Diahemitonic Tonality: A Tension–Release Quarter-Tonal Harmonic System

Abstract

This article expands upon my previous work titled ‘Diahemitonic Modality: A Quarter-Tonal Composition System’ and in which I mapped the diatonic modal system onto quarter-tonality. As tonality derives from modality and modality has been successfully mapped onto quarter-tonality, the prospect of extending this mapping to tonality becomes conceivable. The design process mirrors that of diahemitonic modality, now focusing on tonal harmony and encompassing considerations of scale, mode, key, interval, chord, function, cadence, sequence and modulation. The one-to-one mapping of each aspect of diatonic tonality onto the quarter-tone scale culminates in diahemitonic tonality.

Table of Contents

Introduction

This article is destined for readers familiar with my previous work titled ‘Diahemitonic Modality: A Quarter-Tonal Composition System’.^[1] In that research, I described the mapping of the diatonic modal system onto quarter-tonality, resulting in what I termed diahemitonic modality, a quarter-tonal composition system resting upon the interval of the major fourth and incorporating both pitch and duration dimensions of the modal idiom.

Originating from modal traditions, tonality, as formalised by Jean-Philippe Rameau in 1722, evolved by introducing major and minor modes and defining chord functions, primarily the tonic, dominant and subdominant, upon the basis of acoustic-physics laws.^[2] This pitch organisation, likened to a planetary system by Ivan Wyschnegradsky^[3] and Paul Hindemith,^[4] revolves around a tension–release polarity where tension is embodied by the dominant and release by the tonic.

Since tonality arises from modality and diatonic modality has been effectively mapped onto quarter-tonality, it is now possible to envisage diahemitonic tonality. To explore this uncharted musical territory, I shall apply the same design process which gave rise to diahemitonic modality, now concentrating on tonal harmony. This entails a comprehensive examination of scales, modes, keys, intervals, chords, functions, cadences, sequences and modulations within the diatonic tonal framework. Each of these elements then requires meticulous mapping onto the quarter-tone scale, maintaining the integrity and relationships of the original tonal system. By applying this systematic approach, the present research aims to develop a cohesive tension–release quarter-tonal harmonic system which parallels the established principles of diatonic tonality.

Towards Diahemitonic Tonality

Apart from abandoning the rhythmic modes for reasons of evolution in musical notation, diatonic tonality retains the temporal component of diatonic modality. Indeed, the duple and triple metres, the binary and ternary subdivisions of beats, the hemiola and the octave and perfect-fifth proportions of the appoggiatura are all elements of the musical parameter of duration which apply equally to diatonic modal music and its tonal descendant. Therefore, diahemitonic tonality will rely upon the octuple and undecuple metres, the octonary and undenary subdivisions of beats, the triogdoola and the octave and major-fourth proportions of the appoggiatura inherited from its modal ancestor, allowing the present research to focus exclusively on the musical parameter of pitch.

The Major–Minor Mode System

There are seven diatonic modes, corresponding to the seven pitch classes of the diatonic scale. During the transition from the modal era to the tonal era, two of these modes were retained as the major and minor modes. Vincent d’Indy teaches us that the major mode originates from the harmonic series and the minor mode from the subharmonic series.^[5] The diatonic scale, in its original state, constitutes the C mode. By building upwards a chord made of the first six harmonics inherent to diatonicism on this mode’s tonic, one notices that the root of this chord is C (see Figure 1).

Figure 1. First six harmonics of C with C as the root.

When inverted, the diatonic scale constitutes the E mode. By building downwards a chord made of the first six subharmonics on this mode’s tonic, one notices that the root of this chord is A (see Figure 2).

Figure 2. First six subharmonics of E with A as the root.

Therefore, the C mode is the diatonic major mode and the A mode the diatonic minor mode, with a minor third separating their tonics. They are respectively located as the second from the top and third from the bottom in the hierarchy of the diatonic modes (see Figure 3).

Figure 3. C and A modes in the hierarchy of the diatonic modes.

If one places the C triad next to the A triad, then one notices that the former is major and the latter minor and that they share all their pitch classes in common except for the highest of the first and the lowest of the second (see Figure 4).

Figure 4. C and A triads.

There are 13 diahemitonic modes, corresponding to the 13 pitch classes of the diahemitonic scale. Owing to the fact that diahemitonicism stems from an evolution of diatonicism reaching to the eleventh partial, I shall use the two reference pitch classes C and E to build upwards and downwards chords made of the first 11 harmonics and subharmonics respectively. By building upwards a chord made of the first 11 harmonics of C, one notices that the root of this chord is C (see Figure 5).

Figure 5. First 11 harmonics of C with C as the root.

By building downwards a chord made of the first 11 subharmonics of E, one notices that the root of this chord is B-quarter-tone-flat (see Figure 6).

Figure 6. First 11 subharmonics of E with B-quarter-tone-flat as the root.

Therefore, the C mode is the diahemitonic major mode and the B-quarter-tone-flat mode the diahemitonic minor mode, with a neutral second separating their tonics, an interval which, as will be elucidated later, is diahemitonically analogous to the minor third. Analogously to the preceding, they are respectively located as the second from the top and third from the bottom in the hierarchy of the diahemitonic modes (see Figure 7).

Figure 7. C and B-quarter-tone-flat modes in the hierarchy of the diahemitonic modes.

If one places the C tetrad next to the B-quarter-tone-flat tetrad, then one notices that the former is major and the latter minor and that they share all their pitch classes in common except for the highest of the first and the lowest of the second, which parallels the above (see Figure 8).

Figure 8. C and B-quarter-tone-flat tetrads.

