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Scale 797: "Katocrimic"

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: Katocrimic

Dozenal: Fafian

Analysis

Cardinality Cardinality is the count of how many pitches are in the scale.	6 (hexatonic)
Pitch Class Set The tones in this scale, expressed as numbers from 0 to 11	{0,2,3,4,8,9}
Forte Number A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.	6-Z17
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: 1817
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.	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.	5
Prime Form Describes if this scale is in prime form, using the Rahn/Ring formula.	no prime: 407
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.	[2, 1, 1, 4, 1, 3]
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, 2, 2, 3, 3, 2>
Interval Spectrum The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.	p³m³n²s²d³t²
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> = {1,2,3,4} <2> = {2,3,4,5} <3> = {4,6,8} <4> = {7,8,9,10} <5> = {8,9,10,11}
Spectra Variation Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.	2.667
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.	2.116
Polygon Perimeter Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.	5.699
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.	(12, 10, 57)

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	G♯	{8,0,3}	2	2	1
Minor Triads	am	{9,0,4}	2	2	1
Augmented Triads	C+	{0,4,8}	2	2	1
Diminished Triads	a°	{9,0,3}	2	2	1

The following pitch classes are not present in any of the common triads: {2}

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	2
Radius	2
Self-Centered	yes

Modes

Modes are the rotational transformation of this scale. Scale 797 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode: Scale 1223				Phryptimic
3rd mode: Scale 2659				Katynimic
4th mode: Scale 3377				Phralimic
5th mode: Scale 467				Raga Dhavalangam
6th mode: Scale 2281				Rathimic

Prime

The prime form of this scale is Scale 407

Scale 407

All-Trichord Hexachord

Complement

The hexatonic modal family [797, 1223, 2659, 3377, 467, 2281] (Forte: 6-Z17) is the complement of the hexatonic modal family [359, 907, 1649, 2227, 2501, 3161] (Forte: 6-Z43)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 797 is 1817

Scale 1817

Phrythimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 797 is chiral, and its enantiomorph is scale 1817

Scale 1817

Phrythimic

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>	797	T₀I	<11,0>	1817
T₁	<1,1>	1594	T₁I	<11,1>	3634
T₂	<1,2>	3188	T₂I	<11,2>	3173
T₃	<1,3>	2281	T₃I	<11,3>	2251
T₄	<1,4>	467	T₄I	<11,4>	407
T₅	<1,5>	934	T₅I	<11,5>	814
T₆	<1,6>	1868	T₆I	<11,6>	1628
T₇	<1,7>	3736	T₇I	<11,7>	3256
T₈	<1,8>	3377	T₈I	<11,8>	2417
T₉	<1,9>	2659	T₉I	<11,9>	739
T₁₀	<1,10>	1223	T₁₀I	<11,10>	1478
T₁₁	<1,11>	2446	T₁₁I	<11,11>	2956
Abbrev	Operation	Result	Abbrev	Operation	Result
T₀M	<5,0>	1817	T₀MI	<7,0>	797
T₁M	<5,1>	3634	T₁MI	<7,1>	1594
T₂M	<5,2>	3173	T₂MI	<7,2>	3188
T₃M	<5,3>	2251	T₃MI	<7,3>	2281
T₄M	<5,4>	407	T₄MI	<7,4>	467
T₅M	<5,5>	814	T₅MI	<7,5>	934
T₆M	<5,6>	1628	T₆MI	<7,6>	1868
T₇M	<5,7>	3256	T₇MI	<7,7>	3736
T₈M	<5,8>	2417	T₈MI	<7,8>	3377
T₉M	<5,9>	739	T₉MI	<7,9>	2659
T₁₀M	<5,10>	1478	T₁₀MI	<7,10>	1223
T₁₁M	<5,11>	2956	T₁₁MI	<7,11>	2446

The transformations that map this set to itself are: T₀, T₀MI

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 799				Lolian
Scale 793				Mocritonic
Scale 795				Aeologimic
Scale 789				Zogitonic
Scale 781				Etoian
Scale 813				Larimic
Scale 829				Lygian
Scale 861				Rylian
Scale 925				Chromatic Hypodorian
Scale 541				Demian
Scale 669				Gycrimic
Scale 285				Zaritonic
Scale 1309				Pogimic
Scale 1821				Aeradian
Scale 2845				Baptian

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.