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Scale 3427: "Zacrian"

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: Zacrian

Analysis

Cardinality Cardinality is the count of how many pitches are in the scale.	7 (heptatonic)
Pitch Class Set The tones in this scale, expressed as numbers from 0 to 11	{0,1,5,6,8,10,11}
Forte Number A code assigned by theorist Alan Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.	7-14
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: 2263
Hemitonia A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.	4 (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.	6
Prime Form Describes if this scale is in prime form, using the Rahn/Ring formula.	no prime: 431
Deep Scale A deep scale is one where the interval vector has 6 different digits.	no
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.	[4, 4, 3, 3, 5, 2]
Interval Spectrum The same as the Interval Vector, but expressed in a syntax used by Howard Hansen.	p⁵m³n³s⁴d⁴t²
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} <2> = {2,3,4,5} <3> = {3,4,5,6,7} <4> = {5,6,7,8,9} <5> = {7,8,9,10} <6> = {8,10,11}
Spectra Variation Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.	2.857
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.	2.299
Polygon Perimeter Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.	5.803
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.	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

Harmonic Chords

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♯	{1,5,8}	2	2	1.2
Major Triads	F♯	{6,10,1}	1	4	2
Minor Triads	fm	{5,8,0}	2	3	1.4
Minor Triads	a♯m	{10,1,5}	2	3	1.4
Diminished Triads	f°	{5,8,11}	1	4	2

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	2
Self-Centered	no
Central Vertices	C♯
Peripheral Vertices	f°, F♯

Modes

Modes are the rotational transformation of this scale. Scale 3427 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode: Scale 3761				Raga Madhuri
3rd mode: Scale 491				Aeolyrian
4th mode: Scale 2293				Gorian
5th mode: Scale 1597				Aeolodian
6th mode: Scale 1423				Doptian
7th mode: Scale 2759				Mela Pavani

Prime

The prime form of this scale is Scale 431

Scale 431

Epyrian

Complement

The heptatonic modal family [3427, 3761, 491, 2293, 1597, 1423, 2759] (Forte: 7-14) is the complement of the pentatonic modal family [167, 901, 1249, 2131, 3113] (Forte: 5-14)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3427 is 2263

Scale 2263

Lycrian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3427 is chiral, and its enantiomorph is scale 2263

Scale 2263

Lycrian

Transformations:

T₀	3427	T₀I	2263
T₁	2759	T₁I	431
T₂	1423	T₂I	862
T₃	2846	T₃I	1724
T₄	1597	T₄I	3448
T₅	3194	T₅I	2801
T₆	2293	T₆I	1507
T₇	491	T₇I	3014
T₈	982	T₈I	1933
T₉	1964	T₉I	3866
T₁₀	3928	T₁₀I	3637
T₁₁	3761	T₁₁I	3179

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 3425
Scale 3429				Marian
Scale 3431				Zyptyllic
Scale 3435				Prokofiev
Scale 3443				Verdi's Scala Enigmatica
Scale 3395
Scale 3411				Enigmatic
Scale 3363				Rogimic
Scale 3491				Tharian
Scale 3555				Pylyllic
Scale 3171				Zythimic
Scale 3299				Syptian
Scale 3683				Dycrian
Scale 3939				Dogyllic
Scale 2403				Lycrimic
Scale 2915				Aeolydian
Scale 1379				Kycrimic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. 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.