The Leading Note and the Subtonic

There is a functional difference between the major and minor modes in the last scale degree, which is a leading note in the former and a subtonic in the latter. The leading note is not assigned a harmonic function, which is why it is called the leading note, a note rather than a chord. This fact is illustrated by the leading-note triad and seventh chord, both appearing to be rooted on it but whose voice leadings suggest them to actually be rootless dominant seventh and ninth chords respectively. This seventh degree of the diatonic major scale is a semitone below the tonic. In the diatonic minor scale, the seventh degree is a whole tone below the tonic, making it a subtonic.

In the diahemitonic major scale, the last degree is separated from the first by a quarter-tone, whereas in the diahemitonic minor scale, this interval stretches to a semitone. This intervallic analogy with diatonicism allows the identification of the leading note and subtonic as the 13^th degrees of the diahemitonic major and minor scales respectively.

Primary Chord Functions

In the diatonic tonal idiom, the primary chord functions are the tonic, dominant and subdominant. The tonic is defined as the pole of stability because it corresponds to the first degree of the scale. The harmonic character of the dominant does not spring from its position as the first degree of the second tetrachord in the diatonic scale, which defines it melodically, but from the fact that it corresponds to the third harmonic from which the semitone scale originates. It contains the last scale degree within its triad, whereas the subdominant, which corresponds to the third subharmonic, contains the first.

In the diahemitonic tonal idiom, the primary chord functions must necessarily be the same, but it turns out that the dominant and subdominant are reversed. This is because it is not the dominant, the first degree of the second heptachord in the diahemitonic scale, which corresponds to the eleventh harmonic from which the quarter-tone scale originates but the scale degree just below it, namely the subdominant. It contains the last scale degree within its tetrad, whereas the dominant, which corresponds to the eleventh subharmonic, contains the first. Therefore, it is the subdominant here which plays the harmonic role of the dominant and vice versa. This swap is justified by the fact that while adding a perfect fifth above a given pitch and below its octave to perform the first step in the derivation of the diatonic scale causes a crossing between the two pitches thus generated, performing the same process with a major fourth to derive the diahemitonic scale does not cause such a crossing.

Secondary Chord Functions

Beyond the primary chord functions of tonic, dominant and subdominant, secondary chord functions act as bridges between them. These include the supertonic, mediant, submediant and subtonic, leaving the leading note with the purely melodic role of resolving onto the tonic. The supertonic, as its name suggests, is defined as above the tonic, opposite to the subtonic in the minor scale. The mediant is enclosed within the tonic triad and is halfway between the tonic and dominant. Symmetrically, the submediant is enclosed within the subdominant triad and is halfway between the subdominant and tonic (see Table 1).

Table 1. Diatonic scale degrees and their harmonic functions.

Diatonic Scale Degree	Harmonic Function
First	Tonic
Second	Supertonic
Third	Mediant
Fourth	Subdominant
Fifth	Dominant
Sixth	Submediant
Seventh	Subtonic or Leading Note

The supertonic function must accordingly be assigned to the second degree of the diahemitonic scale positioned just above the tonic and opposite to the subtonic in the minor scale, thereby honouring its name. The tonic tetrad, enclosing two scale degrees halfway between the tonic and subdominant, gives both the function of mediant. Symmetrically, the dominant tetrad, enclosing two scale degrees halfway between the dominant and tonic, gives both the function of submediant (see Table 2).

Table 2. Diahemitonic scale degrees and their harmonic functions.

Diahemitonic Scale Degree	Harmonic Function
First	Tonic
Second	Supertonic
Third	Mediant 1
Fourth
Fifth	Mediant 2
Sixth
Seventh	Subdominant
Eighth	Dominant
Ninth
Tenth	Submediant 1
Eleventh
Twelfth	Submediant 2
Thirteenth	Subtonic or Leading Note

All the diatonic functions have now been assigned to degrees of the diahemitonic scale. The four remaining diahemitonic scale degrees should then be deemed melodic degrees.

Natural, Harmonic and Melodic Minor Scales

As has been stated earlier, the diatonic minor scale, in its natural form, does not have the leading note as its last degree but the subtonic, a whole tone away from the tonic. To remedy this lack of leading-note-to-tonic attraction, the dominant triad from the major mode, comprising the leading note, was embedded into the scale, thus forming the diatonic harmonic minor scale with its characteristic augmented interval (see Figure 9).

Figure 9. Diatonic harmonic minor scale.

Given that this augmented interval has historically caused difficulties for singers, the submediant was raised by a chromatic semitone in order to reduce it, thus forming the diatonic ascending melodic minor scale composed of a minor tetrachord and a major tetrachord (see Figure 10).

Figure 10. Diatonic ascending melodic minor scale.

The diatonic minor scale, when traversed descendingly, has no need for a leading note, which is why the diatonic descending melodic minor scale is identical to its natural state (see Figure 11).

Figure 11. Diatonic descending melodic minor scale.

Like the diatonic natural minor scale, the diahemitonic natural minor scale does not have the leading note as its last degree but the subtonic, a semitone away from the tonic. Applying the same remedy as above translates into embedding the subdominant tetrad from the major mode into the scale. By doing so, one forms the diahemitonic harmonic minor scale with analogous augmented intervals (see Figure 12).

Figure 12. Diahemitonic harmonic minor scale.

To reduce them, the submediants need to be raised by a chromatic quarter-tone, thus forming the analogous diahemitonic ascending melodic minor scale composed of a minor heptachord and a major heptachord (see Figure 13).

Figure 13. Diahemitonic ascending melodic minor scale.

The diahemitonic minor scale, when traversed descendingly, also has no need for a leading note. Therefore, I shall leave it in its natural state (see Figure 14).

Figure 14. Diahemitonic descending melodic minor scale.

