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Scale 1973: "Spanish Octamode 6th Rotation"

Scale 1973: Spanish Octamode 6th Rotation, 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

Western
Spanish Octamode 6th Rotation
Zeitler
Zyryllic
Dozenal
MAPian

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

 {0,2,4,5,7,8,9,10}

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: 1469

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 includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.

 8

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.

 [2, 2, 1, 2, 1, 1, 1, 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>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0, 0.667, 0.25, 0.333, 0.667, 0>

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)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.

0.741

Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.

0.301

Generator

This scale has no generator.

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}242.1
F{5,9,0}341.9
A♯{10,2,5}242.1
Minor Triadsdm{2,5,9}341.9
fm{5,8,0}341.9
gm{7,10,2}242.3
am{9,0,4}242.1
Augmented TriadsC+{0,4,8}341.9
Diminished Triadsd°{2,5,8}242.1
e°{4,7,10}242.3
Parsimonious Voice Leading Between Common Triads of Scale 1973. Created by Ian Ring ©2019 C C C+ C+ C->C+ e° e° C->e° fm fm C+->fm am am C+->am d° d° dm dm d°->dm d°->fm F F dm->F A# A# dm->A# gm gm e°->gm fm->F F->am gm->A#

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 1973 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1517		Scale 1517: Spanish Octamode 4th Rotation, Ian Ring Music TheorySpanish Octamode 4th Rotation
3rd mode:
Scale 1403		Scale 1403: Espla's Scale, Ian Ring Music TheoryEspla's Scale
4th mode:
Scale 2749		Scale 2749: Spanish Octamode 1st Rotation, Ian Ring Music TheorySpanish Octamode 1st Rotation
5th mode:
Scale 1711		Scale 1711: Adonai Malakh, Ian Ring Music TheoryAdonai Malakh
6th mode:
Scale 2903		Scale 2903: Spanish Octamode 10th Rotation, Ian Ring Music TheorySpanish Octamode 10th Rotation
7th mode:
Scale 3499		Scale 3499: Hamel, Ian Ring Music TheoryHamel
8th mode:
Scale 3797		Scale 3797: Spanish Octamode 8th Rotation, Ian Ring Music TheorySpanish Octamode 8th Rotation

Prime

The prime form of this scale is Scale 1391

Scale 1391Scale 1391: Aeradyllic, Ian Ring Music TheoryAeradyllic

Complement

The octatonic modal family [1973, 1517, 1403, 2749, 1711, 2903, 3499, 3797] (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 1973 is 1469

Scale 1469Scale 1469: Epiryllic, Ian Ring Music TheoryEpiryllic

Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

kHierarchizabilityBreakdown PatternDiagram
111010110111101973k = 1h = 1
211010110111101973k = 2h = 1
330([1]0[1])([1]0[1])11([1]0[1])1973k = 3h = 3
43010([1][1]0[1])1([1][1]0[1])1973k = 4h = 3
53010([1][1]0[1])1([1][1]0[1])1973k = 5h = 3

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1973 is chiral, and its enantiomorph is scale 1469

Scale 1469Scale 1469: Epiryllic, Ian Ring Music TheoryEpiryllic

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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 1973       T0I <11,0> 1469
T1 <1,1> 3946      T1I <11,1> 2938
T2 <1,2> 3797      T2I <11,2> 1781
T3 <1,3> 3499      T3I <11,3> 3562
T4 <1,4> 2903      T4I <11,4> 3029
T5 <1,5> 1711      T5I <11,5> 1963
T6 <1,6> 3422      T6I <11,6> 3926
T7 <1,7> 2749      T7I <11,7> 3757
T8 <1,8> 1403      T8I <11,8> 3419
T9 <1,9> 2806      T9I <11,9> 2743
T10 <1,10> 1517      T10I <11,10> 1391
T11 <1,11> 3034      T11I <11,11> 2782
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3863      T0MI <7,0> 3359
T1M <5,1> 3631      T1MI <7,1> 2623
T2M <5,2> 3167      T2MI <7,2> 1151
T3M <5,3> 2239      T3MI <7,3> 2302
T4M <5,4> 383      T4MI <7,4> 509
T5M <5,5> 766      T5MI <7,5> 1018
T6M <5,6> 1532      T6MI <7,6> 2036
T7M <5,7> 3064      T7MI <7,7> 4072
T8M <5,8> 2033      T8MI <7,8> 4049
T9M <5,9> 4066      T9MI <7,9> 4003
T10M <5,10> 4037      T10MI <7,10> 3911
T11M <5,11> 3979      T11MI <7,11> 3727

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 1975Scale 1975: Ionocrygic, Ian Ring Music TheoryIonocrygic
Scale 1969Scale 1969: Zorian, Ian Ring Music TheoryZorian
Scale 1971Scale 1971: Aerynyllic, Ian Ring Music TheoryAerynyllic
Scale 1977Scale 1977: Dagyllic, Ian Ring Music TheoryDagyllic
Scale 1981Scale 1981: Houseini, Ian Ring Music TheoryHouseini
Scale 1957Scale 1957: Pyrian, Ian Ring Music TheoryPyrian
Scale 1965Scale 1965: Raga Mukhari, Ian Ring Music TheoryRaga Mukhari
Scale 1941Scale 1941: Aeranian, Ian Ring Music TheoryAeranian
Scale 2005Scale 2005: Gygyllic, Ian Ring Music TheoryGygyllic
Scale 2037Scale 2037: Sythygic, Ian Ring Music TheorySythygic
Scale 1845Scale 1845: Mixolydian Augmented, Ian Ring Music TheoryMixolydian Augmented
Scale 1909Scale 1909: Epicryllic, Ian Ring Music TheoryEpicryllic
Scale 1717Scale 1717: Mixolydian, Ian Ring Music TheoryMixolydian
Scale 1461Scale 1461: Major-Minor, Ian Ring Music TheoryMajor-Minor
Scale 949Scale 949: Mela Mararanjani, Ian Ring Music TheoryMela Mararanjani
Scale 2997Scale 2997: Major Bebop, Ian Ring Music TheoryMajor Bebop
Scale 4021Scale 4021: Raga Pahadi, Ian Ring Music TheoryRaga Pahadi

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.