The Emergence

It was observed that there was a unique and powerful relationship inherent in the perfect fifth hence the relationship was considered in pureness and evenness. The result was staggering; a trivial point had become a rich space with unbounded scope for harmony.

Amongst the set of all 4096 possible subsets in the 12 note cycle it is possible to identify classes of those with special properties.

Major scales

The first class of scales to consider, widely regarded as the most important scales, is the class of major scales whose elements satisfy the property that it is a maximal segment of the cycle of fifths such that there are no three adjacent notes. Because of this relationship between the cycle of fifths, it is very easy to take the 7 note segment of the cycle which represents one scale and extend it in one direction around the cycle and retract it in the other thereby creating a new major scale. This pattern can be repeated 12 times to generate the 12 distinct major scales in sequence and return to the origin thereby creating a cycle of fifths of major scales. It is conventional to denote a major scale by the second fifth in the scale meaning that is possible to rise by one in the cycle and fall by 5 in the cycle whilst retaining the original tonic within the scale. It is also remarkable that, because a seven place revolution of the cycle of fifths results in a semi-tone, the difference between two scales adjacent major scales in the cycle of fifth is one semi-tone - a sharpening of the 11th in the ascent and a flattening of the seventh in the decent.

Core scales

The second class of scales, the core scales, is a range of scales familiar in music theory and is defined by the maximal sets of notes such that there are no three notes adjacent. Since this is part of the definition of a major scale it is trivial to see that the major scales are part of this class. The other members of the class include the twelve harmonic minor scales and the twelve ascending melodic minor scales (from now on simply called the minor scale) which are commonly used in classical music. There are also a number of symmetric scales which are used for effect in modern music and can be defined by a cycle shorter than an octave - the two whole tone scales, the three diminished scales and the four augmented scales. There is also one more sub class of scales which is fundamental to establishing an elegant theory of core scales - the seven harmonic major scales.

A harmonic major scale has a similar relationship to the major scale as the harmonic minor scale has to the minor scale; it is essentially a major scale with a flattened 13th and also relates to the other two types of seven note scales since sharpening the 11th in a minor scale or sharpening the 3rd in a harmonic minor scale produces a harmonic major scale. A harmonic major scale shares with other seven note core scales the property that it contains a unique dominant seventh chord which can be resolved to a note one perfect fifth below the root of the dominant 7th chord, which is a generalized reason for naming the dominant 7th chord since the dominant in classical theory is the fifth degree of the scale. This has the benefit of identifying a note - tonic - to identify a specific harmonic major scale which is consistent across all seven note core scale - the note of resolution identified above.

An reassuringly equivalent definition of the class of seven note core scales is the sets notes generated by a seven cycle of major and minor thirds. It is worth noting that the major scale appears to have the most even spacing of major and thirds singling it out from an other perspective. If M and m denote a major and minor third respectively then the various generating sequences are:

• (M m M m m M m) generates the major scales;
• (m M M m m M m) generates the minor scales;
  
• (M m M m m m M) generates the harmonic major scales;
  
• (m M M m m m M) generates the harmonic minor scales.
  

Each sequence on major and minor thirds is equivalent to the generating sequences of minor, major and augmented seconds denoted 1,2 and 3 respectively. Once again the major scale appears to be the most evenly spaced:

• (2 2 1 2 2 2 1) generates the major scales;
• (2 1 2 2 2 2 1) generates the minor scales;
• (2 2 1 2 1 3 1) generates the harmonic major scales;
• (2 1 2 2 1 3 1) generates the harmonic minor scales.

It is remarkable to note that, with both the cycles of seconds and the cycles of thirds, if the cycle is reversed for the major scale and the minor scale then the same cycle is produced but with the harmonic minor the reversal of the cycles of seconds or the cycle of thirds produces the harmonic major scale instead and vice versa. The uniqueness of this relationship between the harmonic major and the harmonic minor is reinforced by observing that none of the other core scales produce a different cycle upon reversing their sequence of seconds as demonstrated by providing their generating sequences:

• (1 3 1 3 1 3) generates the augmented scale;
• (1 2 1 2 1 2 1 2) generates the diminished scale;
• (2 2 2 2 2 2) generates the whole tone scale.

