I 951
T6 3804  T6I 1902
T7 3513  T7I 3804
T8 2931  T8I 3513
T9 1767  T9I 2931
T10 3534  T10I 1767
T11 2973  T11I 3534

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 1849Scale 1849: Chromatic Hypodorian Inverse, Ian Ring Music TheoryChromatic Hypodorian Inverse
Scale 1853Scale 1853: Maryllic, Ian Ring Music TheoryMaryllic
Scale 1855Scale 1855: Gaptygic, Ian Ring Music TheoryGaptygic
Scale 1843Scale 1843: Ionygian, Ian Ring Music TheoryIonygian
Scale 1847Scale 1847: Thacryllic, Ian Ring Music TheoryThacryllic
Scale 1835Scale 1835: Byptian, Ian Ring Music TheoryByptian
Scale 1819Scale 1819: Pydian, Ian Ring Music TheoryPydian
Scale 1883Scale 1883, Ian Ring Music Theory
Scale 1915Scale 1915: Thydygic, Ian Ring Music TheoryThydygic
Scale 1979Scale 1979: Aeradygic, Ian Ring Music TheoryAeradygic
Scale 1595Scale 1595: Dacrian, Ian Ring Music TheoryDacrian
Scale 1723Scale 1723: JG Octatonic, Ian Ring Music TheoryJG Octatonic
Scale 1339Scale 1339: Kycrian, Ian Ring Music TheoryKycrian
Scale 827Scale 827: Mixolocrian, Ian Ring Music TheoryMixolocrian
Scale 2875Scale 2875: Ganyllic, Ian Ring Music TheoryGanyllic
Scale 3899Scale 3899: Katorygic, Ian Ring Music TheoryKatorygic

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. 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.