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# Scale 2557: "Dothygic" ### 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).

Zeitler
Dothygic
Dozenal
Pobian

## Analysis

#### Cardinality

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

9 (enneatonic)

#### Pitch Class Set

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

{0,2,3,4,5,6,7,8,11}

#### Forte Number

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

9-3

#### 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: 2035

#### Hemitonia

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

7 (multihemitonic)

#### Cohemitonia

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

5 (multicohemitonic)

#### 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.

8

#### Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

no
prime: 895

#### 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.

[2, 1, 1, 1, 1, 1, 1, 3, 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.

<7, 6, 7, 7, 6, 3>

#### Interval Spectrum

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

p6m7n7s6d7t3

#### 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}
<3> = {3,4,5,6}
<4> = {4,5,6,7}
<5> = {5,6,7,8}
<6> = {6,7,8,9}
<7> = {8,9,10}
<8> = {9,10,11}

#### Spectra Variation

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

2.222

#### 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.

2.683

#### Polygon Perimeter

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

6.038

#### 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.

(64, 107, 194)

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

E{4,8,11}442.24
G{7,11,2}342.35
G♯{8,0,3}342.24
B{11,3,6}342.35
em{4,7,11}342.24
fm{5,8,0}342.53
g♯m{8,11,3}442.12
bm{11,2,6}342.53
D♯+{3,7,11}542
{2,5,8}242.76
{5,8,11}252.71
g♯°{8,11,2}242.59
{11,2,5}242.76

view full size

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 5 4 no cm, C, C+, d°, D♯+, em, E, fm, G, g♯°, g♯m, G♯, b°, bm, B c°, f°

## Modes

Modes are the rotational transformation of this scale. Scale 2557 can be rotated to make 8 other scales. The 1st mode is itself.

 2nd mode:Scale 1663 Lydygic 3rd mode:Scale 2879 Stadygic 4th mode:Scale 3487 Byptygic 5th mode:Scale 3791 Stodygic 6th mode:Scale 3943 Zynygic 7th mode:Scale 4019 Lonygic 8th mode:Scale 4057 Phrygic 9th mode:Scale 1019 Aeranygic

## Prime

The prime form of this scale is Scale 895

 Scale 895 Aeolathygic

## Complement

The enneatonic modal family [2557, 1663, 2879, 3487, 3791, 3943, 4019, 4057, 1019] (Forte: 9-3) is the complement of the tritonic modal family [19, 769, 2057] (Forte: 3-3)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2557 is 2035

 Scale 2035 Aerythygic

## Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2557 is chiral, and its enantiomorph is scale 2035

 Scale 2035 Aerythygic

## 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> 2557       T0I <11,0> 2035
T1 <1,1> 1019      T1I <11,1> 4070
T2 <1,2> 2038      T2I <11,2> 4045
T3 <1,3> 4076      T3I <11,3> 3995
T4 <1,4> 4057      T4I <11,4> 3895
T5 <1,5> 4019      T5I <11,5> 3695
T6 <1,6> 3943      T6I <11,6> 3295
T7 <1,7> 3791      T7I <11,7> 2495
T8 <1,8> 3487      T8I <11,8> 895
T9 <1,9> 2879      T9I <11,9> 1790
T10 <1,10> 1663      T10I <11,10> 3580
T11 <1,11> 3326      T11I <11,11> 3065
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3547      T0MI <7,0> 2935
T1M <5,1> 2999      T1MI <7,1> 1775
T2M <5,2> 1903      T2MI <7,2> 3550
T3M <5,3> 3806      T3MI <7,3> 3005
T4M <5,4> 3517      T4MI <7,4> 1915
T5M <5,5> 2939      T5MI <7,5> 3830
T6M <5,6> 1783      T6MI <7,6> 3565
T7M <5,7> 3566      T7MI <7,7> 3035
T8M <5,8> 3037      T8MI <7,8> 1975
T9M <5,9> 1979      T9MI <7,9> 3950
T10M <5,10> 3958      T10MI <7,10> 3805
T11M <5,11> 3821      T11MI <7,11> 3515

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 2559 Decatonic Chromatic 2 Scale 2553 Aeolaptyllic Scale 2555 Bythygic Scale 2549 Rydyllic Scale 2541 Algerian Scale 2525 Aeolaryllic Scale 2493 Manyllic Scale 2429 Kadyllic Scale 2301 Bydyllic Scale 2813 Zolygic Scale 3069 Maqam Shawq Afza Scale 3581 Epocryllian Scale 509 Ionothyllic Scale 1533 Katycrygic

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.