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Scale 1517: "Sagyllic"

Scale 1517: Sagyllic, 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
Sagyllic

Analysis

Cardinality

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

8 (octatonic)

Pitch Class Set

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

{0,2,3,5,6,7,8,10}

Forte Number

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

8-22

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: 1781

Hemitonia

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

4 (multihemitonic)

Cohemitonia

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

2 (dicohemitonic)

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.

7

Prime Form

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

no
prime: 1391

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, 2, 1, 1, 1, 2, 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.

<4, 6, 5, 5, 6, 2>

Interval Spectrum

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

p6m5n5s6d4t2

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

Spectra Variation

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

1.75

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.

yes

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.

2.732

Polygon Perimeter

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

6.071

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

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}341.9
G♯{8,0,3}242.1
A♯{10,2,5}242.1
Minor Triadscm{0,3,7}341.9
d♯m{3,6,10}341.9
fm{5,8,0}242.3
gm{7,10,2}242.1
Augmented TriadsD+{2,6,10}341.9
Diminished Triads{0,3,6}242.1
{2,5,8}242.3
Parsimonious Voice Leading Between Common Triads of Scale 1517. Created by Ian Ring ©2019 cm cm c°->cm d#m d#m c°->d#m D# D# cm->D# G# G# cm->G# fm fm d°->fm A# A# d°->A# D+ D+ D+->d#m gm gm D+->gm D+->A# d#m->D# D#->gm fm->G#

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
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 1517 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 1403
Scale 1403: Espla's Scale, Ian Ring Music TheoryEspla's Scale
3rd mode:
Scale 2749
Scale 2749: Katagyllic, Ian Ring Music TheoryKatagyllic
4th mode:
Scale 1711
Scale 1711: Adonai Malakh, Ian Ring Music TheoryAdonai Malakh
5th mode:
Scale 2903
Scale 2903: Gothyllic, Ian Ring Music TheoryGothyllic
6th mode:
Scale 3499
Scale 3499: Hamel, Ian Ring Music TheoryHamel
7th mode:
Scale 3797
Scale 3797: Rocryllic, Ian Ring Music TheoryRocryllic
8th mode:
Scale 1973
Scale 1973: Zyryllic, Ian Ring Music TheoryZyryllic

Prime

The prime form of this scale is Scale 1391

Scale 1391Scale 1391: Aeradyllic, Ian Ring Music TheoryAeradyllic

Complement

The octatonic modal family [1517, 1403, 2749, 1711, 2903, 3499, 3797, 1973] (Forte: 8-22) is the complement of the tetratonic modal family [149, 673, 1061, 1289] (Forte: 4-22)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1517 is 1781

Scale 1781Scale 1781: Gocryllic, Ian Ring Music TheoryGocryllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1517 is chiral, and its enantiomorph is scale 1781

Scale 1781Scale 1781: Gocryllic, Ian Ring Music TheoryGocryllic

Transformations:

T0 1517  T0I 1781
T1 3034  T1I 3562
T2 1973  T2I 3029
T3 3946  T3I 1963
T4 3797  T4I 3926
T5 3499  T5I 3757
T6 2903  T6I 3419
T7 1711  T7I 2743
T8 3422  T8I 1391
T9 2749  T9I 2782
T10 1403  T10I 1469
T11 2806  T11I 2938

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 1519Scale 1519: Locrian/Aeolian Mixed, Ian Ring Music TheoryLocrian/Aeolian Mixed
Scale 1513Scale 1513: Stathian, Ian Ring Music TheoryStathian
Scale 1515Scale 1515: Phrygian/Locrian Mixed, Ian Ring Music TheoryPhrygian/Locrian Mixed
Scale 1509Scale 1509: Ragian, Ian Ring Music TheoryRagian
Scale 1525Scale 1525: Sodyllic, Ian Ring Music TheorySodyllic
Scale 1533Scale 1533: Katycrygic, Ian Ring Music TheoryKatycrygic
Scale 1485Scale 1485: Minor Romani, Ian Ring Music TheoryMinor Romani
Scale 1501Scale 1501: Stygyllic, Ian Ring Music TheoryStygyllic
Scale 1453Scale 1453: Aeolian, Ian Ring Music TheoryAeolian
Scale 1389Scale 1389: Minor Locrian, Ian Ring Music TheoryMinor Locrian
Scale 1261Scale 1261: Modified Blues, Ian Ring Music TheoryModified Blues
Scale 1773Scale 1773: Blues Scale II, Ian Ring Music TheoryBlues Scale II
Scale 2029Scale 2029: Kiourdi, Ian Ring Music TheoryKiourdi
Scale 493Scale 493: Rygian, Ian Ring Music TheoryRygian
Scale 1005Scale 1005: Radyllic, Ian Ring Music TheoryRadyllic
Scale 2541Scale 2541: Algerian, Ian Ring Music TheoryAlgerian
Scale 3565Scale 3565: Aeolorygic, Ian Ring Music TheoryAeolorygic

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.