Unique Harmonic Relationships:
A Consideration of Progressions Between
Major Triads, Minor Triads,
Augmented Triads, Fully-Diminished-7th Chords,
Major-Minor-7th Chords, and Half-Diminished-7th Chords
by Jody Nagel
July 27-29, 2004
Go Directly to the Chart of Harmonic Relationship-Types.
When classifying distinctive, unique harmonic relationships between the common triads and 7th-chords of 12-pitch-per-octave tonal composition, one observes the following:
1. A transposition of some particular harmonic relationship-type possesses (at least to those of us without perfect pitch) the same harmonic effect (e.g., C-Major progressing to Eb-Major sounds harmonically identical to F#-Major progressing to A-Major). A unique harmonic relationship-type can be identified by the unique set of "closest voice-leading" fragments that comprises the particular relationship-type.
2. The same harmonic relationship-type exists regardless of the order of the two chords in question. "Chord-A" progressing to "Chord-B" has the same harmonic effect as "Chord-B" progressing to "Chord-A." Melodic directionality will be effected by reversing the order of two chords, but, in the specifically harmonic sense, the relationship-type will remain identical. Furthermore, when other-than-closest voice-leading is used between two chords, the melodic/contrapuntal effects will obviously differ, but the listener's awareness of which harmonic relationship-type is being presented will remain the same as if it were closest voice-leading. The ear, harmonically speaking, will connect the pitches of the first set, within octave space, to the pitches of the second set, regardless of what the actual voices, actually do. Closest voice-leading, then (harmonically speaking again), is not so much about rules for composers, as it is about what happens in the minds of listeners in spite of composers!
3. A relationship between two transpositions of the same asymmetrical pitch-structure (e.g., such as between two different major triads) might be one of six unique relationship-types. These are the "root relationships" of major and minor 2nds and 3rds; perfect 4ths; and tritones. The inversions of these six intervals, when describing root relationships, do not change the harmonic effect of the non-inverted analogous interval (e.g., C-major "up" to E-major has the same harmonic effect as C-major "down" to E-major. C-Major to Ab-Major is a transposition of E-Major to C-Major, and is also, therefore, the same relationship-type.) Furthermore, the addition of one or more octaves to these basic intervals, when describing root relationships, does not change the harmonic effect.
4. A relationship between two different asymmetrical pitch-structures (e.g., such as between one major and one minor triad) might be one of twelve unique relationship-types. These are the "root relationships" of major and minor 2nds, 3rds, 6ths and 7ths; perfect unisons, 4ths and 5ths; and tritones. When examining classes of two different asymmetrical pitch-structures, one sees there is a missing "up-down" symmetry, which two transpositions of the same asymmetrical pitch-structure have (e.g., C-Major "up" to C#-Major has an identical set of "closest voice-leading" intervals as does C-Major "down" to B-Major: namely three voices moving by semitone in the same direction. However, C-Major "up" to C#-Minor differs from C-Major "down" to B-Minor: in the former, two voices move by semitone in the same direction and the 3rd voice "moves" by common tone; in the latter, two voices move by semitone in the same direction and the 3rd voice moves in the same direction by whole-tone.)
5. If one limits harmonic relationship-types to just those between major and minor triads, there will be found: (1) six Major---Major relationships; (2) six Minor---Minor relationships; (3) twelve Major---Minor relationships. This is a total of 24 Major/Minor triadic relationships, which, interestingly, is the same as the number of Major and Minor keys. In all of these 24 two-triad relationships, the relationship can be "polarized" so that either one of the two chords sounds like the "tonic" triad, and the remaining chord sounds like a departure from tonic (e.g., of the relationship between two major triads with roots a perfect 4th apart, say C-Major and F-Major, either chord can be the "I" chord while the remaining chord is "other." If C-Major is "I", then F-Major is "IV." If F-Major is "I", then C-Major is "V".)