Core Chords

All core scales contain chords which help indicate the harmonic qualities of the scale. If we are considering subsets of the 12 note cycle to be scales then a chord is a scale which is generally quite sparsely populated. A neat definition for a core chord is a maximal set of notes such that there is a gap of at least 2 between each adjacent note which for a 12 note cycle gives an excellent set of core chords that are expected in music theory: The major triad, minor triad, augmented triad and diminished chord. It is worth noting that the half diminished triad - the diminished chord with one note omitted - is not part of this classification which is good since it is generally avoided or appears implicitly in music theory.

We can therefore describe the harmonic properties of a core scale in terms of the core chords which it contains. The major scales are the only scales which do not contain an augmented chord and neither the major scales nor the minor scales contain the diminished chord. However, the harmonic major and harmonic minor scales contain all four core chords and, since they are compliments of each other and the major and minor triads are compliments of each other, it is reasonable to expect that they have the same composition with the number of major and minor chords interchanged.

The major scales have a major triad starting on the tonic, sub-dominant and dominant and a minor triad on the super-tonic, mediant and sub-mediant. The minor scales have a major triad starting on the sub-dominant and dominant, a minor chord on the tonic and super-tonic and an augmented triad on the mediant, dominant and leading note. However, these augmented triads that feature in a minor scale are in fact the same chord so, because of the symmetric nature of both the augmented triad and the diminished chord therefore it is possible start building up the chord in thirds from any point in the chord, only one chord need be listed. In the case of the minor scales, an augmented triad on the mediant will suffice.

A harmonic minor scale contains a major triad starting on the dominant and sub-mediant, a minor triad on the tonic and sub-dominant, an augmented triad on the mediant and a diminished chord on the leading note. As suggested earlier a harmonic major scale has a similar composition containing a major chord starting on the tonic instead of on the sub-mediant, a minor chord on the mediant instead of the tonic and the augmented triad on the sub-mediant instead of on the mediant. Both the harmonic major and the harmonic minor scales have an additional chord which has the same root as another chord - a harmonic major scale has an additional major triad on its mediant and the harmonic minor has an additional minor chord with its root on the sub-mediant.

Since the whole-tone, augmented and diminished scales consist of either less notes, in the case of the whole-tone and augmented scales, or more notes, in the case of the diminished scale, and have internal cycles, a tonic which uniquely defines a scale is more difficult to define. However, they can still be described in terms of their underlying core chords. A whole tone scale contains only augmented triads, of which there are two which are distinct, each separated by a tone. An augmented scale contains two adjacent augmented triads and three major and minor triads where a major and a minor triad occurs starting on each of the notes of the upper of the two adjacent augmented triads. Similarly, a diminished chord scale contains two adjacent diminished chords with four major and minor triads with a pair or major and minor chords starting on the each note in the lower of the two adjacent diminished chords. It is interesting to note that the harmonic major and harmonic minor scales have the closest properties to the augmented and diminished scales since they both contain augmented triads, diminished chords and over lapping major and minor chords. A suitable pairing might be the harmonic minor scales with the diminished scales, since the major/minor pair coincides with a diminished chord, and the harmonic major scales with the augmented scales, since the major/minor pair coincide with an augmented triad.

Core Modes

The harmonic properties indicated by a choice of underlying chord and a scale provide a good foundation for describing the concept of a core mode. Modes have, so far, not been considered to be a part of the this theory since a scale is a set of notes without any concept of an order or an emphasis. A core mode could, however, be defined as a pair consisting of a core scale and a core chord. What will please early modal theorists is that the Locrian mode, which was considered to be the devils mode, along with other similarly disturbing modes which occur in the other core scales and is a motive for defining a core mode as a pair consisting of a core chord and a core scale rather the more obvious definition of a pair consisting of a core scale and an individual note.

