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Scale 2397: "Stagian"

Scale 2397: Stagian, 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
Stagian

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

{0,2,3,4,6,8,11}

Forte Number

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

7-26

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: 1875

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.

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: 699

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.

[2, 1, 1, 2, 2, 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, 4, 5, 3, 2>

Interval Spectrum

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

p3m5n4s4d3t2

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

Spectra Variation

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

2.286

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.

(13, 34, 98)

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 TriadsE{4,8,11}231.75
G♯{8,0,3}331.5
B{11,3,6}331.5
Minor Triadsg♯m{8,11,3}421.25
bm{11,2,6}242
Augmented TriadsC+{0,4,8}242
Diminished Triads{0,3,6}231.75
g♯°{8,11,2}231.75
Parsimonious Voice Leading Between Common Triads of Scale 2397. Created by Ian Ring ©2019 G# G# c°->G# B B c°->B C+ C+ E E C+->E C+->G# g#m g#m E->g#m g#° g#° g#°->g#m bm bm g#°->bm g#m->G# g#m->B bm->B

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♯m
Peripheral VerticesC+, bm

Modes

Modes are the rotational transformation of this scale. Scale 2397 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 1623
Scale 1623: Lothian, Ian Ring Music TheoryLothian
3rd mode:
Scale 2859
Scale 2859: Phrycrian, Ian Ring Music TheoryPhrycrian
4th mode:
Scale 3477
Scale 3477: Kyptian, Ian Ring Music TheoryKyptian
5th mode:
Scale 1893
Scale 1893: Ionylian, Ian Ring Music TheoryIonylian
6th mode:
Scale 1497
Scale 1497: Mela Jyotisvarupini, Ian Ring Music TheoryMela Jyotisvarupini
7th mode:
Scale 699
Scale 699: Aerothian, Ian Ring Music TheoryAerothianThis is the prime mode

Prime

The prime form of this scale is Scale 699

Scale 699Scale 699: Aerothian, Ian Ring Music TheoryAerothian

Complement

The heptatonic modal family [2397, 1623, 2859, 3477, 1893, 1497, 699] (Forte: 7-26) is the complement of the pentatonic modal family [309, 849, 1101, 1299, 2697] (Forte: 5-26)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2397 is 1875

Scale 1875Scale 1875: Persichetti Scale, Ian Ring Music TheoryPersichetti Scale

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2397 is chiral, and its enantiomorph is scale 1875

Scale 1875Scale 1875: Persichetti Scale, Ian Ring Music TheoryPersichetti Scale

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> 2397       T0I <11,0> 1875
T1 <1,1> 699      T1I <11,1> 3750
T2 <1,2> 1398      T2I <11,2> 3405
T3 <1,3> 2796      T3I <11,3> 2715
T4 <1,4> 1497      T4I <11,4> 1335
T5 <1,5> 2994      T5I <11,5> 2670
T6 <1,6> 1893      T6I <11,6> 1245
T7 <1,7> 3786      T7I <11,7> 2490
T8 <1,8> 3477      T8I <11,8> 885
T9 <1,9> 2859      T9I <11,9> 1770
T10 <1,10> 1623      T10I <11,10> 3540
T11 <1,11> 3246      T11I <11,11> 2985
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1497      T0MI <7,0> 885
T1M <5,1> 2994      T1MI <7,1> 1770
T2M <5,2> 1893      T2MI <7,2> 3540
T3M <5,3> 3786      T3MI <7,3> 2985
T4M <5,4> 3477      T4MI <7,4> 1875
T5M <5,5> 2859      T5MI <7,5> 3750
T6M <5,6> 1623      T6MI <7,6> 3405
T7M <5,7> 3246      T7MI <7,7> 2715
T8M <5,8> 2397       T8MI <7,8> 1335
T9M <5,9> 699      T9MI <7,9> 2670
T10M <5,10> 1398      T10MI <7,10> 1245
T11M <5,11> 2796      T11MI <7,11> 2490

The transformations that map this set to itself are: T0, T8M

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 2399Scale 2399: Zanyllic, Ian Ring Music TheoryZanyllic
Scale 2393Scale 2393: Zathimic, Ian Ring Music TheoryZathimic
Scale 2395Scale 2395: Zoptian, Ian Ring Music TheoryZoptian
Scale 2389Scale 2389: Eskimo Hexatonic 2, Ian Ring Music TheoryEskimo Hexatonic 2
Scale 2381Scale 2381: Takemitsu Linea Mode 1, Ian Ring Music TheoryTakemitsu Linea Mode 1
Scale 2413Scale 2413: Locrian Natural 2, Ian Ring Music TheoryLocrian Natural 2
Scale 2429Scale 2429: Kadyllic, Ian Ring Music TheoryKadyllic
Scale 2333Scale 2333: Stynimic, Ian Ring Music TheoryStynimic
Scale 2365Scale 2365: Sythian, Ian Ring Music TheorySythian
Scale 2461Scale 2461: Sagian, Ian Ring Music TheorySagian
Scale 2525Scale 2525: Aeolaryllic, Ian Ring Music TheoryAeolaryllic
Scale 2141Scale 2141, Ian Ring Music Theory
Scale 2269Scale 2269: Pygian, Ian Ring Music TheoryPygian
Scale 2653Scale 2653: Sygian, Ian Ring Music TheorySygian
Scale 2909Scale 2909: Mocryllic, Ian Ring Music TheoryMocryllic
Scale 3421Scale 3421: Aerothyllic, Ian Ring Music TheoryAerothyllic
Scale 349Scale 349: Borimic, Ian Ring Music TheoryBorimic
Scale 1373Scale 1373: Storian, Ian Ring Music TheoryStorian

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.