6. Concerning the Augmented Triad and the Fully-Diminished-7th Chord, the symmetry of these pitch-structures reduces the number of harmonic relationship-types that can exist just between themselves, or between themselves and major or minor triads. An Augmented Triad divides octave pitch-space into three equal 4-semitone-sized major-3rds, and so there are only four unique Augmented Triads. A Fully-Diminished-7th Chord divides octave pitch-space into four equal 3-semitone-sized minor-3rds, and so there are only three unique Fully-Diminished-7th Chords. When an Augmented Triad progresses, say, to a Major Triad, there are only four available harmonic relationship-types. These are the cases of one tone of the Augmented Triad moving by common-tone; ascending semitone; descending semitone; or whole-tone to the root of some Major triad. When a Fully-Diminished-7th Chord progresses, say, to a Major Triad, there are only three available harmonic relationship-types. These are the cases of one tone of the Fully-Diminished-7th Chord moving by common-tone; ascending semitone; or descending semitone to the root of some Major triad. Two of these three harmonic relationship-types are traditionally known as "Leading-tone" Fully-Diminished-7ths and "Common-tone" Fully-Diminished-7ths. The remaining type, that of one tone of the Fully-Diminished-7th Chord moving by descending semitone to the root of some Major triad, is playfully labeled, by this author, as "Diminished-7th Chords of the Third Kind." In many cases, this "progression" sounds like what Walter Piston would have called a "dominant minor-9th without a root," where the 9th falls by semitone to the root of a dominant chord. However, if the 7th of the original rootless dominant 9th ascends by whole-step to the root, in contrary motion to the descending 9th (and oppositely to the way 7ths usually move), then this "Phrygian Half-Cadence-like" voice-leading creates a distinctive harmony which sounds much stronger than merely one chord with a non-harmonic tone that resolves.
7. Since there are only three Fully-Diminished-7th Chords, when one of them "progresses" to another by ascending or descending minor-3rd, tritone, or major-6th, the result is to maintain the original set of four pitch-classes, and this is not really a progression, from one chord to another, at all. If one Fully-Diminished-7th Chord moves to another by ascending semitone, major-3rd, perfect-5th, or minor-7th, or by descending whole-tone, perfect-4th, minor-6th, or major-7th, the resultant harmonic relationships will all be of the same type, and a listener will notice a distinct sense that the first Fully-Diminished-7th Chord is rising by semitone to the second one. If one Fully-Diminished-7th Chord moves to another by descending semitone, major-3rd, perfect-5th, or minor-7th, or by ascending whole-tone, perfect-4th, minor-6th, or major-7th, the resultant harmonic relationships will again all be of the same type, and a listener will notice a distinct sense that the first Fully-Diminished-7th Chord is falling by semitone to the second one. Referring to Paragraph  above, one recalls that the order of two chords does not change their fundamental harmonic relationship-type, so, when one Fully-Diminished-7th Chord progresses to another, assuming it does indeed move to one of the other two, then whatever it does will result in the same harmonic relationship-type.
8. Reasoning in a similar manner for Augmented Triads, when one Augmented Triad "progresses" to another by ascending or descending major-3rd, or minor-6th, the result is to maintain the original set of three pitch-classes, and this also is not really a progression. If one Augmented Triad moves to another by ascending semitone, perfect-4th, or major-6th, or by descending minor-3rd, perfect-5th, or major-7th, the resultant harmonic relationships will all be of the same type, and a listener will notice a distinct sense that the first Augmented Triad is rising by semitone to the second one. If one Augmented Triad moves to another by descending semitone, perfect-4th, or major-6th, or by ascending minor-3rd, perfect-5th, or major-7th, the resultant harmonic relationships will again all be of the same type, and a listener will notice a distinct sense that the first Augmented Triad is falling by semitone to the second one. So, if any two Augmented Triads are related by semitone, then they are of one basic harmonic relationship-type, and the overall sense will be "Hexatonic" in nature. (The Hexatonic Scale is an alternation of semitones and minor-3rds.) The second and only other relationship-type between two Augmented Triads is if one moves to the other by ascending or descending whole-tone, or by tritone. In this case, the overall sense will be "Whole-tone" in nature.
