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Scale 2899: "Kagian"

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: Kagian

Dozenal: Sefian

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	{0,1,4,6,8,9,11}
Forte Number A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.	7-27
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: 2395
Hemitonia A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.	3 (trihemitonic)
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.	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 Starr/Rahn algorithm.	no prime: 695
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, 3, 2, 2, 1, 2, 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.	<3, 4, 4, 4, 5, 1>
Interval Spectrum The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.	p⁵m⁴n⁴s⁴d³t
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> = {4,5,6,7} <4> = {5,6,7,8} <5> = {7,8,9,10} <6> = {9,10,11}
Spectra Variation Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.	2.286
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.	2.549
Polygon Perimeter Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.	5.967
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.	(9, 35, 98)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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	E	{4,8,11}	1	4	2.14
Major Triads	A	{9,1,4}	3	3	1.43
Minor Triads	c♯m	{1,4,8}	2	3	1.57
	f♯m	{6,9,1}	2	4	1.86
	am	{9,0,4}	3	2	1.29
Augmented Triads	C+	{0,4,8}	3	3	1.43
Diminished Triads	f♯°	{6,9,0}	2	3	1.71

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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	am
Peripheral Vertices	E, f♯m

Modes

Modes are the rotational transformation of this scale. Scale 2899 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode: Scale 3497				Phrolian
3rd mode: Scale 949				Mela Mararanjani
4th mode: Scale 1261				Modified Blues
5th mode: Scale 1339				Kycrian
6th mode: Scale 2717				Epygian
7th mode: Scale 1703				Mela Vanaspati

Prime

The prime form of this scale is Scale 695

Scale 695

Sarian

Complement

The heptatonic modal family [2899, 3497, 949, 1261, 1339, 2717, 1703] (Forte: 7-27) is the complement of the pentatonic modal family [299, 689, 1417, 1573, 2197] (Forte: 5-27)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2899 is 2395

Scale 2395

Zoptian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2899 is chiral, and its enantiomorph is scale 2395

Scale 2395

Zoptian

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>	2899	T₀I	<11,0>	2395
T₁	<1,1>	1703	T₁I	<11,1>	695
T₂	<1,2>	3406	T₂I	<11,2>	1390
T₃	<1,3>	2717	T₃I	<11,3>	2780
T₄	<1,4>	1339	T₄I	<11,4>	1465
T₅	<1,5>	2678	T₅I	<11,5>	2930
T₆	<1,6>	1261	T₆I	<11,6>	1765
T₇	<1,7>	2522	T₇I	<11,7>	3530
T₈	<1,8>	949	T₈I	<11,8>	2965
T₉	<1,9>	1898	T₉I	<11,9>	1835
T₁₀	<1,10>	3796	T₁₀I	<11,10>	3670
T₁₁	<1,11>	3497	T₁₁I	<11,11>	3245
Abbrev	Operation	Result	Abbrev	Operation	Result
T₀M	<5,0>	1009	T₀MI	<7,0>	505
T₁M	<5,1>	2018	T₁MI	<7,1>	1010
T₂M	<5,2>	4036	T₂MI	<7,2>	2020
T₃M	<5,3>	3977	T₃MI	<7,3>	4040
T₄M	<5,4>	3859	T₄MI	<7,4>	3985
T₅M	<5,5>	3623	T₅MI	<7,5>	3875
T₆M	<5,6>	3151	T₆MI	<7,6>	3655
T₇M	<5,7>	2207	T₇MI	<7,7>	3215
T₈M	<5,8>	319	T₈MI	<7,8>	2335
T₉M	<5,9>	638	T₉MI	<7,9>	575
T₁₀M	<5,10>	1276	T₁₀MI	<7,10>	1150
T₁₁M	<5,11>	2552	T₁₁MI	<7,11>	2300

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 2897				Rycrimic
Scale 2901				Lydian Augmented
Scale 2903				Gothyllic
Scale 2907				Magen Abot 2
Scale 2883				Savian
Scale 2891				Phrogian
Scale 2915				Aeolydian
Scale 2931				Zathyllic
Scale 2835				Ionygimic
Scale 2867				Socrian
Scale 2963				Bygian
Scale 3027				Rythyllic
Scale 2643				Raga Hamsanandi
Scale 2771				Marva That
Scale 2387				Paptimic
Scale 3411				Enigmatic
Scale 3923				Stoptyllic
Scale 851				Raga Hejjajji
Scale 1875				Persichetti Scale

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.