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Scale 1251: "Sylimic"

Scale 1251: Sylimic, 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
Sylimic
Dozenal
Hoxian

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

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

Forte Number

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

6-18

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: 2277

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.

5

Prime Form

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

no
prime: 423

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 Structure

Defines the scale as the sequence of intervals between one tone and the next.

[1, 4, 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, 2, 2, 2, 4, 2>

Interval Spectrum

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

p4m2n2s2d3t2

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

Spectra Variation

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

2.333

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.

2.116

Polygon Perimeter

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

5.699

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.

(6, 12, 57)

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 TriadsF♯{6,10,1}210.67
Minor Triadsa♯m{10,1,5}121
Diminished Triads{7,10,1}121

The following pitch classes are not present in any of the common triads: {0}

Parsimonious Voice Leading Between Common Triads of Scale 1251. Created by Ian Ring ©2019 F# F# F#->g° a#m a#m F#->a#m

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.

Diameter2
Radius1
Self-Centeredno
Central VerticesF♯
Peripheral Verticesg°, a♯m

Modes

Modes are the rotational transformation of this scale. Scale 1251 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2673
Scale 2673: Mythimic, Ian Ring Music TheoryMythimic
3rd mode:
Scale 423
Scale 423: Sogimic, Ian Ring Music TheorySogimicThis is the prime mode
4th mode:
Scale 2259
Scale 2259: Raga Mandari, Ian Ring Music TheoryRaga Mandari
5th mode:
Scale 3177
Scale 3177: Rothimic, Ian Ring Music TheoryRothimic
6th mode:
Scale 909
Scale 909: Katarimic, Ian Ring Music TheoryKatarimic

Prime

The prime form of this scale is Scale 423

Scale 423Scale 423: Sogimic, Ian Ring Music TheorySogimic

Complement

The hexatonic modal family [1251, 2673, 423, 2259, 3177, 909] (Forte: 6-18) is the complement of the hexatonic modal family [423, 909, 1251, 2259, 2673, 3177] (Forte: 6-18)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1251 is 2277

Scale 2277Scale 2277: Kagimic, Ian Ring Music TheoryKagimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1251 is chiral, and its enantiomorph is scale 2277

Scale 2277Scale 2277: Kagimic, Ian Ring Music TheoryKagimic

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> 1251       T0I <11,0> 2277
T1 <1,1> 2502      T1I <11,1> 459
T2 <1,2> 909      T2I <11,2> 918
T3 <1,3> 1818      T3I <11,3> 1836
T4 <1,4> 3636      T4I <11,4> 3672
T5 <1,5> 3177      T5I <11,5> 3249
T6 <1,6> 2259      T6I <11,6> 2403
T7 <1,7> 423      T7I <11,7> 711
T8 <1,8> 846      T8I <11,8> 1422
T9 <1,9> 1692      T9I <11,9> 2844
T10 <1,10> 3384      T10I <11,10> 1593
T11 <1,11> 2673      T11I <11,11> 3186
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2151      T0MI <7,0> 3267
T1M <5,1> 207      T1MI <7,1> 2439
T2M <5,2> 414      T2MI <7,2> 783
T3M <5,3> 828      T3MI <7,3> 1566
T4M <5,4> 1656      T4MI <7,4> 3132
T5M <5,5> 3312      T5MI <7,5> 2169
T6M <5,6> 2529      T6MI <7,6> 243
T7M <5,7> 963      T7MI <7,7> 486
T8M <5,8> 1926      T8MI <7,8> 972
T9M <5,9> 3852      T9MI <7,9> 1944
T10M <5,10> 3609      T10MI <7,10> 3888
T11M <5,11> 3123      T11MI <7,11> 3681

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 1249Scale 1249: Howian, Ian Ring Music TheoryHowian
Scale 1253Scale 1253: Zolimic, Ian Ring Music TheoryZolimic
Scale 1255Scale 1255: Chromatic Mixolydian, Ian Ring Music TheoryChromatic Mixolydian
Scale 1259Scale 1259: Stadian, Ian Ring Music TheoryStadian
Scale 1267Scale 1267: Katynian, Ian Ring Music TheoryKatynian
Scale 1219Scale 1219: Hidian, Ian Ring Music TheoryHidian
Scale 1235Scale 1235: Messiaen Truncated Mode 2, Ian Ring Music TheoryMessiaen Truncated Mode 2
Scale 1187Scale 1187: Kokin-joshi, Ian Ring Music TheoryKokin-joshi
Scale 1123Scale 1123: Iwato, Ian Ring Music TheoryIwato
Scale 1379Scale 1379: Kycrimic, Ian Ring Music TheoryKycrimic
Scale 1507Scale 1507: Zynian, Ian Ring Music TheoryZynian
Scale 1763Scale 1763: Katalian, Ian Ring Music TheoryKatalian
Scale 227Scale 227: Bician, Ian Ring Music TheoryBician
Scale 739Scale 739: Rorimic, Ian Ring Music TheoryRorimic
Scale 2275Scale 2275: Messiaen Mode 5, Ian Ring Music TheoryMessiaen Mode 5
Scale 3299Scale 3299: Syptian, Ian Ring Music TheorySyptian

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, and George Howlett for assistance with the Carnatic ragas.