The modes of the seven note core scales which are generated are listed bellow in order from lightest to darkest where the order is determined by sorting them, in order of importance, by the pitch of the mediant, dominant, leading note, super-tonic, sub-dominant, sub-mediant. The listing contains a name for the scale in terms of an alteration of the extension (the 9th, 11th and 13th degrees of the scale) of a well known mode, the name of the class of scales on which it is based and the degree of the scale on which to begin to obtain the mode. There are nine major modes, nine minor modes, three augmented modes and two diminished modes:

Name	Scale	Note
augmented #9 #11	harmonic major	sub-mediant
augmented #11	minor	mediant
augmented	harmonic minor	mediant
Lydian #9	harmonic minor	sub-mediant
Lydian	major	sub-dominant
Ionian	major	tonic
Ionian b13	harmonic major	tonic
Mixolydian #11	minor	sub-dominant
Mixolydian	major	dominant
Mixolydian b13	minor	dominant
Mixolydian b9	harmonic major	dominant
Mixolydian b9 b13	harmonic minor	dominant
minor #11	harmonic major	sub-dominant
minor	minor	tonic
minor b13	harmonic minor	tonic
Dorian #11	harmonic minor	sub-dominant
Dorian	major	super-tonic
Aeolian	major	sub-mediant
Dorian b9	minor	super-tonic
Phrygian	major	mediant
Phrygian b11	harmonic major	mediant
diminished b9 b13	harmonic major	leading note
diminished b9 b11 b13	harmonic minor	leading note

The use of modes gives an incredible textural wealth of 276 distinct sounds that can be exploited in music. The remaining modes that are not listed are all half diminished modes which are not included since there is no core chord starting on the root of these scales, or if there is then the same chord points towards another stronger mode. The half diminished modes are in one to one correspondence with five Mixolydian modes which should be used in their place for a strong effect. Half diminished chords do however complete the diatonic cycles of fifths within each of the seven note core scales but do not stand up as a harmonic center and can generally be explained as an inversion of another chord.

The the Phrygian b11 modes and the Lydian #9 modes, respectively the darkest minor modes and the brightest major modes, are represented twice in our model of a mode since they contain both a major and minor triad starting on the same degree of the scale which provides the mode. This gives neat correlation between the two apparently opposing modes. Similarly, the darkest diminished mode and the brightest augmented mode contain a diminished and augmented chord starting on the same degree of the scale which provides the mode.

Many observations can be made about this well defined ordering of modes, regarding the distribution of base core scales and core chords, which give insight into chord progressions and modulations. The harmonic major scales have the most extreme minor modes where as the harmonic minor scales have the most extreme major modes. The dominant modes are the only modes on a particular degree of the scale which are grouped together in this ordering. Flattening the root of a diminished b9 b11 b13 mode gives the Mixolydian mode and flattening the root of an Aeolian mode gives an augmented #9 #11 mode. Similarly, flattening the roots of a diminished b9 b13, Phrygian b11 or Phrygian mode gives a Mixolydian #11, augmented or augmented #11 mode respectively. All the modes beginning on the tonics, sub-dominants and dominants, the primary modes, appear together in a symmetric pattern between the Dorian #11 and the Lydian modes with the dominants in the center, the tonics paired at either side and the sub-dominants alternating with these blocks of dominants and tonics. The secondary chords at the bright end of the spectrum also have a symmetry consisting of a pair of mediants with a sub-mediant at either side.

Interestingly, if the half diminished modes are inserted between the minor and diminished modes then the a symmetric pattern - super-tonic, sub-mediant, super-tonic, mediant, mediant, super-tonic, sub-mediant, super-tonic - emerges which is a condensed version of the symmetric sequence of primary modes shifted down a minor third. The inclusion of the half-diminished modes also introduces a cluster of the four modes based on the leading note which emphasizes further the relationship between the leading note and the dominant, with the former being a first inversion of the later. These two observations justify the acknowledgment of the half-diminished modes for the purpose of progressions but do not justify them as a tonal center. They are listed below using the same ordering as with the other modes.