9. When some particular Fully-Diminished-7th Chord progresses directly to one of the four Augmented Triads, it is observed that, in all cases, the closest voice-leading involves one common tone, one asecending semitone, one descending semitone, and a voice that can optionally ascend and/or descend by whole-tone to create a doubling (in the Augmented Triad) with one of the semitone-moving voices. When some particular Augmented Triad progresses directly to one of the three Fully-Diminished-7th Chords, it is observed that, in all cases, the closest voice-leading involves one common tone, one asecending semitone, one descending semitone, and either (1) one doubled voice (in the Augmented Triad) that can also ascend by whole-tone or (2) one doubled voice (in the Augmented Triad) that can also descend by whole-tone. In other words, when using closest voice-leading, the same set of voice-leading fragments occurs no matter which transposition of the Augmented Triad or which transposition of the Fully-Diminished-7th Chord is involved. There is, then, merely one relationship-type that exists between Augmented Triads and Fully-Diminished-7th Chords.
10. Consider the asymmetrical Major-Minor-7th Chord or the asymmetrical Half-Diminished-7th Chord progressing to a major or minor triad. For each of these four cases, there are twelve possible harmonic relationship-types. However, the twelve relationship-types group themselves into three sub-classifications based on with which of the three Fully-Diminished-7th Chords the particular Major-Minor-7th Chord or the particular Half-Diminished-7th Chord shares three of its four pitch-classes. If any one of the four pitch-classes of a Fully-Diminished-7th Chord is lowered one semitone, the resultant pitch-structure will be a Major-Minor-7th Chord. If any one of the four pitch-classes of a Fully-Diminished-7th Chord is raised one semitone, the resultant pitch-structure will be a Half-Diminished-7th Chord. Recalling the names of the relationship-types involving a Fully-Diminished-7th Chord progressing to a major or minor triad [L.T. = "Leading-Tone Fully-Diminished-7th Chord"; C.T. = "Common-Tone Fully-Diminished-7th Chord"; 3.K. = "Third-Kind Fully-Diminished-7th Chord"], we can classify four transpositions of the Major-Minor-7th Chord (moving to some specified major or minor triad) as "Almost-L.T." Major-Minor-7ths; four other transpositions of the Major-Minor-7th Chord (moving to the same specified major or minor triad) as "Almost-C.T." Major-Minor-7ths; and the remaining four transpositions of the Major-Minor-7th Chord (moving to the same specified major or minor triad) as "Almost-3.K." Major-Minor-7ths. In like manner, we can classify "Almost-L.T." Half-Diminished-7ths; "Almost-C.T." Half-Diminished-7ths; and "Almost-3.K." Half-Diminished-7ths (e.g., if an Half-Diminished-7th Chord is to progress to a tonic G-Minor Triad, then the chords
will each be an "Almost-L.T." Half-Diminished-7th Chord. It should be emphasized that these four chord-progressions, though of the same sub-classification, are indeed four different unique harmonic relationship-types.)
11. There are six unique harmonic relationship-types involving two transpositions of the Major-Minor-7th Chord. And again, there are six unique harmonic relationship-types involving two transpositions of the Half-Diminished-7th Chord. There are twelve unique harmonic relationship-types involving one Major-Minor-7th Chord and some transposition of one Half-Diminished-7th Chord. This is, of course, identical in principle to the situation involving major and minor triads. The asymmetrical major triad is inversely related to the minor triad, and, as has been said, there are 24 major/minor triadic relationship-types. The asymmetrical Major-Minor-7th Chord and the asymmetrical Half-Diminished-7th Chord are also inversely related (both having the "Prime-Form" pitch structure [0, 2, 5, 8]), and once again we see that there are 24 relationship-types that employ only Major-Minor and/or Half-Diminished-7th Chords.
12. There are four unique harmonic relationship-types involving a Half-Diminished-7th Chord progressing to an Augmented Triad, and four unique harmonic relationship-types involving a Major-Minor-7th Chord progressing to an Augmented Triad. There are three unique harmonic relationship-types involving a Half-Diminished-7th Chord progressing to a Fully-Diminished-7th Chord, and three unique harmonic relationship-types involving a Major-Minor-7th Chord progressing to a Fully-Diminished-7th Chord.
