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Scale 3089: "Tirian"

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

Dozenal: Tirian

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	{0,4,10,11}
Forte Number A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.	4-5
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: 263
Hemitonia A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.	2 (dihemitonic)
Cohemitonia A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.	1 (uncohemitonic)
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 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.	3
Prime Form Describes if this scale is in prime form, using the Rahn/Ring formula.	no prime: 71
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.	no
Interval Structure Defines the scale as the sequence of intervals between one tone and the next.	[4, 6, 1, 1]
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.	<2, 1, 0, 1, 1, 1>
Interval Spectrum The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.	pmsd²t
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> = {1,4,6} <2> = {2,5,7,10} <3> = {6,8,11}
Spectra Variation Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.	4.5
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.	0.933
Polygon Perimeter Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.	4.767
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
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.	(6, 1, 16)

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.

Modes

Modes are the rotational transformation of this scale. Scale 3089 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode: Scale 449	Cujian
3rd mode: Scale 71	Aloian	This is the prime mode
4th mode: Scale 2083	Mofian

Prime

The prime form of this scale is Scale 71

Scale 71

Aloian

Complement

The tetratonic modal family [3089, 449, 71, 2083] (Forte: 4-5) is the complement of the octatonic modal family [479, 1991, 2287, 3043, 3191, 3569, 3643, 3869] (Forte: 8-5)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3089 is 263

Scale 263

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3089 is chiral, and its enantiomorph is scale 263

Scale 263

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

Abbrev	Operation	Result	Abbrev	Operation	Result
T₀	<1,0>	3089	T₀I	<11,0>	263
T₁	<1,1>	2083	T₁I	<11,1>	526
T₂	<1,2>	71	T₂I	<11,2>	1052
T₃	<1,3>	142	T₃I	<11,3>	2104
T₄	<1,4>	284	T₄I	<11,4>	113
T₅	<1,5>	568	T₅I	<11,5>	226
T₆	<1,6>	1136	T₆I	<11,6>	452
T₇	<1,7>	2272	T₇I	<11,7>	904
T₈	<1,8>	449	T₈I	<11,8>	1808
T₉	<1,9>	898	T₉I	<11,9>	3616
T₁₀	<1,10>	1796	T₁₀I	<11,10>	3137
T₁₁	<1,11>	3592	T₁₁I	<11,11>	2179
Abbrev	Operation	Result	Abbrev	Operation	Result
T₀M	<5,0>	389	T₀MI	<7,0>	1073
T₁M	<5,1>	778	T₁MI	<7,1>	2146
T₂M	<5,2>	1556	T₂MI	<7,2>	197
T₃M	<5,3>	3112	T₃MI	<7,3>	394
T₄M	<5,4>	2129	T₄MI	<7,4>	788
T₅M	<5,5>	163	T₅MI	<7,5>	1576
T₆M	<5,6>	326	T₆MI	<7,6>	3152
T₇M	<5,7>	652	T₇MI	<7,7>	2209
T₈M	<5,8>	1304	T₈MI	<7,8>	323
T₉M	<5,9>	2608	T₉MI	<7,9>	646
T₁₀M	<5,10>	1121	T₁₀MI	<7,10>	1292
T₁₁M	<5,11>	2242	T₁₁MI	<7,11>	2584

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 3091				Tisian
Scale 3093				Buqian
Scale 3097				Tiwian
Scale 3073				Tritonic Chromatic Descending
Scale 3081				Temian
Scale 3105				Tibian
Scale 3121				Tilian
Scale 3153				Zathitonic
Scale 3217				Molitonic
Scale 3345				Zylitonic
Scale 3601				Wilian
Scale 2065				Motian
Scale 2577				Punian
Scale 1041				Hitian

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. 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.