Name	Scale	Note
half diminished	harmonic major	super-tonic
half diminished b13	minor	sub-mediant
half diminished b9	harmonic minor	super-tonic
Locrian	major	leading note
Locrian b11	minor	leading note

Base scales and chords

There appears to be a logical manner to extend the definition of both a core chord and a core scale in a sensible way. Let an nth order chord be a set of notes such that there exists a gap of at least n notes between any two notes. Similarly, let an nth order scale be a set of notes such that at most n notes are adjacent. It can be noted immediately that the second order scales and chords are, by definition, the core scales and chords.

It is also significant that the first order scales and the first order chords coincide exactly there by suggesting that they fulfil the role of either. This class of scales consists of the pentatonic scales, the dominant 9th chords, the whole tone scales and the diminished chords. Two of these scales, the whole tone scale and the diminished chord, are already covered by the definitions of a second order scale and chord there by demonstrating that the generalised definition does not produce disjoint classes of scales.

Similarly to a major scale, which is a maximal segment of the cycle of the fifths which is also a second order scale, a pentatonic scale is a maximal segment of the cycle of fifths which is also a first order scale. A pentatonic scale contains one major triad and one minor triad whose root is a minor third lower than the root of the major triad an therefore over lapping in two places. It thereby embodies the strong progression which occurs in harmony which is the move between the major and its relative minor. It is reasonable to designate the root of either triad as the tonic of the scale but we will select the root of the major triad.

A dominant 9th chord is a maximal sequence of major and minor thirds such that it is also a first order chord (or any order of chord for that matter). A dominant ninth chord contains a major triad and a minor triad whose root coincides with the fifth of the major triad, which is one of the strongest pairs of chords used in chord progressions and is known as the II-V progression. It has the significant property that it is the only chord of an order greater than or equal to one which contains a dominant 7th chord, which is acknowledged as perhaps the strongest chord in the theory of music. It is reasonable to designate the root of the major triad contained within the dominant 9th chord as the root.

Furthermore, drawing upon a fundamental property which may explain its strength, the notes of a dominant ninth chord coincide with the notes in the maximal initial segment of the harmonic series which constitute a first order chord. The first ten intervals of the harmonic series, along with their associated ratios to the root frequency are listed below:

Ratio	Normalised ratio	Interval
1	1	root
2	1	root
3	3/2	perfect 5th
4	1	root
5	5/4	major 3rd
6	3/2	perfect 5th
7	7/4	minor 7th
8	1	root
9	9/8	major 2nd
10	5/4	major 3rd
(11)	(11/8)	(diminished 5th)

Both the pentatonic scales and the dominant 9th chords are contained within the major and minor scales. A major scale contains one dominant 9th chord beginning on its dominant and three distinct pentatonic scales beginning on its tonic, sub-dominant and dominant coinciding with all its contingent major triads and minor triads. A minor scale consists of only one pentatonic scale beginning on its sub-dominant and two distinct dominant 9th chords beginning on its sub-dominant and dominant. With the exception of the augmented triad in the minor scale, the harmonic qualities of these scales are characterised entirely by its contingent first order chords and their harmonic qualities.

However, neither the harmonic major nor the harmonic minor scales contain a pentatonic scale or a dominant 9th chord. They are only characterised at this level by a contingent diminished chord which emphasises the diminished chords' role as the back bone to these scales. The lack of any base scale in the harmonic major and harmonic minor scales may not be surprising since all base scales have the property that a reversal of their sequence of intervals yields the same scale.


Higher order chords

For a twelve note cycle the generalised definition of a chord yields a non-empty scale up until to its twelfth order and single notes from its sixth order to its eleventh order inclusive. The first and second order chords have been covered as the base and core chords respectively hence the third to fifth order chords are worth describing.

Interestingly, as the more densely populated diminished chord overlapped with the first and second order chords, the augmented triad overlaps with the second order and third order chords. In the class of third order chords, it resides alongside the twelve perfect intervals and the six tritones. These are all the foundation of some second order chord. The major and minor triads contain a perfect interval each and the diminished triad contains two tritones.