13. Major-Minor-7th, Fully-Diminished-7th and Half-Diminished-7th Chords and Diminished and Augmented Triads do not seem to be able to function that easily as "tonic" sonorities. This is most likely due to the fact that these sonorities possess the symmetrical tritone, or symmetrically divide the octave into thirds (in the case of the Augmented Triad). Pitch symmetry tends to create an inability to orient aurally to a sound, and this is opposite to the nature of a "tonic" sound. A final "tonic" sound, almost by definition, creates the sensation of "return" to familiar orientation, and the ability to orient aurally to a sound requires that its pitch-structure be asymmetrical, and, better yet, contain no subsets that could be construed symmetrically (e.g., the tritone within a Major-Minor-7th Chord). Of course, the Major and Minor Triads are the perfect candidates for "tonic" sonorities. Moreover, consider the nature of the asymmetricality of these triads: an octave is divided unequally into perfect-5th and perfect-4th (7 + 5 semitones), and the larger of these two intervals is subdivided further into a major-3rd and a minor-3rd (4 + 3 semitones, or 3 + 4 semitones). Adding up to an octave are minor-3rd, major-3rd, and perfect-4th (3 + 4 + 5 semitones). This set of proportions is the smallest of the Pythagorean Triples (i.e., 32 + 42 = 52). A Pythagorean "Right" triangle, unlike Equilateral or Isosceles triangles, is asymmetrical in each of the three dimensions that it might be rotated. Whether it is spun around or flipped, one can "always tell" where the right angle is and where the hypotenuse is. In the case of Major and Minor Triads, one can place them in root position, 1st or 2nd inversion, open or close position, and one can still "always tell" which note is the root, which is the third, and which is the fifth.
14. There are other non-Pythagorean asymmetrical sets of numbers that add up to 12, such as (1 + 2 + 9), representing the intervals between, say, the pitches C, Db, Eb, and C. Other possibilities that lack a tritone subset include (1 + 3 + 8), (1 + 4 + 7), and (2 + 3 + 7). The asymmetry need not be limited to 3-interval divisions. An example of an asymmetrical 4-interval division is (1 + 2 + 4 + 5). The one remaining advantage that (3 + 4 + 5) has over these others is that it contains the maximally widest and most evenly separated set of pitches, allowing the ear maximal clarity in keeping track of the pitches. The only "better" distribution in terms of evenness of size, would be the augmented triad (4 + 4 + 4), but because it is so even as to be perfectly symmetrical it utterly fails at being a "tonic" sound.
15. When considering two chords continually alternating (A - B - A - B - A - B - A . . .), if just one of the two pitch-structures is a major or minor triad, then the major or minor triad will sound like the "tonic" sound. If both chords are a major or minor triad, then the relationship-type can be polarized either way, depending on extra-harmonic issues such as rhythm and accent. An alternation of half notes and quarter notes coupled with the alternating triads will probably make the chord with the longer duration "the tonic." If the alternation gradually slows down from chord to chord, in some cases, such as two major triads with roots a tritone apart or a perfect-4th apart, the strange sensation occurs that whichever chord was most recently sounded is the one that sounds "tonic." If neither of the chords are major or minor triads, it is difficult for either sonority to sound convincingly "tonic." However, it can become possible for the root, third and fifth of the Major-Minor-7th Chord (the Major Triad subset), or the third, fifth and seventh of the Half-Diminished-7th Chord (the Minor Triad subset) to dominate the sonority, and then for one of these chords to give the impression of being "tonic-like" (with the "other" 4th note more of a non-chord-tone than a chord-tone. The author considers this to be the situation when, in some jazz pieces, the lowered 7th is added to a final tonic Major Triad.)