The Notion of Key

The key is defined as the formation of a major or minor mode starting from a specific pitch class of the chromatic scale. Thus, the original mode is transposed, except for C major and A minor in the diatonic context and C major and B-quarter-tone-flat minor in the diahemitonic context. Keys are related to each other according to their degree of proximity. Two keys of different modes but identical key signature are said to be in a relative relationship. Two keys of different modes but identical tonic are said to be in a parallel relationship. In the diatonic context, whose theory is founded upon the circle of fifths, what are called closely related keys correspond to the keys located a fifth above and below the main key as well as the relative modal keys of these three, minor if the main key is major and major if the main key is minor. For example, the five closely related keys of C diatonic–major are G diatonic–major, F diatonic–major, A diatonic–minor, E diatonic–minor and D diatonic–minor (see Figure 15).

Figure 15. Closely related keys of C diatonic–major.

When transposing the main diatonic key into that of the dominant, the former’s subdominant is raised by a chromatic semitone to become the latter’s leading note (see Figure 16).

Figure 16. Transposition of C diatonic–major into G diatonic–major.

Inversely, when transposing the main diatonic key into that of the subdominant, the former’s leading note is lowered by a chromatic semitone to become the latter’s subdominant (see Figure 17).

Figure 17. Transposition of C diatonic–major into F diatonic–major.

In the diahemitonic context, whose theory is founded upon the circle of major fourths, closely related keys must accordingly correspond to the keys located a major fourth above and below the main key as well as the relative modal keys of these three. Consequently, the five closely related keys of C diahemitonic–major are F-quarter-tone-sharp diahemitonic–major, G-quarter-tone-flat diahemitonic–major, B-quarter-tone-flat diahemitonic–minor, E diahemitonic–minor and F diahemitonic–minor (see Figure 18).

Figure 18. Closely related keys of C diahemitonic–major.

When transposing the main diahemitonic key into that of the dominant, the former’s subdominant is raised by a chromatic quarter-tone to become the latter’s leading note, in analogy with the above (see Figure 19).

Figure 19. Transposition of C diahemitonic–major into G-quarter-tone-flat diahemitonic–major.

Inversely, when transposing the main diahemitonic key into that of the subdominant, the former’s leading note is lowered by a chromatic quarter-tone to become the latter’s subdominant (see Figure 20).

Figure 20. Transposition of C diahemitonic–major into F-quarter-tone-sharp diahemitonic–major.

Modal mixture consists in the borrowing of pitch classes from the parallel key. Two types are distinguished: the Picardy third and the lowered submediant. The former type implies the majorisation of the tonic triad in an otherwise minor key, usually occurring at the end of a movement or piece. The latter type implies the minorisation of the subdominant triad in an otherwise major key (see Figure 21).

Figure 21. Minorised subdominant triad in C diatonic–major.

This results in a diatonic scale composed of a major tetrachord and a harmonic–minor tetrachord and which may be labelled as Moll–Dur according to Germanic terminology (see Figure 22).

Figure 22. Diatonic Moll–Dur scale.

Since a tetrad embraces two pitch classes, a second and a third respectively from the root, between its endpoints, the term Picardy second and third appropriately designates the majorisation of the tonic tetrad in an otherwise minor key. The lowered submediants, whose plural terminology is due to the presence of two submediants within the diahemitonic scale, naturally imply the minorisation of the dominant tetrad in an otherwise major key (see Figure 23).

Figure 23. Minorised dominant tetrad in C diahemitonic–major.

Analogously, this results in a diahemitonic scale composed of a major heptachord and a harmonic–minor heptachord and which may also be labelled as Moll–Dur (see Figure 24).

Figure 24. Diahemitonic Moll–Dur scale.

Intervals

It is now appropriate to establish analogies between intervals from the diatonic and diahemitonic domains. Let us start with those which are neither augmented nor diminished. The unison remains a unison. The smallest non-diminished conjunct interval is the minor second in the diatonic scale and the semi-diminished second in the diahemitonic scale. The largest non-augmented conjunct interval is the major second in the diatonic scale and the minor second in the diahemitonic scale. The third, in the diatonic scale, corresponds to the interval between the tonic and mediant, so it is either minor or major depending upon the mode in use. In the diahemitonic scale, since there are two mediants, there are two candidate intervals for this status: the second, either neutral or major, and the third, also either neutral or major. Consequently, the minor third in the diatonic aesthetic has as its diahemitonic analogues the neutral second and neutral third and the major third in the diatonic aesthetic the major second and major third. The perfect fourth, in the diatonic scale, corresponds to the interval between the tonic and subdominant, which must be translated as the interval between the tonic and dominant in the diahemitonic scale, namely the minor fifth. The remaining intervals are the inversions of the aforementioned ones (see Table 3).

Table 3. Analogies between non-altered diatonic and diahemitonic intervals.

Let us now proceed to the augmented or diminished intervals made of whole tones. The ditonic fourth corresponds to the interval between the mediant and leading note in the diatonic minor scale. Since there are two diahemitonic analogues of the mediant, there will necessarily be two diahemitonically analogous intervals: the monotonic third and ditonic fourth. The tritonic fourth is identical between the diatonic and diahemitonic realms. The remaining intervals are the inversions of the aforementioned ones (see Table 4).

Table 4. Analogies between whole-tonal diatonic and diahemitonic intervals.

