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Scale 97: "Athian"

Scale 97: Athian, 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

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
Athian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

3 (tritonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,5,6}

Forte Number

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

3-5

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: 193

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

1 (unhemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)

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.

2

Prime Form

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

no
prime: 67

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.

[5, 1, 6]

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.

<1, 0, 0, 0, 1, 1>

Interval Spectrum

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

pdt

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> = {1,5,6}
<2> = {6,7,11}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

3.333

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.

0.5

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

4.449

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".

Proper

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.

(0, 1, 6)

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 97 can be rotated to make 2 other scales. The 1st mode is itself.

2nd mode:
Scale 131
Scale 131: Atoian, Ian Ring Music TheoryAtoian
3rd mode:
Scale 2113
Scale 2113: Muxian, Ian Ring Music TheoryMuxian

Prime

The prime form of this scale is Scale 67

Scale 67Scale 67: Abrian, Ian Ring Music TheoryAbrian

Complement

The tritonic modal family [97, 131, 2113] (Forte: 3-5) is the complement of the enneatonic modal family [991, 1999, 2543, 3047, 3319, 3571, 3707, 3833, 3901] (Forte: 9-5)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 97 is 193

Scale 193Scale 193: Raga Ongkari, Ian Ring Music TheoryRaga Ongkari

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 97 is chiral, and its enantiomorph is scale 193

Scale 193Scale 193: Raga Ongkari, Ian Ring Music TheoryRaga Ongkari

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> 97       T0I <11,0> 193
T1 <1,1> 194      T1I <11,1> 386
T2 <1,2> 388      T2I <11,2> 772
T3 <1,3> 776      T3I <11,3> 1544
T4 <1,4> 1552      T4I <11,4> 3088
T5 <1,5> 3104      T5I <11,5> 2081
T6 <1,6> 2113      T6I <11,6> 67
T7 <1,7> 131      T7I <11,7> 134
T8 <1,8> 262      T8I <11,8> 268
T9 <1,9> 524      T9I <11,9> 536
T10 <1,10> 1048      T10I <11,10> 1072
T11 <1,11> 2096      T11I <11,11> 2144
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 67      T0MI <7,0> 2113
T1M <5,1> 134      T1MI <7,1> 131
T2M <5,2> 268      T2MI <7,2> 262
T3M <5,3> 536      T3MI <7,3> 524
T4M <5,4> 1072      T4MI <7,4> 1048
T5M <5,5> 2144      T5MI <7,5> 2096
T6M <5,6> 193      T6MI <7,6> 97
T7M <5,7> 386      T7MI <7,7> 194
T8M <5,8> 772      T8MI <7,8> 388
T9M <5,9> 1544      T9MI <7,9> 776
T10M <5,10> 3088      T10MI <7,10> 1552
T11M <5,11> 2081      T11MI <7,11> 3104

The transformations that map this set to itself are: T0, T6MI

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 99Scale 99: Iprian, Ian Ring Music TheoryIprian
Scale 101Scale 101: Apoian, Ian Ring Music TheoryApoian
Scale 105Scale 105, Ian Ring Music Theory
Scale 113Scale 113, Ian Ring Music Theory
Scale 65Scale 65: Tritone, Ian Ring Music TheoryTritone
Scale 81Scale 81: Disian, Ian Ring Music TheoryDisian
Scale 33Scale 33: Honchoshi, Ian Ring Music TheoryHonchoshi
Scale 161Scale 161: Raga Sarvasri, Ian Ring Music TheoryRaga Sarvasri
Scale 225Scale 225: Bibian, Ian Ring Music TheoryBibian
Scale 353Scale 353: Cebian, Ian Ring Music TheoryCebian
Scale 609Scale 609: Docian, Ian Ring Music TheoryDocian
Scale 1121Scale 1121: Guwian, Ian Ring Music TheoryGuwian
Scale 2145Scale 2145: Messiaen Truncated Mode 5 Inverse, Ian Ring Music TheoryMessiaen Truncated Mode 5 Inverse

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.