The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 1947: "Byptyllic"

Scale 1947: Byptyllic, 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
Byptyllic
Dozenal
Luzian

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

{0,1,3,4,7,8,9,10}

Forte Number

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

8-18

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: 2877

Hemitonia

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

5 (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.

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.

7

Prime Form

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

no
prime: 879

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, 2, 1, 3, 1, 1, 1, 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.

<5, 4, 6, 5, 5, 3>

Interval Spectrum

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

p5m5n6s4d5t3

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

2.616

Polygon Perimeter

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

6.002

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.

(12, 59, 138)

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 TriadsC{0,4,7}441.92
D♯{3,7,10}342.23
G♯{8,0,3}342.08
A{9,1,4}342.15
Minor Triadscm{0,3,7}342
c♯m{1,4,8}342.08
am{9,0,4}342.08
Augmented TriadsC+{0,4,8}441.85
Diminished Triadsc♯°{1,4,7}242.31
{4,7,10}242.31
{7,10,1}242.46
{9,0,3}242.46
a♯°{10,1,4}242.38
Parsimonious Voice Leading Between Common Triads of Scale 1947. Created by Ian Ring ©2019 cm cm C C cm->C D# D# cm->D# G# G# cm->G# C+ C+ C->C+ c#° c#° C->c#° C->e° c#m c#m C+->c#m C+->G# am am C+->am c#°->c#m A A c#m->A D#->e° D#->g° a#° a#° g°->a#° G#->a° a°->am am->A A->a#°

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 1947 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3021
Scale 3021: Stodyllic, Ian Ring Music TheoryStodyllic
3rd mode:
Scale 1779
Scale 1779: Zynyllic, Ian Ring Music TheoryZynyllic
4th mode:
Scale 2937
Scale 2937: Phragyllic, Ian Ring Music TheoryPhragyllic
5th mode:
Scale 879
Scale 879: Aeranyllic, Ian Ring Music TheoryAeranyllicThis is the prime mode
6th mode:
Scale 2487
Scale 2487: Dothyllic, Ian Ring Music TheoryDothyllic
7th mode:
Scale 3291
Scale 3291: Lygyllic, Ian Ring Music TheoryLygyllic
8th mode:
Scale 3693
Scale 3693: Stadyllic, Ian Ring Music TheoryStadyllic

Prime

The prime form of this scale is Scale 879

Scale 879Scale 879: Aeranyllic, Ian Ring Music TheoryAeranyllic

Complement

The octatonic modal family [1947, 3021, 1779, 2937, 879, 2487, 3291, 3693] (Forte: 8-18) is the complement of the tetratonic modal family [147, 609, 777, 2121] (Forte: 4-18)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 1947 is 2877

Scale 2877Scale 2877: Phrylyllic, Ian Ring Music TheoryPhrylyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 1947 is chiral, and its enantiomorph is scale 2877

Scale 2877Scale 2877: Phrylyllic, Ian Ring Music TheoryPhrylyllic

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> 1947       T0I <11,0> 2877
T1 <1,1> 3894      T1I <11,1> 1659
T2 <1,2> 3693      T2I <11,2> 3318
T3 <1,3> 3291      T3I <11,3> 2541
T4 <1,4> 2487      T4I <11,4> 987
T5 <1,5> 879      T5I <11,5> 1974
T6 <1,6> 1758      T6I <11,6> 3948
T7 <1,7> 3516      T7I <11,7> 3801
T8 <1,8> 2937      T8I <11,8> 3507
T9 <1,9> 1779      T9I <11,9> 2919
T10 <1,10> 3558      T10I <11,10> 1743
T11 <1,11> 3021      T11I <11,11> 3486
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2877      T0MI <7,0> 1947
T1M <5,1> 1659      T1MI <7,1> 3894
T2M <5,2> 3318      T2MI <7,2> 3693
T3M <5,3> 2541      T3MI <7,3> 3291
T4M <5,4> 987      T4MI <7,4> 2487
T5M <5,5> 1974      T5MI <7,5> 879
T6M <5,6> 3948      T6MI <7,6> 1758
T7M <5,7> 3801      T7MI <7,7> 3516
T8M <5,8> 3507      T8MI <7,8> 2937
T9M <5,9> 2919      T9MI <7,9> 1779
T10M <5,10> 1743      T10MI <7,10> 3558
T11M <5,11> 3486      T11MI <7,11> 3021

The transformations that map this set to itself are: T0, T0MI

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 1945Scale 1945: Zarian, Ian Ring Music TheoryZarian
Scale 1949Scale 1949: Mathyllic, Ian Ring Music TheoryMathyllic
Scale 1951Scale 1951: Marygic, Ian Ring Music TheoryMarygic
Scale 1939Scale 1939: Dathian, Ian Ring Music TheoryDathian
Scale 1943Scale 1943: Luxian, Ian Ring Music TheoryLuxian
Scale 1931Scale 1931: Stogian, Ian Ring Music TheoryStogian
Scale 1963Scale 1963: Epocryllic, Ian Ring Music TheoryEpocryllic
Scale 1979Scale 1979: Aeradygic, Ian Ring Music TheoryAeradygic
Scale 2011Scale 2011: Raphygic, Ian Ring Music TheoryRaphygic
Scale 1819Scale 1819: Pydian, Ian Ring Music TheoryPydian
Scale 1883Scale 1883: Lomian, Ian Ring Music TheoryLomian
Scale 1691Scale 1691: Kathian, Ian Ring Music TheoryKathian
Scale 1435Scale 1435: Makam Huzzam, Ian Ring Music TheoryMakam Huzzam
Scale 923Scale 923: Ultraphrygian, Ian Ring Music TheoryUltraphrygian
Scale 2971Scale 2971: Aeolynyllic, Ian Ring Music TheoryAeolynyllic
Scale 3995Scale 3995: Ionygic, Ian Ring Music TheoryIonygic

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.