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Scale 1217

Scale 1217, 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).

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

4 (tetratonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,6,7,10}

Forte Number

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

4-Z15

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: 101

Hemitonia

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

1 (unhemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)

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.

3

Prime Form

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

no
prime: 83

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.

[6, 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.

<1, 1, 1, 1, 1, 1>

Interval Spectrum

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

pmnsdt

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

Spectra Variation

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

3.5

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.

1.183

Polygon Perimeter

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

4.932

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.

(4, 1, 18)

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.

Modes

Modes are the rotational transformation of this scale. Scale 1217 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 83
Scale 83, Ian Ring Music TheoryThis is the prime mode
3rd mode:
Scale 2089
Scale 2089, Ian Ring Music Theory
4th mode:
Scale 773
Scale 773, Ian Ring Music Theory

Prime

The prime form of this scale is Scale 83

Scale 83Scale 83, Ian Ring Music Theory

Complement

The tetratonic modal family [1217, 83, 2089, 773] (Forte: 4-Z15) is the complement of the octatonic modal family [863, 1523, 1997, 2479, 2809, 3287, 3691, 3893] (Forte: 8-Z15)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1217 is 101

Scale 101Scale 101, Ian Ring Music Theory

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1217 is chiral, and its enantiomorph is scale 101

Scale 101Scale 101, Ian Ring Music Theory

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> 1217       T0I <11,0> 101
T1 <1,1> 2434      T1I <11,1> 202
T2 <1,2> 773      T2I <11,2> 404
T3 <1,3> 1546      T3I <11,3> 808
T4 <1,4> 3092      T4I <11,4> 1616
T5 <1,5> 2089      T5I <11,5> 3232
T6 <1,6> 83      T6I <11,6> 2369
T7 <1,7> 166      T7I <11,7> 643
T8 <1,8> 332      T8I <11,8> 1286
T9 <1,9> 664      T9I <11,9> 2572
T10 <1,10> 1328      T10I <11,10> 1049
T11 <1,11> 2656      T11I <11,11> 2098
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2117      T0MI <7,0> 1091
T1M <5,1> 139      T1MI <7,1> 2182
T2M <5,2> 278      T2MI <7,2> 269
T3M <5,3> 556      T3MI <7,3> 538
T4M <5,4> 1112      T4MI <7,4> 1076
T5M <5,5> 2224      T5MI <7,5> 2152
T6M <5,6> 353      T6MI <7,6> 209
T7M <5,7> 706      T7MI <7,7> 418
T8M <5,8> 1412      T8MI <7,8> 836
T9M <5,9> 2824      T9MI <7,9> 1672
T10M <5,10> 1553      T10MI <7,10> 3344
T11M <5,11> 3106      T11MI <7,11> 2593

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 1219Scale 1219, Ian Ring Music Theory
Scale 1221Scale 1221: Epyritonic, Ian Ring Music TheoryEpyritonic
Scale 1225Scale 1225: Raga Samudhra Priya, Ian Ring Music TheoryRaga Samudhra Priya
Scale 1233Scale 1233: Ionoditonic, Ian Ring Music TheoryIonoditonic
Scale 1249Scale 1249, Ian Ring Music Theory
Scale 1153Scale 1153, Ian Ring Music Theory
Scale 1185Scale 1185: Genus Primum Inverse, Ian Ring Music TheoryGenus Primum Inverse
Scale 1089Scale 1089, Ian Ring Music Theory
Scale 1345Scale 1345, Ian Ring Music Theory
Scale 1473Scale 1473, Ian Ring Music Theory
Scale 1729Scale 1729, Ian Ring Music Theory
Scale 193Scale 193: Raga Ongkari, Ian Ring Music TheoryRaga Ongkari
Scale 705Scale 705, Ian Ring Music Theory
Scale 2241Scale 2241, Ian Ring Music Theory
Scale 3265Scale 3265, Ian Ring Music Theory

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.