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Scale 1403: "Espla's Scale"

Scale 1403: Espla's Scale, Ian Ring Music Theory

Named after the great Spanish composer, Óscar Esplá y Triay (1886 – 1976). Óscar studied in Barcelona with Max Reger, and Paris with Camille Saint Saëns, winning the coveted Vienna prize in 1911.


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

Named After Composers
Espla's Scale
Exoticisms
Eight-tone Spanish
Zeitler
Epinyllic
Dozenal
Ipuian

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

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

Forte Number

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

8-22

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: 3029

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.

2 (dicohemitonic)

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.

7

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

no
prime: 1391

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, an indicator of maximum hierarchization.

no

Interval Structure

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

[1, 2, 1, 1, 1, 2, 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, 6, 5, 5, 6, 2>

Interval Spectrum

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

p6m5n5s6d4t2

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

1.75

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 Distribution Spectra have 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.

(10, 59, 137)

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}341.9
F♯{6,10,1}242.1
G♯{8,0,3}242.1
Minor Triadsc♯m{1,4,8}341.9
d♯m{3,6,10}242.3
fm{5,8,0}242.1
a♯m{10,1,5}341.9
Augmented TriadsC+{0,4,8}341.9
Diminished Triads{0,3,6}242.3
a♯°{10,1,4}242.1
Parsimonious Voice Leading Between Common Triads of Scale 1403. Created by Ian Ring ©2019 d#m d#m c°->d#m G# G# c°->G# C+ C+ c#m c#m C+->c#m fm fm C+->fm C+->G# C# C# c#m->C# a#° a#° c#m->a#° C#->fm a#m a#m C#->a#m F# F# d#m->F# 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 1403 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 2749
Scale 2749: Katagyllic, Ian Ring Music TheoryKatagyllic
3rd mode:
Scale 1711
Scale 1711: Adonai Malakh, Ian Ring Music TheoryAdonai Malakh
4th mode:
Scale 2903
Scale 2903: Gothyllic, Ian Ring Music TheoryGothyllic
5th mode:
Scale 3499
Scale 3499: Hamel, Ian Ring Music TheoryHamel
6th mode:
Scale 3797
Scale 3797: Rocryllic, Ian Ring Music TheoryRocryllic
7th mode:
Scale 1973
Scale 1973: Zyryllic, Ian Ring Music TheoryZyryllic
8th mode:
Scale 1517
Scale 1517: Sagyllic, Ian Ring Music TheorySagyllic

Prime

The prime form of this scale is Scale 1391

Scale 1391Scale 1391: Aeradyllic, Ian Ring Music TheoryAeradyllic

Complement

The octatonic modal family [1403, 2749, 1711, 2903, 3499, 3797, 1973, 1517] (Forte: 8-22) is the complement of the tetratonic modal family [149, 673, 1061, 1289] (Forte: 4-22)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1403 is 3029

Scale 3029Scale 3029: Ionocryllic, Ian Ring Music TheoryIonocryllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1403 is chiral, and its enantiomorph is scale 3029

Scale 3029Scale 3029: Ionocryllic, Ian Ring Music TheoryIonocryllic

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> 1403       T0I <11,0> 3029
T1 <1,1> 2806      T1I <11,1> 1963
T2 <1,2> 1517      T2I <11,2> 3926
T3 <1,3> 3034      T3I <11,3> 3757
T4 <1,4> 1973      T4I <11,4> 3419
T5 <1,5> 3946      T5I <11,5> 2743
T6 <1,6> 3797      T6I <11,6> 1391
T7 <1,7> 3499      T7I <11,7> 2782
T8 <1,8> 2903      T8I <11,8> 1469
T9 <1,9> 1711      T9I <11,9> 2938
T10 <1,10> 3422      T10I <11,10> 1781
T11 <1,11> 2749      T11I <11,11> 3562
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 383      T0MI <7,0> 4049
T1M <5,1> 766      T1MI <7,1> 4003
T2M <5,2> 1532      T2MI <7,2> 3911
T3M <5,3> 3064      T3MI <7,3> 3727
T4M <5,4> 2033      T4MI <7,4> 3359
T5M <5,5> 4066      T5MI <7,5> 2623
T6M <5,6> 4037      T6MI <7,6> 1151
T7M <5,7> 3979      T7MI <7,7> 2302
T8M <5,8> 3863      T8MI <7,8> 509
T9M <5,9> 3631      T9MI <7,9> 1018
T10M <5,10> 3167      T10MI <7,10> 2036
T11M <5,11> 2239      T11MI <7,11> 4072

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 1401Scale 1401: Pagian, Ian Ring Music TheoryPagian
Scale 1405Scale 1405: Goryllic, Ian Ring Music TheoryGoryllic
Scale 1407Scale 1407: Tharygic, Ian Ring Music TheoryTharygic
Scale 1395Scale 1395: Locrian Dominant, Ian Ring Music TheoryLocrian Dominant
Scale 1399Scale 1399: Syryllic, Ian Ring Music TheorySyryllic
Scale 1387Scale 1387: Locrian, Ian Ring Music TheoryLocrian
Scale 1371Scale 1371: Superlocrian, Ian Ring Music TheorySuperlocrian
Scale 1339Scale 1339: Kycrian, Ian Ring Music TheoryKycrian
Scale 1467Scale 1467: Spanish Phrygian, Ian Ring Music TheorySpanish Phrygian
Scale 1531Scale 1531: Styptygic, Ian Ring Music TheoryStyptygic
Scale 1147Scale 1147: Epynian, Ian Ring Music TheoryEpynian
Scale 1275Scale 1275: Stagyllic, Ian Ring Music TheoryStagyllic
Scale 1659Scale 1659: Maqam Shadd'araban, Ian Ring Music TheoryMaqam Shadd'araban
Scale 1915Scale 1915: Thydygic, Ian Ring Music TheoryThydygic
Scale 379Scale 379: Aeragian, Ian Ring Music TheoryAeragian
Scale 891Scale 891: Ionilyllic, Ian Ring Music TheoryIonilyllic
Scale 2427Scale 2427: Katoryllic, Ian Ring Music TheoryKatoryllic
Scale 3451Scale 3451: Garygic, Ian Ring Music TheoryGarygic

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.