Two pairs of altered intervals frequently encountered in diatonic tonality are the augmented second and the diminished third, along with their inversions. The augmented second and its inversion are naturally present in the diatonic harmonic minor scale. As seen in the section on minor scales, their respective diahemitonic analogues are what may be termed the sesquiaugmented^[6] unison, made of three quarter-tones, and its inversion. The diminished third and its inversion, typically found following a Neapolitan-sixth chord, are formed through a Phrygian alteration of the diatonic minor scale. Indeed, with the chromatic lowering of its second degree, the pair of intervals formed with the leading note are the diminished third and its inversion. The Phrygian mode is the E mode, located at the opposite end of the hierarchy of the diatonic modes from the foundational C mode. The diahemitonic mode located at the opposite end of the hierarchy of the diahemitonic modes from the foundational C mode is the F mode. Their structures are analogous: the E diatonic mode is constituted of two identical tetrachords separated by a whole tone and composed of a semitone followed by whole tones; the F diahemitonic mode is constituted of two identical heptachords separated by a semitone and composed of a quarter-tone followed by semitones. Applying an F-diahemitonic-modal alteration to the diahemitonic minor scale analogously means chromatically lowering its second degree, forming the interval pair of the minor second and its inversion with the leading note.

Let us conclude this section with an analysis of the remaining diahemitonic intervals: the semi-augmented second, the minor third, the semi-augmented third, the perfect fourth and their inversions. They all, without exception, constitute frequency ratios foreign to the quarter-tonal world. In order: 22:19, 6:5, 22:17, 4:3 and their inversions (see Table 5). For this reason, I recommend composers use them with great caution.

Table 5. Diahemitonic intervals associated with harmonically non-quarter-tonal frequency ratios.

Harmonic Cadences

While modality recognises fewer cadences than there are modes and these cadences are of contrapuntal essence, tonality covers several distinct cadences both in the major and minor modes, drawing their substance from harmonic functions. The perfect cadence consists of the progression from the dominant triad to the tonic triad with both chords in root position and, typically, with the conjunct melodic motion from the leading note to the tonic in the soprano (see Figure 25).

Figure 25. Perfect cadence in C diatonic–major.

Given that it is the subdominant of the diahemitonic scale which embodies the harmonic function of the dominant, in order to realise a perfect cadence in the diahemitonic system, it is the subdominant tetrad which must progress to the tonic tetrad, with an analogous conjunct melodic motion from the leading note to the tonic in the soprano (see Figure 26).

Figure 26. Perfect cadence in C diahemitonic–major.

The imperfect cadence consists of the progression from a chord, often the tonic triad, to the dominant triad with at least the latter chord in root position (see Figure 27).

Figure 27. Imperfect cadence in C diatonic–major.

For the same reason as mentioned above, in order to realise an imperfect cadence in the diahemitonic system, it is to the subdominant tetrad that the first chord must progress (see Figure 28).

Figure 28. Imperfect cadence in C diahemitonic–major.

The Phrygian imperfect cadence, exclusive to the minor mode, consists of a Phrygian cadence to the dominant (see Figure 29).

Figure 29. Phrygian imperfect cadence in C diatonic–minor.

Considering that the diahemitonic Phrygian mode is the F mode and that the diahemitonic function which fulfils the harmonic role of the dominant is the subdominant, in order to realise a Phrygian imperfect cadence in the diahemitonic system, one must perform an F-diahemitonic-modal cadence to the subdominant (see Figure 30).

Figure 30. Phrygian imperfect cadence in C diahemitonic–minor.

The plagal cadence consists of the progression from the subdominant triad to the tonic triad with both chords in root position (see Figure 31).

Figure 31. Plagal cadence in C diatonic–major.

Given that it is the dominant of the diahemitonic scale which embodies the harmonic function of the subdominant, in order to realise a plagal cadence in the diahemitonic system, it is the dominant tetrad which must progress to the tonic tetrad (see Figure 32).

Figure 32. Plagal cadence in C diahemitonic–major.

The flat-submediant-to-tonic cadence consists of the progression from the submediant triad of the minor mode to the tonic triad of the major mode with both chords in root position (see Figure 33).

Figure 33. Flat-submediant-to-tonic cadence in C diatonic–major.

Accordingly, in order to realise a quarter-tone-flat-submediant-to-tonic cadence in the diahemitonic system, either of the two submediant tetrads of the minor mode must progress to the tonic tetrad of the major mode (see Figures 34a and 34b).

Figure 34a. Quarter-tone-flat-submediant-1-to-tonic cadence in C diahemitonic–major.

Figure 34b. Quarter-tone-flat-submediant-2-to-tonic cadence in C diahemitonic–major.

The interrupted cadence is analysable in two different ways: either as a progression from the dominant triad to the submediant triad or from the dominant triad to the next scale degree triad, in both cases with both chords in root position. Emulating the first pattern diahemitonically would cause a harmonically non-quarter-tonal melodic interval in the bass. Doing so with the second would cause improper parallels and a tonally non-idiomatic chord progression. Therefore, the interrupted cadence is not mappable onto the diahemitonic scale and should be viewed as idiomatic to diatonicism.

The evaded cadence involves the dominant seventh chord, which has not yet been studied at this stage of the present research. It will be discussed in a later section.

Harmonic Sequences

A harmonic sequence is a cyclic chord progression. Two are noteworthy: the descending-fifths and descending-thirds sequences. The former consists of root position triads descending diatonically in the circle of fifths until reaching a perfect cadence (see Figure 35).

Figure 35. Diatonic descending-fifths harmonic sequence.

The latter consists of root position triads descending by thirds with inverted chords interpolated between them, so that the bassline proceeds by conjunct motion (see Figure 36).

Figure 36. Diatonic descending-thirds harmonic sequence.

The interval diahemitonically analogous to the perfect fifth is the major fourth. Initiating a diahemitonic descent of fourths chordally engenders a chain of tetrads analogous to the diatonic descending-fifths sequence and which will also ultimately end in a perfect cadence (see Figure 37).

Figure 37. Diahemitonic descending-fourths harmonic sequence.

The intervals diahemitonically analogous to the third are the second and the third. To help choose between the two in view of the diahemitonic analogue of the descending-thirds sequence, I shall rely upon the requirement to use a passing inverted chord between each disjunct bass motion, which excludes the second option as it would necessitate multiple transitional tetrads. Sequencing root position tetrads descending by seconds with inverted chords interpolated between them results in the bassline proceeding by conjunct motion, in conformity with this requirement (see Figure 38).

