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# Scale 3159: "Stocrian" ### 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
Stocrian
Dozenal
Tujian

## Analysis

#### Cardinality

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

7 (heptatonic)

#### Pitch Class Set

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

{0,1,2,4,6,10,11}

#### Forte Number

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

7-9

#### 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: 3399

#### 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.

3 (tricohemitonic)

#### 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.

6

#### Prime Form

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

no
prime: 351

#### 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.

[1, 1, 2, 2, 4, 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, 3, 4, 3, 2>

#### Interval Spectrum

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

p3m4n3s5d4t2

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

#### Spectra Variation

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

3.429

#### 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.

2.299

#### Polygon Perimeter

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

5.803

#### 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.

(46, 40, 104)

## Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony. 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

The following pitch classes are not present in any of the common triads: {0}

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 3 2 no D+, F♯ a♯°, bm

## Modes

Modes are the rotational transformation of this scale. Scale 3159 can be rotated to make 6 other scales. The 1st mode is itself.

 2nd mode:Scale 3627 Kalian 3rd mode:Scale 3861 Phroptian 4th mode:Scale 1989 Dydian 5th mode:Scale 1521 Stanian 6th mode:Scale 351 Epanian This is the prime mode 7th mode:Scale 2223 Konian

## Prime

The prime form of this scale is Scale 351

 Scale 351 Epanian

## Complement

The heptatonic modal family [3159, 3627, 3861, 1989, 1521, 351, 2223] (Forte: 7-9) is the complement of the pentatonic modal family [87, 1473, 1797, 2091, 3093] (Forte: 5-9)

## Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3159 is 3399

 Scale 3399 Zonian

## Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3159 is chiral, and its enantiomorph is scale 3399

 Scale 3399 Zonian

## 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> 3159       T0I <11,0> 3399
T1 <1,1> 2223      T1I <11,1> 2703
T2 <1,2> 351      T2I <11,2> 1311
T3 <1,3> 702      T3I <11,3> 2622
T4 <1,4> 1404      T4I <11,4> 1149
T5 <1,5> 2808      T5I <11,5> 2298
T6 <1,6> 1521      T6I <11,6> 501
T7 <1,7> 3042      T7I <11,7> 1002
T8 <1,8> 1989      T8I <11,8> 2004
T9 <1,9> 3978      T9I <11,9> 4008
T10 <1,10> 3861      T10I <11,10> 3921
T11 <1,11> 3627      T11I <11,11> 3747
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1509      T0MI <7,0> 1269
T1M <5,1> 3018      T1MI <7,1> 2538
T2M <5,2> 1941      T2MI <7,2> 981
T3M <5,3> 3882      T3MI <7,3> 1962
T4M <5,4> 3669      T4MI <7,4> 3924
T5M <5,5> 3243      T5MI <7,5> 3753
T6M <5,6> 2391      T6MI <7,6> 3411
T7M <5,7> 687      T7MI <7,7> 2727
T8M <5,8> 1374      T8MI <7,8> 1359
T9M <5,9> 2748      T9MI <7,9> 2718
T10M <5,10> 1401      T10MI <7,10> 1341
T11M <5,11> 2802      T11MI <7,11> 2682

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 3157 Zyptimic Scale 3155 Ladimic Scale 3163 Rogian Scale 3167 Thynyllic Scale 3143 Polimic Scale 3151 Pacrian Scale 3175 Eponian Scale 3191 Bynyllic Scale 3095 Tivian Scale 3127 Topian Scale 3223 Thyphian Scale 3287 Phrathyllic Scale 3415 Ionaptyllic Scale 3671 Aeonyllic Scale 2135 Nakian Scale 2647 Dadian Scale 1111 Sycrimic

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.