The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 1259: "Stadian"

Scale 1259: Stadian, 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
Stadian

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

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

Forte Number

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

7-29

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: 2789

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.

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.

6

Prime Form

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

no
prime: 727

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, 2, 1, 1, 3, 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.

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

Interval Spectrum

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

p5m3n4s4d3t2

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}
<4> = {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

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

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 TriadsD♯{3,7,10}331.43
F♯{6,10,1}331.43
Minor Triadscm{0,3,7}241.86
d♯m{3,6,10}321.29
a♯m{10,1,5}142.14
Diminished Triads{0,3,6}231.71
{7,10,1}231.57
Parsimonious Voice Leading Between Common Triads of Scale 1259. Created by Ian Ring ©2019 cm cm c°->cm d#m d#m c°->d#m D# D# cm->D# d#m->D# F# F# d#m->F# D#->g° F#->g° a#m a#m F#->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
Radius2
Self-Centeredno
Central Verticesd♯m
Peripheral Verticescm, a♯m

Modes

Modes are the rotational transformation of this scale. Scale 1259 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 2677
Scale 2677: Thodian, Ian Ring Music TheoryThodian
3rd mode:
Scale 1693
Scale 1693: Dogian, Ian Ring Music TheoryDogian
4th mode:
Scale 1447
Scale 1447: Mela Ratnangi, Ian Ring Music TheoryMela Ratnangi
5th mode:
Scale 2771
Scale 2771: Marva That, Ian Ring Music TheoryMarva That
6th mode:
Scale 3433
Scale 3433: Thonian, Ian Ring Music TheoryThonian
7th mode:
Scale 941
Scale 941: Mela Jhankaradhvani, Ian Ring Music TheoryMela Jhankaradhvani

Prime

The prime form of this scale is Scale 727

Scale 727Scale 727: Phradian, Ian Ring Music TheoryPhradian

Complement

The heptatonic modal family [1259, 2677, 1693, 1447, 2771, 3433, 941] (Forte: 7-29) is the complement of the pentatonic modal family [331, 709, 1201, 1577, 2213] (Forte: 5-29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1259 is 2789

Scale 2789Scale 2789: Zolian, Ian Ring Music TheoryZolian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1259 is chiral, and its enantiomorph is scale 2789

Scale 2789Scale 2789: Zolian, Ian Ring Music TheoryZolian

Transformations:

T0 1259  T0I 2789
T1 2518  T1I 1483
T2 941  T2I 2966
T3 1882  T3I 1837
T4 3764  T4I 3674
T5 3433  T5I 3253
T6 2771  T6I 2411
T7 1447  T7I 727
T8 2894  T8I 1454
T9 1693  T9I 2908
T10 3386  T10I 1721
T11 2677  T11I 3442

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 1257Scale 1257: Blues Scale, Ian Ring Music TheoryBlues Scale
Scale 1261Scale 1261: Modified Blues, Ian Ring Music TheoryModified Blues
Scale 1263Scale 1263: Stynyllic, Ian Ring Music TheoryStynyllic
Scale 1251Scale 1251: Sylimic, Ian Ring Music TheorySylimic
Scale 1255Scale 1255: Chromatic Mixolydian, Ian Ring Music TheoryChromatic Mixolydian
Scale 1267Scale 1267: Katynian, Ian Ring Music TheoryKatynian
Scale 1275Scale 1275: Stagyllic, Ian Ring Music TheoryStagyllic
Scale 1227Scale 1227: Thacrimic, Ian Ring Music TheoryThacrimic
Scale 1243Scale 1243: Epylian, Ian Ring Music TheoryEpylian
Scale 1195Scale 1195: Raga Gandharavam, Ian Ring Music TheoryRaga Gandharavam
Scale 1131Scale 1131: Honchoshi Plagal Form, Ian Ring Music TheoryHonchoshi Plagal Form
Scale 1387Scale 1387: Locrian, Ian Ring Music TheoryLocrian
Scale 1515Scale 1515: Phrygian/Locrian Mixed, Ian Ring Music TheoryPhrygian/Locrian Mixed
Scale 1771Scale 1771, Ian Ring Music Theory
Scale 235Scale 235, Ian Ring Music Theory
Scale 747Scale 747: Lynian, Ian Ring Music TheoryLynian
Scale 2283Scale 2283: Aeolyptian, Ian Ring Music TheoryAeolyptian
Scale 3307Scale 3307: Boptyllic, Ian Ring Music TheoryBoptyllic

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. 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.