Figure 38. Diahemitonic descending-seconds harmonic sequence.

Dominant Seventh and Ninth Chords

The origin of the dominant seventh chord is contrapuntal, with the seventh formed by a passing note descending from the dominant’s octave doubling (see Figure 39).^[7]

Figure 39. Passing note connecting the dominant triad to the tonic triad in C diatonic–major.

The elision of the original octave yields the dominant seventh chord (see Figure 40).

Figure 40. Dominant seventh chord and its resolution in C diatonic–major.

When the unelided melodic line occurs in the bass, it constitutes an evaded cadence by sidestepping towards an inversion of the tonic triad (see Figure 41).

Figure 41. Evaded cadence in C diatonic–major.

The inverted dominant seventh chord is easily approached from the subtonic triad in the minor mode, with a characteristic chromatic motion in the bass (see Figure 42).

Figure 42. Inverted dominant seventh chord approached by the subtonic triad in C diatonic–minor.

Doubling the root of the subdominant tetrad at the octave and initiating a conjunct descent results in an analogous pattern (see Figure 43).

Figure 43. Passing note connecting the subdominant tetrad to the tonic tetrad in C diahemitonic–major.

The elision of the original octave yields the diahemitonic subdominant seventh chord (see Figure 44).

Figure 44. Subdominant seventh chord and its resolution in C diahemitonic–major.

The evaded cadence is analogously constituted when the unelided melodic line occurs in the bass (see Figure 45).

Figure 45. Evaded cadence in C diahemitonic–major.

The inverted diahemitonic subdominant seventh chord is equally easily approached from the subtonic tetrad in the minor mode, with an analogous chromatic motion in the bass (see Figure 46).

Figure 46. Inverted subdominant seventh chord approached by the subtonic tetrad in C diahemitonic–minor.

The dominant ninth chord is defined by the addition of the ninth to the dominant seventh chord. It resolves downwards onto the top pitch class of the tonic triad, in parallel motion with the seventh and contrary motion with the leading note (see Figure 47).

Figure 47. Dominant ninth chord and its resolution in C diatonic–major.

The rootless dominant ninth chord rests upon the leading note and requires the same voice leading as the standard dominant ninth chord (see Figure 48).

Figure 48. Rootless dominant ninth chord and its resolution in C diatonic–major.

Adding a ninth to the diahemitonic subdominant seventh chord yields the diahemitonic subdominant ninth chord and does not prevent equally respecting the considerations of voice leading (see Figure 49).

Figure 49. Subdominant ninth chord and its resolution in C diahemitonic–major.

To turn it into the diahemitonic analogue of the rootless dominant ninth chord, depriving it of its root alone would not suffice to have it rest upon the leading note; its second and third must also be removed (see Figure 50).

Figure 50. Root-, second- and thirdless subdominant ninth chord and its resolution in C diahemitonic–major.

Non-Dominant Seventh Chords

There are two seventh chords not founded upon the dominant: the supertonic and subdominant seventh chords. They share their contrapuntal origin with the dominant seventh chord, which is why their sevenths also resolve downwards. Both have a pre-dominant function and are thus grammatically interchangeable. The supertonic seventh chord is most often encountered in an inverted position (see Figure 51).

Figure 51. Inverted supertonic seventh chord and its resolution in C diatonic–major.

Unlike the dominant seventh chord, its seventh needs not only to be resolved but also to be prepared (see Figure 52).

Figure 52. Chord progression involving the inverted supertonic seventh chord in C diatonic–major.

The subdominant seventh chord is most often encountered in root position (see Figure 53).

Figure 53. Subdominant seventh chord and its resolution in C diatonic–major.

Like the supertonic seventh chord, its seventh needs not only to be resolved but also to be prepared (see Figure 54).

Figure 54. Chord progression involving the subdominant seventh chord in C diatonic–major.

Building an inverted supertonic tetrad in the diahemitonic scale and surmounting it with a descending line from the elided root’s octave leads analogously to the subdominant, preferably a diahemitonic subdominant seventh chord for reasons of voice leading (see Figure 55).

Figure 55. Inverted supertonic seventh chord and its resolution in C diahemitonic–major.

Its seventh needs, naturally, to be prepared (see Figure 56).

Figure 56. Chord progression involving the inverted supertonic seventh chord in C diahemitonic–major.

Building a root position tetrad upon the dominant and surmounting it with a descending line from the elided root’s octave leads analogously to the subdominant, preferably a diahemitonic subdominant seventh or ninth chord for reasons of voice leading (see Figure 57).

Figure 57. Dominant seventh chord and its resolution in C diahemitonic–major.

Its seventh needs, naturally, also to be prepared (see Figure 58).

Figure 58. Chord progression involving the dominant seventh chord in C diahemitonic–major.

Dominant Chords with Added Bass

The dominant seventh chord with added bass originates from the supposition of the mediant under the dominant seventh chord, most commonly in the minor mode (see Figure 59).^[8]

Figure 59. Dominant seventh chord with added bass in C diatonic–minor.

Idiomatically, this chord is preceded by a supertonic seventh chord preparing the seventh and approaching the other notes by conjunct motion. Its resolution follows conventional dominant seventh chord voice leading while the bass moves down by a minor third (see Figure 60).^[9]

Figure 60. Chord progression involving the dominant seventh chord with added bass in C diatonic–minor.

A variant of this chord consists of the replacement of the dominant seventh chord with a rootless dominant ninth chord (see Figure 61).^[10]

Figure 61. Rootless dominant ninth chord with added bass in C diatonic–minor.

