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Scale 1521: "Stanian"

Scale 1521: Stanian, Ian Ring Music Theory

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).

Common Names

Zeitler
Stanian
Dozenal
Jiyian

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

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

Forte Number

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

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: 501

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.

[4, 1, 1, 1, 1, 2, 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.

<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.

Common Triads

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

Triad TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsC{0,4,7}221
Minor Triadsfm{5,8,0}131.5
Augmented TriadsC+{0,4,8}221
Diminished Triads{4,7,10}131.5

The following pitch classes are not present in any of the common triads: {6}

Parsimonious Voice Leading Between Common Triads of Scale 1521. Created by Ian Ring ©2019 C C C+ C+ C->C+ C->e° fm fm C+->fm

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.

Diameter3
Radius2
Self-Centeredno
Central VerticesC, C+
Peripheral Verticese°, fm

Modes

Modes are the rotational transformation of this scale. Scale 1521 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 351
Scale 351: Epanian, Ian Ring Music TheoryEpanianThis is the prime mode
3rd mode:
Scale 2223
Scale 2223: Konian, Ian Ring Music TheoryKonian
4th mode:
Scale 3159
Scale 3159: Stocrian, Ian Ring Music TheoryStocrian
5th mode:
Scale 3627
Scale 3627: Kalian, Ian Ring Music TheoryKalian
6th mode:
Scale 3861
Scale 3861: Phroptian, Ian Ring Music TheoryPhroptian
7th mode:
Scale 1989
Scale 1989: Dydian, Ian Ring Music TheoryDydian

Prime

The prime form of this scale is Scale 351

Scale 351Scale 351: Epanian, Ian Ring Music TheoryEpanian

Complement

The heptatonic modal family [1521, 351, 2223, 3159, 3627, 3861, 1989] (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 1521 is 501

Scale 501Scale 501: Katylian, Ian Ring Music TheoryKatylian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1521 is chiral, and its enantiomorph is scale 501

Scale 501Scale 501: Katylian, Ian Ring Music TheoryKatylian

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

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 1523Scale 1523: Zothyllic, Ian Ring Music TheoryZothyllic
Scale 1525Scale 1525: Sodyllic, Ian Ring Music TheorySodyllic
Scale 1529Scale 1529: Kataryllic, Ian Ring Music TheoryKataryllic
Scale 1505Scale 1505: Jepian, Ian Ring Music TheoryJepian
Scale 1513Scale 1513: Stathian, Ian Ring Music TheoryStathian
Scale 1489Scale 1489: Raga Jyoti, Ian Ring Music TheoryRaga Jyoti
Scale 1457Scale 1457: Raga Kamalamanohari, Ian Ring Music TheoryRaga Kamalamanohari
Scale 1393Scale 1393: Mycrimic, Ian Ring Music TheoryMycrimic
Scale 1265Scale 1265: Pynimic, Ian Ring Music TheoryPynimic
Scale 1777Scale 1777: Saptian, Ian Ring Music TheorySaptian
Scale 2033Scale 2033: Stolyllic, Ian Ring Music TheoryStolyllic
Scale 497Scale 497: Kadimic, Ian Ring Music TheoryKadimic
Scale 1009Scale 1009: Katyptian, Ian Ring Music TheoryKatyptian
Scale 2545Scale 2545: Thycrian, Ian Ring Music TheoryThycrian
Scale 3569Scale 3569: Aeoladyllic, Ian Ring Music TheoryAeoladyllic

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.