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Scale 1531: "Styptygic"

Scale 1531: Styptygic, 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
Styptygic

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

{0,1,3,4,5,6,7,8,10}

Forte Number

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

9-7

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: 3061

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.

4 (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.

2

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 Rahn/Ring formula.

no
prime: 1471

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 Formula

Defines the scale as the sequence of intervals between one tone and the next.

[1, 2, 1, 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.

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

Interval Spectrum

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

p7m6n7s7d6t3

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

Spectra Variation

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

1.778

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.

2.799

Polygon Perimeter

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

6.106

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.

(25, 109, 196)

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}442.13
C♯{1,5,8}342.44
D♯{3,7,10}442.31
F♯{6,10,1}342.44
G♯{8,0,3}242.38
Minor Triadscm{0,3,7}442.19
c♯m{1,4,8}442.31
d♯m{3,6,10}342.44
fm{5,8,0}242.56
a♯m{10,1,5}342.44
Augmented TriadsC+{0,4,8}442.19
Diminished Triads{0,3,6}242.56
c♯°{1,4,7}242.44
{4,7,10}242.44
{7,10,1}242.56
a♯°{10,1,4}242.56
Parsimonious Voice Leading Between Common Triads of Scale 1531. Created by Ian Ring ©2019 cm cm c°->cm d#m d#m c°->d#m C C cm->C D# D# cm->D# G# G# cm->G# C+ C+ C->C+ c#° c#° C->c#° C->e° c#m c#m C+->c#m fm fm C+->fm C+->G# c#°->c#m C# C# c#m->C# a#° a#° c#m->a#° C#->fm a#m a#m C#->a#m d#m->D# F# F# d#m->F# D#->e° D#->g° F#->g° F#->a#m a#°->a#m

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.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1531 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 2813
Scale 2813: Zolygic, Ian Ring Music TheoryZolygic
3rd mode:
Scale 1727
Scale 1727: Sydygic, Ian Ring Music TheorySydygic
4th mode:
Scale 2911
Scale 2911: Katygic, Ian Ring Music TheoryKatygic
5th mode:
Scale 3503
Scale 3503: Zyphygic, Ian Ring Music TheoryZyphygic
6th mode:
Scale 3799
Scale 3799: Aeralygic, Ian Ring Music TheoryAeralygic
7th mode:
Scale 3947
Scale 3947: Ryptygic, Ian Ring Music TheoryRyptygic
8th mode:
Scale 4021
Scale 4021: Raga Pahadi, Ian Ring Music TheoryRaga Pahadi
9th mode:
Scale 2029
Scale 2029: Kiourdi, Ian Ring Music TheoryKiourdi

Prime

The prime form of this scale is Scale 1471

Scale 1471Scale 1471: Radygic, Ian Ring Music TheoryRadygic

Complement

The enneatonic modal family [1531, 2813, 1727, 2911, 3503, 3799, 3947, 4021, 2029] (Forte: 9-7) is the complement of the tritonic modal family [37, 641, 1033] (Forte: 3-7)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1531 is 3061

Scale 3061Scale 3061: Apinygic, Ian Ring Music TheoryApinygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1531 is chiral, and its enantiomorph is scale 3061

Scale 3061Scale 3061: Apinygic, Ian Ring Music TheoryApinygic

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> 1531       T0I <11,0> 3061
T1 <1,1> 3062      T1I <11,1> 2027
T2 <1,2> 2029      T2I <11,2> 4054
T3 <1,3> 4058      T3I <11,3> 4013
T4 <1,4> 4021      T4I <11,4> 3931
T5 <1,5> 3947      T5I <11,5> 3767
T6 <1,6> 3799      T6I <11,6> 3439
T7 <1,7> 3503      T7I <11,7> 2783
T8 <1,8> 2911      T8I <11,8> 1471
T9 <1,9> 1727      T9I <11,9> 2942
T10 <1,10> 3454      T10I <11,10> 1789
T11 <1,11> 2813      T11I <11,11> 3578
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2431      T0MI <7,0> 4051
T1M <5,1> 767      T1MI <7,1> 4007
T2M <5,2> 1534      T2MI <7,2> 3919
T3M <5,3> 3068      T3MI <7,3> 3743
T4M <5,4> 2041      T4MI <7,4> 3391
T5M <5,5> 4082      T5MI <7,5> 2687
T6M <5,6> 4069      T6MI <7,6> 1279
T7M <5,7> 4043      T7MI <7,7> 2558
T8M <5,8> 3991      T8MI <7,8> 1021
T9M <5,9> 3887      T9MI <7,9> 2042
T10M <5,10> 3679      T10MI <7,10> 4084
T11M <5,11> 3263      T11MI <7,11> 4073

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 1529Scale 1529: Kataryllic, Ian Ring Music TheoryKataryllic
Scale 1533Scale 1533: Katycrygic, Ian Ring Music TheoryKatycrygic
Scale 1535Scale 1535: Mixodyllian, Ian Ring Music TheoryMixodyllian
Scale 1523Scale 1523: Zothyllic, Ian Ring Music TheoryZothyllic
Scale 1527Scale 1527: Aeolyrigic, Ian Ring Music TheoryAeolyrigic
Scale 1515Scale 1515: Phrygian/Locrian Mixed, Ian Ring Music TheoryPhrygian/Locrian Mixed
Scale 1499Scale 1499: Bebop Locrian, Ian Ring Music TheoryBebop Locrian
Scale 1467Scale 1467: Spanish Phrygian, Ian Ring Music TheorySpanish Phrygian
Scale 1403Scale 1403: Espla's Scale, Ian Ring Music TheoryEspla's Scale
Scale 1275Scale 1275: Stagyllic, Ian Ring Music TheoryStagyllic
Scale 1787Scale 1787: Mycrygic, Ian Ring Music TheoryMycrygic
Scale 2043Scale 2043: Maqam Tarzanuyn, Ian Ring Music TheoryMaqam Tarzanuyn
Scale 507Scale 507: Moryllic, Ian Ring Music TheoryMoryllic
Scale 1019Scale 1019: Aeranygic, Ian Ring Music TheoryAeranygic
Scale 2555Scale 2555: Bythygic, Ian Ring Music TheoryBythygic
Scale 3579Scale 3579: Zyphyllian, Ian Ring Music TheoryZyphyllian

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.