In this case, the supertonic seventh chord prepares both its seventh and its ninth (see Figure 62).

Figure 62. Chord progression involving the rootless dominant ninth chord with added bass in C diatonic–minor.

Supposing a mediant under the diahemitonic subdominant seventh chord yields the diahemitonic subdominant seventh chord with added bass (see Figure 63).

Figure 63. Subdominant seventh chord with added bass in C diahemitonic–minor.

By opting for mediant 1, one can prepare the seventh and approach the other notes by conjunct motion from a diahemitonic supertonic seventh chord. The resolution occurs while the bass moves down by a neutral second, an interval diahemitonically analogous to the minor third (see Figure 64).

Figure 64. Chord progression involving the subdominant seventh chord with added bass in C diahemitonic–minor.

Replacing the diahemitonic subdominant seventh chord with a root-, second- and thirdless subdominant ninth chord yields an analogous variant of this chord (see Figure 65).

Figure 65. Root-, second- and thirdless subdominant ninth chord with added bass in C diahemitonic–minor.

In this case, the diahemitonic supertonic seventh chord prepares both its seventh and its ninth, as required (see Figure 66).

Figure 66. Chord progression involving the root-, second- and thirdless subdominant ninth chord with added bass in C diahemitonic–minor.

The Cadential Six–Four Chord

Looking identical to a last-inverted tonic triad, the cadential six–four chord is first and foremost a dominant chord embellished with a double appoggiatura. Its voice leading requires resolution onto a dominant chord by conjunct motion for its fourth and sixth while the bass, doubled if there is doubling, remains stationary (see Figure 67).

Figure 67. Cadential six–four chord and its resolution in C diatonic–major.

Its fourth and sixth are approached by conjunct or oblique motion, making the supertonic and subdominant triads good candidates for its preparation (see Figure 68).

Figure 68. Chord progression involving the cadential six–four chord in C diatonic–major.

The last-inverted tonic tetrad progresses analogously to the subdominant, provided it is a diahemitonic subdominant seventh or ninth chord (see Figure 69).

Figure 69. Cadential six–four chord and its resolution in C diahemitonic–major.

While it cannot be approached by the dominant tetrad due to an inevitable disjunct motion, it can be approached by conjunct motion by the supertonic tetrad (see Figure 70).

Figure 70. Chord progression involving the cadential six–four chord in C diahemitonic–major.

The Neapolitan-Sixth Chord

The Neapolitan-sixth chord is an inverted supertonic triad borrowed from the Phrygian mode. It is used in the minor mode and serves the same function as the ordinary supertonic, that of a pre-dominant. When doubling occurs, it is usually its third which is doubled. Its progression to a dominant chord is marked by its characteristic diminished interval in the soprano (see Figure 71).

Figure 71. Neapolitan-sixth chord progressing to the dominant in C diatonic–minor.

It is common practice to insert a cadential six–four chord between the Neapolitan-sixth chord and the subsequent dominant chord to smooth the soprano line with a conjunct chromatic motion (see Figure 72).

Figure 72. Cadential six–four chord inserted into a Neapolitan progression in C diatonic–minor.

The diahemitonic Phrygian mode is the F mode. Borrowing the supertonic tetrad from this mode and inverting it while respecting the doubling guidelines mentioned above yields the diahemitonic Neapolitan-sixth chord. When it progresses to the subdominant, preferably a diahemitonic subdominant seventh or ninth chord for reasons of voice leading, it produces an analogous diminished interval in the soprano (see Figure 73).

Figure 73. Neapolitan-sixth chord progressing to the subdominant in C diahemitonic–minor.

Inserting a diahemitonic cadential six–four chord between the diahemitonic Neapolitan-sixth chord and the subsequent diahemitonic subdominant chord analogously smooths the soprano line with a conjunct chromatic motion (see Figure 74).

Figure 74. Cadential six–four chord inserted into a Neapolitan progression in C diahemitonic–minor.

Augmented-Sixth Chords

The origin of the augmented-sixth chord is contrapuntal, with the augmented sixth formed by an ascending chromatic passing note.^[11] There are three types of augmented-sixth chords: Italian, German and French. They are used in the minor mode and serve a pre-dominant function. The Italian augmented-sixth chord originates from a Phrygian imperfect cadence embellished with a chromatic passing note in the soprano (see Figure 75).

Figure 75. Chromatic passing note in the soprano of a Phrygian imperfect cadence in C diatonic–minor.

The elision of the original harmonic note yields the Italian augmented-sixth chord (see Figure 76).

Figure 76. Italian augmented-sixth chord and its resolution in C diatonic–minor.

If one decides to regard its bass as a root and completes the chord with its missing fifth, then one obtains the German augmented-sixth chord (see Figure 77).

Figure 77. German augmented-sixth chord and its resolution in C diatonic–minor.

When examining the composition of this latter chord, it becomes apparent that it is enharmonically equivalent to the dominant seventh chord, allowing modulation into the raised-tonic key (see Figure 78).

Figure 78. Enharmonic modulation using the German augmented-sixth chord.

Realising a Phrygian imperfect cadence with the soprano rising by conjunct chromatic motion results in an analogous pattern (see Figure 79).

Figure 79. Chromatic passing note in the soprano of a Phrygian imperfect cadence in C diahemitonic–minor.

The elision of the original harmonic note yields the diahemitonic Italian augmented-sixth chord (see Figure 80).

Figure 80. Italian augmented-sixth chord and its resolution in C diahemitonic–minor.

If one decides to regard its bass as a root and completes the chord with its missing fourth, then one obtains the diahemitonic German augmented-sixth chord (see Figure 81).

Figure 81. German augmented-sixth chord and its resolution in C diahemitonic–minor.

