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# Scale 2033: "Stolyllic" ### 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
Stolyllic
Dozenal
MIZian

## Analysis

#### Cardinality

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

8 (octatonic)

#### Pitch Class Set

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

{0,4,5,6,7,8,9,10}

#### Forte Number

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

8-2

#### 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: 509

#### Hemitonia

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

6 (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.

4

#### 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.

7

#### Prime Form

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

no
prime: 383

#### 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.

[4, 1, 1, 1, 1, 1, 1, 2]

#### 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.

<6, 6, 5, 5, 4, 2>

#### Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.667, 0.667, 0.25, 0.333, 0, 0>

#### Interval Spectrum

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

p4m5n5s6d6t2

#### 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> = {1,2,4}
<2> = {2,3,5,6}
<3> = {3,4,6,7}
<4> = {4,5,7,8}
<5> = {5,6,8,9}
<6> = {6,7,9,10}
<7> = {8,10,11}

#### Spectra Variation

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

3.25

#### 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.

2.366

#### Polygon Perimeter

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

5.838

#### 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.

(90, 48, 126)

#### Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.481

#### Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.357

## Generator

This scale has no generator.

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

F{5,9,0}341.71
am{9,0,4}231.57
f♯°{6,9,0}152.43

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 3 no C+, fm, am e°, f♯°

## Modes

Modes are the rotational transformation of this scale. Scale 2033 can be rotated to make 7 other scales. The 1st mode is itself.

 2nd mode:Scale 383 Logyllic This is the prime mode 3rd mode:Scale 2239 Dacryllic 4th mode:Scale 3167 Thynyllic 5th mode:Scale 3631 Gydyllic 6th mode:Scale 3863 Eparyllic 7th mode:Scale 3979 Dynyllic 8th mode:Scale 4037 Ionyllic

## Prime

The prime form of this scale is Scale 383

 Scale 383 Logyllic

## Complement

The octatonic modal family [2033, 383, 2239, 3167, 3631, 3863, 3979, 4037] (Forte: 8-2) is the complement of the tetratonic modal family [23, 1793, 2059, 3077] (Forte: 4-2)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2033 is 509

 Scale 509 Ionothyllic

## Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2033 is chiral, and its enantiomorph is scale 509

 Scale 509 Ionothyllic

## 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> 2033       T0I <11,0> 509
T1 <1,1> 4066      T1I <11,1> 1018
T2 <1,2> 4037      T2I <11,2> 2036
T3 <1,3> 3979      T3I <11,3> 4072
T4 <1,4> 3863      T4I <11,4> 4049
T5 <1,5> 3631      T5I <11,5> 4003
T6 <1,6> 3167      T6I <11,6> 3911
T7 <1,7> 2239      T7I <11,7> 3727
T8 <1,8> 383      T8I <11,8> 3359
T9 <1,9> 766      T9I <11,9> 2623
T10 <1,10> 1532      T10I <11,10> 1151
T11 <1,11> 3064      T11I <11,11> 2302
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2903      T0MI <7,0> 3419
T1M <5,1> 1711      T1MI <7,1> 2743
T2M <5,2> 3422      T2MI <7,2> 1391
T3M <5,3> 2749      T3MI <7,3> 2782
T4M <5,4> 1403      T4MI <7,4> 1469
T5M <5,5> 2806      T5MI <7,5> 2938
T6M <5,6> 1517      T6MI <7,6> 1781
T7M <5,7> 3034      T7MI <7,7> 3562
T8M <5,8> 1973      T8MI <7,8> 3029
T9M <5,9> 3946      T9MI <7,9> 1963
T10M <5,10> 3797      T10MI <7,10> 3926
T11M <5,11> 3499      T11MI <7,11> 3757

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 2035 Aerythygic Scale 2037 Sythygic Scale 2041 Aeolacrygic Scale 2017 MEQian Scale 2025 Mixolydyllic Scale 2001 Gydian Scale 1969 Zorian Scale 1905 Katacrian Scale 1777 Saptian Scale 1521 Stanian Scale 1009 Katyptian Scale 3057 Phroryllic Scale 4081 Nonatonic Chromatic Descending

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.