The fourth and fifth order chords are the same as the 3rd order chords with the augmented triad and the perfect interval omitted respectively. The existence of the tritones down to the highest level of chord with more than two notes and the fact that it transcends three orders of chord gives a further explanation to the significance of the dominant 7th chord in the seven note core scales which is partly down to the apparent need to resolve its contingent tritone.


Higher Order Scales

The higher the order of a class of scales, the greater the density of notes contained in a scale in the class. Most new scales of interest which are introduced above the second order scales are in the third order which produces melodically chromaticised jazzy scales and the fourth order scales which are intense. The fifth to eleventh order scales introduces only a single class of similar scales each after which no new scales are introduced.

The third order scales consists of three classes of similar scales which can be produced by carefully adding a single note to in of the seven note core scales. These can be denoted the blues major, blues harmonic major and blues harmonic minor, to achieve a unified naming convention. A blues major scale contains a major scale and harmonic major sharing the same tonic and a harmonic minor with its tonic coinciding with the sub-mediant of the major scale. The blues harmonic major and blues harmonic minor scales can be produced in a similar manner by superimposing a harmonic major and harmonic minor scale respectively onto a minor scale, in the unique manner, such that there is only one note difference between the two scales. The result is the following three eight note scales:

• (2 2 1 2 1 1 2 1) generates the blues major scales;
• (2 2 1 2 1 2 1 1) generates the blues harmonic major scales; and
• (2 1 2 2 1 1 2 1) generates the blues harmonic minor scales.
  

Implied in these scales are four disjoint paths, which cover the seven note core scales and can be produced by only moving between scales that are both contained in one of the above third order scales. These paths are as follows:

• A major, F# harmonic minor, F# minor, C# harmonic major, C# major, A# harmonic minor, A# minor, F harmonic major, F major, D harmonic minor, D minor, A harmonic major.
• E major, C# harmonic minor, C# minor, G# harmonic major, G# major, E# harmonic minor, E# minor, C harmonic major, C major, A harmonic minor, A minor, E harmonic major.
• B major, G# harmonic minor, G# minor, A# harmonic major, A# major, C harmonic minor, C minor, G harmonic major, G major, E harmonic minor, E minor, B harmonic major.
• F# major, D# harmonic minor, D# minor, F harmonic major, F major, G harmonic minor, G minor, D harmonic major, D major, B harmonic minor, B minor, F# harmonic major.

The further three classes of scale can be denoted the blues augmented scales, the blues whole tone scales and, simply, the blues scales. The four blues augmented scales each contain two augmented scales starting on adjacent notes and, as a result, a whole tone scale. The six blues whole tone scales are each whole tone scales with two extra notes a diminished fifth apart. Finally, a blues scale contains an extended version of, what is considered to be, the classic blues scale. The only second order scale that a blues scale contains is an augmented scale, but it contains the basic set of pentatonic scales and chords that many musicians appeal to. The interval sequences are as follows:

• (2 1 1 2 1 1 2 1 1) generates the blues augmented scales;
• (2 2 1 1 2 2 1 1) generates the blues whole tone scales; and
  
• (2 1 2 1 1 3 1 1) generates the blues scales.
  

The fourth order scales include the major triad compliment and the minor triad compliment which are the all notes of the twelve note cycle without a major or minor triad respectively. These contain a blues major scale and blues scales each in addition to a blues harmonic minor for the major triad compliment and blues harmonic major for the minor triad compliment. The super major contains a three major scales positioned at the tonic, sub-dominant and dominant with an additional minor scale at the dominant and the super diminished contains only a diminished chord from the lower order scales. Each is as follows:

• (2 1 1 1 2 1 1 2 1) generates the major triad compliment;
• (2 1 1 1 2 1 2 1 1) generates the minor triad compliment;
• (2 2 1 1 1 2 1 1 1) generates the super major; and
• (3 1 1 1 3 1 1 1) generates the super diminished.
  

Beyond the fourth order scales there are three classes of note. The new sixth and seventh order scales have the property, which is shared only by the familiar seven note core scales, is that they can be generated without repetition from sequences of major and minor thirds. These ten note sequences are as follows, each missing two notes a 4th and major 3rd apart respectively:

• MmMMmMMmMm for the sixth order scales; and
• MmMMmMMmmM for the seventh order scales.