When examining the composition of this latter chord, it becomes apparent that it is enharmonically equivalent to the diahemitonic subdominant seventh chord, allowing an analogous modulation into the raised-tonic key (see Figure 82).

Figure 82. Enharmonic modulation using the diahemitonic German augmented-sixth chord.

The French augmented-sixth chord follows the same process as that which gave rise to its Italian counterpart but starts from an inverted supertonic seventh chord (see Figure 83).

Figure 83. Chromatic passing note in the soprano of an inverted-supertonic-seventh-chord-to-dominant progression in C diatonic–minor.

The elision of the original harmonic note yields the French augmented-sixth chord (see Figure 84).

Figure 84. French augmented-sixth chord and its resolution in C diatonic–minor.

Realising an inverted-supertonic-seventh-chord-to-subdominant progression with the soprano rising by conjunct chromatic motion results in an analogous pattern (see Figure 85).

Figure 85. Chromatic passing note in the soprano of an inverted-supertonic-seventh-chord-to-subdominant progression in C diahemitonic–minor.

The elision of the original harmonic note yields the diahemitonic French augmented-sixth chord, which preferably resolves onto a diahemitonic subdominant seventh chord for reasons of voice leading (see Figure 86).

Figure 86. French augmented-sixth chord and its resolution in C diahemitonic–minor.

Passing Chords

Passing chords are chords which play a more contrapuntal role than a harmonic one, with two such chords: the passing six–four chord and the passing leading-note triad. The former serves to transition from one chord to another. A typical case of its use is its interpolation between an inverted supertonic seventh chord and an augmented-sixth chord as part of a voice exchange to avoid false relations (see Figure 87).

Figure 87. Passing six–four chord as part of a voice exchange avoiding false relations in C diatonic–minor.

Inverted, the latter serves to transition conjunctly between a tonic triad in root position and its inversion, or conversely (see Figure 88).

Figure 88. Passing leading-note triad in C diatonic–major.

Interpolating a diahemitonic six–four chord between an inverted diahemitonic supertonic seventh chord and a diahemitonic augmented-sixth chord also exhibits a voice exchange and analogously avoids false relations (see Figure 89).

Figure 89. Passing six–four chord as part of a voice exchange avoiding false relations in C diahemitonic–minor.

When inverted, the leading-note tetrad fits equally well between the tonic tetrad in root position and its inversion, involving analogous conjunct motions (see Figure 90).

Figure 90. Passing leading-note tetrad in C diahemitonic–major.

The Pedal Six–Four Chord

The pedal six–four chord is a neighbouring chord built on a stationary bass. It is preceded and followed by a root position triad with which it shares the lowest-voice note and proceeds by conjunct motion.

Interpolating a diahemitonic six–four chord between two identical root position tetrads over an oblique bass does not result in conjunct motion in the upper voices. For this reason, the pedal six–four chord cannot be mapped onto the diahemitonic scale and should be regarded as idiomatic to diatonicism.

The Consonant Six–Four Chord

The consonant six–four chord functions as a substitute for the perfect triad in root position. Idiomatically, it is often a tonic triad succeeding the same triad in root position.

The last-inverted perfect tetrad is naturally eligible to substitute its root position form. It thus constitutes the diahemitonic consonant six–four chord and can be used just as idiomatically as in the diatonic sphere.

Modulations

Modulation is the process of transitioning from one key to another. Several techniques exist, the most well-known of which are the pivot chord, pivot note, chromatic and enharmonic modulations. Pivot chord modulation uses one or more chords shared between the original and target keys (see Figure 91).

Figure 91. Pivot chord modulation from C diatonic–major into A diatonic–minor.

Pivot note modulation follows a similar principle by using a common, often repeated pitch class between the tonic triads of the original and destination keys (see Figure 92).

Figure 92. Pivot note modulation from C diatonic–major into A-flat diatonic–major.

Chromatic modulation rests upon a chord progression from which emerges a conjunct chromatic line leading to the new key (see Figure 93).

Figure 93. Chromatic modulation from C diatonic–major into D diatonic–minor.

Enharmonic modulation exploits the dual function of a chord according to its enharmonic spelling, exemplified by the German augmented-sixth chord of one key functioning enharmonically as the dominant seventh chord of another (see Figure 78).

By choosing a chord which belongs to both the original and target diahemitonic keys, one can use it as a pivot to modulate (see Figure 94).

Figure 94. Pivot chord modulation from C diahemitonic–major into B-quarter-tone-flat diahemitonic–minor.

By selecting a pitch class which belongs to both the tonic tetrad of the original diahemitonic key and that of the destination one, one can use it as a pivot to modulate (see Figure 95).

Figure 95. Pivot note modulation from C diahemitonic–major into B-flat diahemitonic–major.

A diahemitonic chord progression can be designed so that a conjunct chromatic line emerges and leads from the original diahemitonic key to the new one (see Figure 96).

Figure 96. Chromatic modulation from C diahemitonic–major into C-sharp diahemitonic–minor.

The diahemitonic enharmonic German augmented-sixth chord was discussed earlier (see Figure 82).

Non-Harmonic Notes

Non-harmonic notes are melodic notes which are not part of the defined harmony. Several types are distinguished: passing note, neighbour note, escape note, anticipation, suspension, retardation, appoggiatura and pedal note. The passing note conjunctly connects two distinct harmonic notes. The neighbour note briefly departs from the harmonic note by conjunct motion before returning. The escape note departs from the harmonic note by conjunct motion before resolving disjunctly in the opposite direction onto the next harmonic note. The anticipation is heard before the chord of which it is a harmonic note. The suspension and the retardation prolong the harmonic note into the following chord to which they are foreign and then resolve conjunctly, downwards in the case of the suspension and upwards in the case of the retardation. The appoggiatura is sounded simultaneously with the chord to which it is foreign and then resolves onto the harmonic note by conjunct motion. The pedal note is sustained while various chords to which it is foreign, excluding the first and last, progress.

