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# Scale 57: "Ahoian" ### 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).

Dozenal
Ahoian

## 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

{0,3,4,5}

#### Forte Number

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

4-4

#### 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: 897

#### 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: 39

#### 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.

[3, 1, 1, 7]

#### 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, 1, 1, 1, 0>

#### Interval Spectrum

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

pmnsd2

#### 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> = {1,3,7}
<2> = {2,4,8,10}
<3> = {5,9,11}

#### Spectra Variation

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

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.

0.75

#### Polygon Perimeter

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

4.381

#### 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.

(7, 0, 16)

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 57 can be rotated to make 3 other scales. The 1st mode is itself.

 2nd mode:Scale 519 Deyian 3rd mode:Scale 2307 Ocoian 4th mode:Scale 3201 Urtian

## Prime

The prime form of this scale is Scale 39

 Scale 39 Afuian

## Complement

The tetratonic modal family [57, 519, 2307, 3201] (Forte: 4-4) is the complement of the octatonic modal family [447, 2019, 2271, 3057, 3183, 3639, 3867, 3981] (Forte: 8-4)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 57 is 897

 Scale 897 Fopian

## Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 57 is chiral, and its enantiomorph is scale 897

 Scale 897 Fopian

## 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> 57       T0I <11,0> 897
T1 <1,1> 114      T1I <11,1> 1794
T2 <1,2> 228      T2I <11,2> 3588
T3 <1,3> 456      T3I <11,3> 3081
T4 <1,4> 912      T4I <11,4> 2067
T5 <1,5> 1824      T5I <11,5> 39
T6 <1,6> 3648      T6I <11,6> 78
T7 <1,7> 3201      T7I <11,7> 156
T8 <1,8> 2307      T8I <11,8> 312
T9 <1,9> 519      T9I <11,9> 624
T10 <1,10> 1038      T10I <11,10> 1248
T11 <1,11> 2076      T11I <11,11> 2496
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 267      T0MI <7,0> 2577
T1M <5,1> 534      T1MI <7,1> 1059
T2M <5,2> 1068      T2MI <7,2> 2118
T3M <5,3> 2136      T3MI <7,3> 141
T4M <5,4> 177      T4MI <7,4> 282
T5M <5,5> 354      T5MI <7,5> 564
T6M <5,6> 708      T6MI <7,6> 1128
T7M <5,7> 1416      T7MI <7,7> 2256
T8M <5,8> 2832      T8MI <7,8> 417
T9M <5,9> 1569      T9MI <7,9> 834
T10M <5,10> 3138      T10MI <7,10> 1668
T11M <5,11> 2181      T11MI <7,11> 3336

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 59 Ahuian Scale 61 Ajuian Scale 49 Aguian Scale 53 Absian Scale 41 Vietnamese Tritonic Scale 25 Ackian Scale 89 Aggian Scale 121 Asoian Scale 185 Becian Scale 313 Goritonic Scale 569 Mothitonic Scale 1081 Goxian Scale 2105 Rigian

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.