Each can be seen as three chromatically descending major triads with the third triad in the sequence extended with a major 7th and minor 7th respectively which points back to the beginning of the cycle. Perhaps surprisingly, interpreting the chords implied by these sequences leads to a bright and harmonically rich space containing several useful chromatic passages.

These sixth order scales, when considered from the root, have a major feel with an interestingly effective augmented dominant. Similarly, the seventh order scales can make use of the resident dominant 9th chord, which naturally resolves to the minor triad starting at the fifth step in the sequence, to achieve a minor texture. Further analysis uncovers several other strangely familiar progressions.

The eleventh order scales, those missing one note only from the twelve note cycle, are the only other scales which can be generated in this manner. The sequences of major and minor thirds, which essentially form three chromatically ascending dominant seventh chords, are permutations of the following:

• MmmmMmmmMmm
  

The six scales introduced as fifth order scales and the twelve introduced each time as eighth, ninth and tenth order scales are most likely to be of interest for the study of the sets of scales accessible under a restricted vocabulary.


Thoughts:

• There is a cycle of modifications of triads that occurs in transforming a harmonic major into a harmonic minor, does this happen else where in transitions between keys and does it indicate any new or equivalent property about the transition?
• What do the hierarchies of chords mean for the voicing of harmony?
  

The Chaos

In the beginning every scale had equal significance. However this led to random wandering between scales resulting in complete chaos.

The concept of a scale is inconsistently defined in music theory but a good starting point to consider might be "a series of notes differing in pitch according to a specific scheme (usually within an octave)" [1].

This informal definition leaves a lot to desire so a better model should be developed. If we model the spectrum of sounds using the strictly positive real line where each note is denoted by a distinct frequency value in Hertz then there are infinitely many differing pitches and an even greater size of infinity of sets of notes. Going on no further information this does not allow you to solve the problem of finding a scale to use for a given situation.

It is reasonable to modify the model such that only cyclic sets of notes are permitted. Where the cycle is defined by some continuous mapping from the real line onto some continuous cyclic set such as the unit circle in the complex plain. Because of the strong sound of doubling a note's frequency to get the octave a reasonable mapping might map any real number x to e^(2*pi*i*log₂(x/a)) where a is some fixed real number which will act as a reference frequency such as 440, which is a solution for f to the equation f(2x)=f(x).

The restriction of a scale to cyclic sets of notes does not eliminate the problem of having infinitely many scales to choose from. It is therefore reasonable to ensure that the number of notes in the cycle is finite. Along with the octave, another strong sound is the perfect fifth which corresponds to a scaling of the frequency by 3/2. What is remarkable is that an ascent of 12 perfect fifths is only slightly greater than an ascent of 7 octaves by a scale of 3^12/2^19, which is a canonical form of (3/2)^12/2^7. It is worth noting that a ratio of 2^(7/12), which is the solution for r of r^12=2^7, would be required to reach exactly seven octaves in 12 equal scalings which is irrational hence it was a reasonable assumption to use the real line rather than the rational numbers only in the initial model. Such adjustment of the seven perfect fifths to fit into an octave is the art of temperament which had a range of solutions an happens naturally in a performance with varying degrees of deliberation.

By assuming that a temperament can be selected for a particular performance we have constructed a model of a scale consisting of 12 possible notes in 2^12=4097 distinct combinations. Selecting a suitable scale without any further information is still an extremely daunting task.

Here are some thoughts worth considering:

• Is the set of useful scales a null set or is there enough space to measure the number of useful scales giving an evenly distributed random scale in a continuous model a chance of sounding good? Can this be answered by considering the intervals between the perfect and even temperaments along the 12 sequence ascent and then consideration the various concepts of a scale still to be defined?
  
• Is m=12 and n=19 the solution to inf|1-3^m/2^n| where m,n are strictly positive integers? Does this equation with 3 and 2 replaced by other prime numbers give rise to other good sounding sound cycles?