Based upon the above definitions, there is no reason why these non-harmonic notes cannot be applied to diahemitonic tonal music. The composer can therefore use them as desired, drawing upon the expertise acquired through their practice of diatonic tonality.

Pitch Range

Diatonic tonal music conventionally spans approximately seven octaves, as exemplified by the piano keyboard. This mirrors the fact that the perfect fifth, from which the semitone scale originates, is made of seven semitones and its unfolded cycle spans the same number of octaves.

The major fourth, from which the quarter-tone scale originates, is made of 11 quarter-tones and its unfolded cycle also spans the same number of octaves. Therefore, diahemitonic tonal music may span 11 octaves, potentially enriching it with infra- and ultrasonic frequencies.

Conclusion

In this article, I have studied the characteristics of diatonic tonal music before mapping them onto the quarter-tone scale, thereby leading to diahemitonic tonality, a tension–release quarter-tonal harmonic system which considers scale, mode, key, interval, chord, function, cadence, sequence and modulation. I have demonstrated that, with the exception of the interrupted cadence and the pedal six–four chord, each facet of diatonic tonality finds its diahemitonic counterpart, marking a significant recurrence of the diatonic–diahemitonic analogy since diahemitonic modality, from which the present system descends. The successful rebuilding of tonal grammar within a quarter-tonal environment not only reaffirms the kinship between diatonicism and diahemitonicism but also highlights fundamental properties of harmony and paves the way for further microtonal explorations, promising deeper insights into musical principles.

Charles Ives, in his article on quarter-tonal music,^[12] referred to the perfect fifth, the interval at the basis of semitonal music, as an ‘inexorable thing […] which apparently no aesthetic principle has yet beaten out.’ He eagerly longed for the day when ‘an Edison, a Dempsey or an Einstein will or will not suppress [it] with a blow from a new natural law.’ The diahemitonic system, by its reliance upon the 11:8 major-fourth ratio acoustically intrinsic to quarter-tonality, seems to answer this call. Such an alignment with Ives’s prophetic vision underscores the necessity of diahemitonic theory for advancing microtonal practices.

Inasmuch as tonality has historically relegated the modal system which precedes it to a preliminary stage, I believe I have discovered, with the advent of diahemitonic tonality, the major component of diahemitonicism, of which the earlier modality may now be viewed as an indispensable first step. This suggests that while, as Alain Bancquart observed, the 20^th century constituted a second Renaissance,^[13] the new level thus reached offers, at least as far as the xenotonal microcosm is concerned, the horizons of a second classicism.

Bibliography

Aldwell, Edward, Carl Schachter and Allen Cadwallader. Harmony and Voice Leading. 5^th edition. Boston: Cengage Learning, 2018.

Bancquart, Alain. Musique : habiter le temps. Lyon: Symétrie, 2003.

D’Indy, Vincent. Cours de composition musicale, premier livre. Paris: Durand, 1912.

Hindemith, Paul. The Craft of Musical Composition – Book 1: Theoretical Part. Translated by Arthur Mendel. 4^th edition. London: Schott, 1984.

Ives, Charles. ‘Some “Quarter-Tone” Impressions’. In Essays Before a Sonata, The Majority, and Other Writings, edited by Howard Boatwright, pp. 107–19. New York: W. W. Norton, 1970.

Kaplan, Simon. ‘Diahemitonic Modality: A Quarter-Tonal Composition System’. Tempo, vol. 77, no. 306 (October 2023): pp. 69–92.

Rameau, Jean-Philippe. Treatise on Harmony. Translated by Philip Gossett. Revised edition. New York: Dover, 1971.

Wyschnegradsky, Ivan. La loi de la pansonorité. 1^st edition. Geneva: Contrechamps, 1996.

^[1] Simon Kaplan, ‘Diahemitonic Modality: A Quarter-Tonal Composition System’, Tempo, vol. 77, no. 306 (Oct. 2023): pp. 69–92.

^[2] Jean-Philippe Rameau, Treatise on Harmony, tr. Philip Gossett (New York: Dover, 1971), pp. 3–55.

^[3] Ivan Wyschnegradsky, La loi de la pansonorité, 1^st edition (Geneva: Contrechamps, 1996), pp. 68–73.

^[4] Paul Hindemith, The Craft of Musical Composition – Book 1: Theoretical Part, tr. Arthur Mendel, 4^th edition (London: Schott, 1984), pp. 57–64.

^[5] Vincent d’Indy, Cours de composition musicale, premier livre (Paris: Durand, 1912), pp. 91–106.

^[6] Coined from the Latin sesquiaugmentatum, meaning ‘(one) and a half (times) augmented’ (from semis ‘half’ + -que ‘and’ + augmentare ‘to augment’ + -atum ‘-ed’), upon the model of ‘augmented’.

^[7] Edward Aldwell, Carl Schachter and Allen Cadwallader, Harmony and Voice Leading, 5^th edition (Boston: Cengage Learning, 2018), pp. 57–58.

^[8] Rameau, Treatise on Harmony, pp. 40–52.

^[9] Rameau, Treatise on Harmony, pp. 88–91.

^[10] Rameau, Treatise on Harmony, pp. 93–95.

^[11] Aldwell, Schachter and Cadwallader, Harmony and Voice Leading, p. 549.

^[12] Charles Ives, ‘Some “Quarter-Tone” Impressions’, in Essays Before a Sonata, The Majority, and Other Writings, ed. Howard Boatwright (New York: W. W. Norton, 1970), pp. 107–19.

^[13] Alain Bancquart, Musique : habiter le temps (Lyon: Symétrie, 2003), pp. 1–8.

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