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Scale 3499: "Hamel"

Scale 3499: Hamel, Ian Ring Music Theory

This scale is named "Hamel", or alternately, the "Peter Hamel Octatonic", after the German composer Peter Hamel (1947-). Peter is associated with the Minimalist style, and the New Simplicity movement of the 1970s.


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

Named After Composers
Hamel
Peter Hamel Octatonic
Zeitler
Lythyllic
Dozenal
Vubian

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

{0,1,3,5,7,8,10,11}

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: 2743

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

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, 2, 2, 1, 2, 1, 1]

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

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.

(10, 59, 137)

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♯{1,5,8}242.1
D♯{3,7,10}242.1
G♯{8,0,3}341.9
Minor Triadscm{0,3,7}242.1
fm{5,8,0}341.9
g♯m{8,11,3}341.9
a♯m{10,1,5}242.3
Augmented TriadsD♯+{3,7,11}341.9
Diminished Triads{5,8,11}242.1
{7,10,1}242.3
Parsimonious Voice Leading Between Common Triads of Scale 3499. Created by Ian Ring ©2019 cm cm D#+ D#+ cm->D#+ G# G# cm->G# C# C# fm fm C#->fm a#m a#m C#->a#m D# D# D#->D#+ D#->g° g#m g#m D#+->g#m f°->fm f°->g#m fm->G# g°->a#m g#m->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 3499 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3797
Scale 3797: Rocryllic, Ian Ring Music TheoryRocryllic
3rd mode:
Scale 1973
Scale 1973: Zyryllic, Ian Ring Music TheoryZyryllic
4th mode:
Scale 1517
Scale 1517: Sagyllic, Ian Ring Music TheorySagyllic
5th mode:
Scale 1403
Scale 1403: Espla's Scale, Ian Ring Music TheoryEspla's Scale
6th mode:
Scale 2749
Scale 2749: Katagyllic, Ian Ring Music TheoryKatagyllic
7th mode:
Scale 1711
Scale 1711: Adonai Malakh, Ian Ring Music TheoryAdonai Malakh
8th mode:
Scale 2903
Scale 2903: Gothyllic, Ian Ring Music TheoryGothyllic

Prime

The prime form of this scale is Scale 1391

Scale 1391Scale 1391: Aeradyllic, Ian Ring Music TheoryAeradyllic

Complement

The octatonic modal family [3499, 3797, 1973, 1517, 1403, 2749, 1711, 2903] (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 3499 is 2743

Scale 2743Scale 2743: Staptyllic, Ian Ring Music TheoryStaptyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3499 is chiral, and its enantiomorph is scale 2743

Scale 2743Scale 2743: Staptyllic, Ian Ring Music TheoryStaptyllic

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> 3499       T0I <11,0> 2743
T1 <1,1> 2903      T1I <11,1> 1391
T2 <1,2> 1711      T2I <11,2> 2782
T3 <1,3> 3422      T3I <11,3> 1469
T4 <1,4> 2749      T4I <11,4> 2938
T5 <1,5> 1403      T5I <11,5> 1781
T6 <1,6> 2806      T6I <11,6> 3562
T7 <1,7> 1517      T7I <11,7> 3029
T8 <1,8> 3034      T8I <11,8> 1963
T9 <1,9> 1973      T9I <11,9> 3926
T10 <1,10> 3946      T10I <11,10> 3757
T11 <1,11> 3797      T11I <11,11> 3419
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2239      T0MI <7,0> 4003
T1M <5,1> 383      T1MI <7,1> 3911
T2M <5,2> 766      T2MI <7,2> 3727
T3M <5,3> 1532      T3MI <7,3> 3359
T4M <5,4> 3064      T4MI <7,4> 2623
T5M <5,5> 2033      T5MI <7,5> 1151
T6M <5,6> 4066      T6MI <7,6> 2302
T7M <5,7> 4037      T7MI <7,7> 509
T8M <5,8> 3979      T8MI <7,8> 1018
T9M <5,9> 3863      T9MI <7,9> 2036
T10M <5,10> 3631      T10MI <7,10> 4072
T11M <5,11> 3167      T11MI <7,11> 4049

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 3497Scale 3497: Phrolian, Ian Ring Music TheoryPhrolian
Scale 3501Scale 3501: Maqam Nahawand, Ian Ring Music TheoryMaqam Nahawand
Scale 3503Scale 3503: Zyphygic, Ian Ring Music TheoryZyphygic
Scale 3491Scale 3491: Tharian, Ian Ring Music TheoryTharian
Scale 3495Scale 3495: Banyllic, Ian Ring Music TheoryBanyllic
Scale 3507Scale 3507: Maqam Hijaz, Ian Ring Music TheoryMaqam Hijaz
Scale 3515Scale 3515: Moorish Phrygian, Ian Ring Music TheoryMoorish Phrygian
Scale 3467Scale 3467: Katonian, Ian Ring Music TheoryKatonian
Scale 3483Scale 3483: Mixotharyllic, Ian Ring Music TheoryMixotharyllic
Scale 3531Scale 3531: Neveseri, Ian Ring Music TheoryNeveseri
Scale 3563Scale 3563: Ionoptygic, Ian Ring Music TheoryIonoptygic
Scale 3371Scale 3371: Aeolylian, Ian Ring Music TheoryAeolylian
Scale 3435Scale 3435: Prokofiev, Ian Ring Music TheoryProkofiev
Scale 3243Scale 3243: Mela Rupavati, Ian Ring Music TheoryMela Rupavati
Scale 3755Scale 3755: Phryryllic, Ian Ring Music TheoryPhryryllic
Scale 4011Scale 4011: Styrygic, Ian Ring Music TheoryStyrygic
Scale 2475Scale 2475: Neapolitan Minor, Ian Ring Music TheoryNeapolitan Minor
Scale 2987Scale 2987: Neapolitan Major and Minor Mixed, Ian Ring Music TheoryNeapolitan Major and Minor Mixed
Scale 1451Scale 1451: Phrygian, Ian Ring Music TheoryPhrygian

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.