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Scale 2475: "Neapolitan Minor"

Scale 2475: Neapolitan Minor, Ian Ring Music Theory

With the distinctive ♭2 of the Neapolitan, and the upper tones of a harmonic minor, this scale is less contentious than its sibling the Neapolitan Major. This scale is constructed to harmonize well with a neapolitan N6-V-i cadence.


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

Western
Neapolitan Minor
Minor Neapolitan
Dozenal
Egrian
Carnatic
Mela Dhenuka
Raga Bhinnasadjam
Dhunibinnashadjam
Unknown / Unsorted
Kirvanti
Takka
Arabic
Maqam Shahnaz Kurdi
Exoticisms
Hungarian Romani
Zeitler
Aerylian
Carnatic Melakarta
Dhenuka
Carnatic Numbered Melakarta
9th Melakarta raga

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

{0,1,3,5,7,8,11}

Forte Number

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

7-30

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: 2739

Hemitonia

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

3 (trihemitonic)

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.

6

Prime Form

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

no
prime: 855

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, 2, 2, 2, 1, 3, 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.

<3, 4, 3, 5, 4, 2>

Interval Spectrum

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

p4m5n3s4d3t2

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

Spectra Variation

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

1.714

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.

2.549

Polygon Perimeter

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

5.967

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.

(2, 22, 86)

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♯{1,5,8}142.14
G♯{8,0,3}321.29
Minor Triadscm{0,3,7}231.71
fm{5,8,0}331.43
g♯m{8,11,3}331.43
Augmented TriadsD♯+{3,7,11}241.86
Diminished Triads{5,8,11}231.57
Parsimonious Voice Leading Between Common Triads of Scale 2475. Created by Ian Ring ©2019 cm cm D#+ D#+ cm->D#+ G# G# cm->G# C# C# fm fm C#->fm g#m g#m D#+->g#m f°->fm f°->g#m fm->G# g#m->G#

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
Radius2
Self-Centeredno
Central VerticesG♯
Peripheral VerticesC♯, D♯+

Modes

Modes are the rotational transformation of this scale. Scale 2475 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3285
Scale 3285: Mela Citrambari, Ian Ring Music TheoryMela Citrambari
3rd mode:
Scale 1845
Scale 1845: Lagian, Ian Ring Music TheoryLagian
4th mode:
Scale 1485
Scale 1485: Minor Romani, Ian Ring Music TheoryMinor Romani
5th mode:
Scale 1395
Scale 1395: Locrian Dominant, Ian Ring Music TheoryLocrian Dominant
6th mode:
Scale 2745
Scale 2745: Mela Sulini, Ian Ring Music TheoryMela Sulini
7th mode:
Scale 855
Scale 855: Porian, Ian Ring Music TheoryPorianThis is the prime mode

Prime

The prime form of this scale is Scale 855

Scale 855Scale 855: Porian, Ian Ring Music TheoryPorian

Complement

The heptatonic modal family [2475, 3285, 1845, 1485, 1395, 2745, 855] (Forte: 7-30) is the complement of the pentatonic modal family [339, 789, 1221, 1329, 2217] (Forte: 5-30)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2475 is 2739

Scale 2739Scale 2739: Mela Suryakanta, Ian Ring Music TheoryMela Suryakanta

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2475 is chiral, and its enantiomorph is scale 2739

Scale 2739Scale 2739: Mela Suryakanta, Ian Ring Music TheoryMela Suryakanta

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> 2475       T0I <11,0> 2739
T1 <1,1> 855      T1I <11,1> 1383
T2 <1,2> 1710      T2I <11,2> 2766
T3 <1,3> 3420      T3I <11,3> 1437
T4 <1,4> 2745      T4I <11,4> 2874
T5 <1,5> 1395      T5I <11,5> 1653
T6 <1,6> 2790      T6I <11,6> 3306
T7 <1,7> 1485      T7I <11,7> 2517
T8 <1,8> 2970      T8I <11,8> 939
T9 <1,9> 1845      T9I <11,9> 1878
T10 <1,10> 3690      T10I <11,10> 3756
T11 <1,11> 3285      T11I <11,11> 3417
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2235      T0MI <7,0> 2979
T1M <5,1> 375      T1MI <7,1> 1863
T2M <5,2> 750      T2MI <7,2> 3726
T3M <5,3> 1500      T3MI <7,3> 3357
T4M <5,4> 3000      T4MI <7,4> 2619
T5M <5,5> 1905      T5MI <7,5> 1143
T6M <5,6> 3810      T6MI <7,6> 2286
T7M <5,7> 3525      T7MI <7,7> 477
T8M <5,8> 2955      T8MI <7,8> 954
T9M <5,9> 1815      T9MI <7,9> 1908
T10M <5,10> 3630      T10MI <7,10> 3816
T11M <5,11> 3165      T11MI <7,11> 3537

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 2473Scale 2473: Raga Takka, Ian Ring Music TheoryRaga Takka
Scale 2477Scale 2477: Harmonic Minor, Ian Ring Music TheoryHarmonic Minor
Scale 2479Scale 2479: Harmonic and Neapolitan Minor Mixed, Ian Ring Music TheoryHarmonic and Neapolitan Minor Mixed
Scale 2467Scale 2467: Raga Padi, Ian Ring Music TheoryRaga Padi
Scale 2471Scale 2471: Mela Ganamurti, Ian Ring Music TheoryMela Ganamurti
Scale 2483Scale 2483: Double Harmonic, Ian Ring Music TheoryDouble Harmonic
Scale 2491Scale 2491: Layllic, Ian Ring Music TheoryLayllic
Scale 2443Scale 2443: Panimic, Ian Ring Music TheoryPanimic
Scale 2459Scale 2459: Ionocrian, Ian Ring Music TheoryIonocrian
Scale 2507Scale 2507: Todi That, Ian Ring Music TheoryTodi That
Scale 2539Scale 2539: Half-Diminished Bebop, Ian Ring Music TheoryHalf-Diminished Bebop
Scale 2347Scale 2347: Raga Viyogavarali, Ian Ring Music TheoryRaga Viyogavarali
Scale 2411Scale 2411: Aeolorian, Ian Ring Music TheoryAeolorian
Scale 2219Scale 2219: Phrydimic, Ian Ring Music TheoryPhrydimic
Scale 2731Scale 2731: Neapolitan Major, Ian Ring Music TheoryNeapolitan Major
Scale 2987Scale 2987: Neapolitan Major and Minor Mixed, Ian Ring Music TheoryNeapolitan Major and Minor Mixed
Scale 3499Scale 3499: Hamel, Ian Ring Music TheoryHamel
Scale 427Scale 427: Raga Suddha Simantini, Ian Ring Music TheoryRaga Suddha Simantini
Scale 1451Scale 1451: Phrygian, Ian Ring Music TheoryPhrygian

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.