 The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

# Scale 3509: "Stogyllic" ### 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
Stogyllic
Dozenal
Wagian

## 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

{0,2,4,5,7,8,10,11}

#### Forte Number

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

8-27

#### 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: 1463

#### Hemitonia

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

4 (multihemitonic)

#### 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.

7

#### Prime Form

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

no
prime: 1463

#### 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, 2, 1, 2, 1, 2, 1, 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.

<4, 5, 6, 5, 5, 3>

#### Interval Spectrum

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

p5m5n6s5d4t3

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

#### Spectra Variation

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

1.25

#### 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.

yes

#### 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.

2.732

#### Polygon Perimeter

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

6.071

#### 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, 24, 103)

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}441.92
G{7,11,2}441.92
A♯{10,2,5}342.23
fm{5,8,0}342.23
gm{7,10,2}342.15
{4,7,10}242.23
{5,8,11}242.31
g♯°{8,11,2}242.15
{11,2,5}242.31

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 4 4 yes

## Modes

Modes are the rotational transformation of this scale. Scale 3509 can be rotated to make 7 other scales. The 1st mode is itself.

 2nd mode:Scale 1901 Ionidyllic 3rd mode:Scale 1499 Bebop Locrian 4th mode:Scale 2797 Stalyllic 5th mode:Scale 1723 JG Octatonic 6th mode:Scale 2909 Mocryllic 7th mode:Scale 1751 Aeolyryllic 8th mode:Scale 2923 Baryllic

## Prime

The prime form of this scale is Scale 1463

 Scale 1463 Ugrian

## Complement

The octatonic modal family [3509, 1901, 1499, 2797, 1723, 2909, 1751, 2923] (Forte: 8-27) is the complement of the tetratonic modal family [293, 593, 649, 1097] (Forte: 4-27)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3509 is 1463

 Scale 1463 Ugrian

## Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3509 is chiral, and its enantiomorph is scale 1463

 Scale 1463 Ugrian

## 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> 3509       T0I <11,0> 1463
T1 <1,1> 2923      T1I <11,1> 2926
T2 <1,2> 1751      T2I <11,2> 1757
T3 <1,3> 3502      T3I <11,3> 3514
T4 <1,4> 2909      T4I <11,4> 2933
T5 <1,5> 1723      T5I <11,5> 1771
T6 <1,6> 3446      T6I <11,6> 3542
T7 <1,7> 2797      T7I <11,7> 2989
T8 <1,8> 1499      T8I <11,8> 1883
T9 <1,9> 2998      T9I <11,9> 3766
T10 <1,10> 1901      T10I <11,10> 3437
T11 <1,11> 3802      T11I <11,11> 2779
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3479      T0MI <7,0> 3383
T1M <5,1> 2863      T1MI <7,1> 2671
T2M <5,2> 1631      T2MI <7,2> 1247
T3M <5,3> 3262      T3MI <7,3> 2494
T4M <5,4> 2429      T4MI <7,4> 893
T5M <5,5> 763      T5MI <7,5> 1786
T6M <5,6> 1526      T6MI <7,6> 3572
T7M <5,7> 3052      T7MI <7,7> 3049
T8M <5,8> 2009      T8MI <7,8> 2003
T9M <5,9> 4018      T9MI <7,9> 4006
T10M <5,10> 3941      T10MI <7,10> 3917
T11M <5,11> 3787      T11MI <7,11> 3739

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 3511 Epolygic Scale 3505 Stygian Scale 3507 Maqam Hijaz Scale 3513 Dydyllic Scale 3517 Epocrygic Scale 3493 Rathian Scale 3501 Maqam Nahawand Scale 3477 Kyptian Scale 3541 Racryllic Scale 3573 Kaptygic Scale 3381 Katanian Scale 3445 Messiaen Mode 6 Inverse Scale 3253 Mela Naganandini Scale 3765 Dominant Bebop Scale 4021 Raga Pahadi Scale 2485 Harmonic Major Scale 2997 Major Bebop Scale 1461 Major-Minor

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.