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Scale 1227: "Thacrimic"

Scale 1227: Thacrimic, 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

41161837294116105072918310504116183
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
Thacrimic

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

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

Forte Number

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

6-Z50

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.

[0.5]

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.

no

Hemitonia

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

2 (dihemitonic)

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.

5

Prime Form

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

no
prime: 723

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

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.

[2, 2, 4, 2, 3, 2]

Interval Spectrum

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

p3m2n4s2d2t2

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

Spectra Variation

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

1.667

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.

2.366

Polygon Perimeter

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

5.864

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.

[1]

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".

Proper

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}321.17
F♯{6,10,1}231.5
Minor Triadscm{0,3,7}231.5
d♯m{3,6,10}321.17
Diminished Triads{0,3,6}231.5
{7,10,1}231.5
Parsimonious Voice Leading Between Common Triads of Scale 1227. 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°

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.

Diameter3
Radius2
Self-Centeredno
Central Verticesd♯m, D♯
Peripheral Verticesc°, cm, F♯, g°

Modes

Modes are the rotational transformation of this scale. Scale 1227 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2661
Scale 2661: Stydimic, Ian Ring Music TheoryStydimic
3rd mode:
Scale 1689
Scale 1689: Lorimic, Ian Ring Music TheoryLorimic
4th mode:
Scale 723
Scale 723: Ionadimic, Ian Ring Music TheoryIonadimicThis is the prime mode
5th mode:
Scale 2409
Scale 2409: Zacrimic, Ian Ring Music TheoryZacrimic
6th mode:
Scale 813
Scale 813: Larimic, Ian Ring Music TheoryLarimic

Prime

The prime form of this scale is Scale 723

Scale 723Scale 723: Ionadimic, Ian Ring Music TheoryIonadimic

Complement

The hexatonic modal family [1227, 2661, 1689, 723, 2409, 813] (Forte: 6-Z50) is the complement of the hexatonic modal family [717, 843, 1203, 1641, 2469, 2649] (Forte: 6-Z29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1227 is 2661

Scale 2661Scale 2661: Stydimic, Ian Ring Music TheoryStydimic

Transformations:

T0 1227  T0I 2661
T1 2454  T1I 1227
T2 813  T2I 2454
T3 1626  T3I 813
T4 3252  T4I 1626
T5 2409  T5I 3252
T6 723  T6I 2409
T7 1446  T7I 723
T8 2892  T8I 1446
T9 1689  T9I 2892
T10 3378  T10I 1689
T11 2661  T11I 3378

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 1225Scale 1225: Raga Samudhra Priya, Ian Ring Music TheoryRaga Samudhra Priya
Scale 1229Scale 1229: Raga Simharava, Ian Ring Music TheoryRaga Simharava
Scale 1231Scale 1231: Logian, Ian Ring Music TheoryLogian
Scale 1219Scale 1219, Ian Ring Music Theory
Scale 1223Scale 1223: Phryptimic, Ian Ring Music TheoryPhryptimic
Scale 1235Scale 1235: Messiaen Truncated Mode 2, Ian Ring Music TheoryMessiaen Truncated Mode 2
Scale 1243Scale 1243: Epylian, Ian Ring Music TheoryEpylian
Scale 1259Scale 1259: Stadian, Ian Ring Music TheoryStadian
Scale 1163Scale 1163: Raga Rukmangi, Ian Ring Music TheoryRaga Rukmangi
Scale 1195Scale 1195: Raga Gandharavam, Ian Ring Music TheoryRaga Gandharavam
Scale 1099Scale 1099: Dyritonic, Ian Ring Music TheoryDyritonic
Scale 1355Scale 1355: Raga Bhavani, Ian Ring Music TheoryRaga Bhavani
Scale 1483Scale 1483: Mela Bhavapriya, Ian Ring Music TheoryMela Bhavapriya
Scale 1739Scale 1739: Mela Sadvidhamargini, Ian Ring Music TheoryMela Sadvidhamargini
Scale 203Scale 203, Ian Ring Music Theory
Scale 715Scale 715: Messiaen Truncated Mode 2, Ian Ring Music TheoryMessiaen Truncated Mode 2
Scale 2251Scale 2251: Zodimic, Ian Ring Music TheoryZodimic
Scale 3275Scale 3275: Mela Divyamani, Ian Ring Music TheoryMela Divyamani

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. The software used to generate this analysis is an open source project at GitHub. 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.