16. All harmonic relationships of "Classical" tonal music are basically one of the above-mentioned relationship-types, perhaps flavored a bit with non-harmonic tones. For example, within the progression I-IV-V-I, the first two chords and the last two chords are examples of "two Major Triads with roots a perfect-4th apart," while the IV-V relationship is an example of "two Major Triads with roots a whole-tone apart." When more than two chords are sounded in succession, there is the question of why the "actual tonic" continues to be felt beyond its connection to the immediately following chord. For example, in the key of C Major, The I-IV-V-I progression is C-Major, F-Major, G-Major, C-Major. The C-Major chord followed by the F-Major chord could easily sound like a I-IV progression. However, if the F-Major chord lingers for awhile, then the progression from F-Major to G-Major might sound like I-II in F-Lydian, or perhaps even as bVII-I in G-Mixolydian. The only way the ear is willing to consider the F-Major-to-G-Major event as "being in C" (i.e., IV-V) is if the initial C-major Chord was sufficiently convincing to the short-term memory that it still prolongs some effect into the moment that the F-Major chord changes to the G-Major chord. If the C-Major chord fails to create this sensation, either because of poor composition techniques, or masterful composition techniques (depending on if the goal is or isn't for C-Major to remain "tonic"), or if the listener isn't paying attention, or if the listener does not have the memory capacity to consider the relationship in the same way that the composer intended, then the moment of F-Major progressing to G-Major will not sound like IV-V in C-Major, but rather as something else.
17. An excellent experiment is to study and listen to Wagner's Prelude to Tristan und Isolde, and carefully to consider which "tonic" triad is being suggested by each chord within the succession of beautiful sonorities. One will realize that the first several minutes of the piece emphasize the keys of A-minor, C-Major, and E-Minor (the outline of the "background" tonic of the entire Prelude) though these triads are never themselves sounded. So, do they or don't they exert an harmonic interpretive effect over the chords that do sound? This writer suspects that, to most folks, there is no sensation of these "tonic" interpretations, but that to those that have trained themselves in expectations of normative harmonic interpretations (whatever these might be to that person), then Wagner's music is hopelessly tonal, and all about the key centers A, C, and E.
18. In a complex succession of triads and 7th-chords, it would seem to be possible that a careful listener would begin to notice more the flow of relationship-types than to notice the chords as they supposedly relate to a background key. Harmony grew out of scales; this is why Rameau used Roman Numerals to identify a triad's root as to where it is found within the scale. But it is possible that harmony, even triadic and 7th-chord-based harmony (nothing terribly "way out," relative to the 20th century), has outgrown its beginnings, and that the succession of relationship-types within music is worthy of consideration independently of scales.
19. The number of total available voice-leading fragments between two chords can be found by multiplying the number of pitch-classes in the first chord by the number of pitch-classes in the second chord. So, for a progression of two triads, one can melodically make use of the fragments: root-root, root-third, root-fifth, third-root, third-third, third-fifth, fifth-root, fifth-third, fifth-fifth. These nine fragments can, of course, melodically ascend or descend. Even in the repetition of a single triad, these nine fragments are implicit; however, the root-root, third-third, and fifth-fifth fragments are generally called "repeated tones" and the other six fragments are referred to as "arpeggiation." Concerning classifying unique harmonic relationship-types (at least, for the chordal vocabulary discussed in this article), it does not matter into which relative register the particular voice-leadings are placed; harmonic identity will be maintained whether a given voice-leading is placed in the soprano, bass, or a middle voice. An example of a potential "problem" (for some folks) with this system would be the Classical Era's use of the I64 chord as a dominant function. For example, a progression involving V/V - I64 is presumably thought to be functioning with a perfect-4th root-relationship. Nevertheless, the closest possible voice-leading of this progression still involves the relationship of two Major Triads with roots a whole-step apart, and the harmonic color of the local event is quite different from what it would have been if V/V - V. To claim that the function of the I64 is "dominant" is to have already a larger commitment to a sense of "key" then this author wishes to assume.
20. In a four-part texture, "closest possible voice-leading" of the individual parts always entails the use of only common-tones, semitones, or whole-tones. There can, however, be anywhere from 3 to 8 of these small intervals available for close voice-leading within various two-chord progressions (where each of the two chords is one of the types of triads or 7th-chords listed in the title of this article). For example, a C-Major Triad progressing to a D-Major Triad contains five possible small-interval voicings [C-D, E-F#, G-A, E-D, and G-F#], but only the first three of these would be necessary in a three-voice texture. Of the 24 major/minor triad relationships (using only complete triads), only the Major-to-Major root relationship of a tritone, the Minor-to-Minor root relationship of a tritone, and the Major-to-Minor root relationship of an ascending minor-7th requires, in three-part writing, one voice to move by three semitones.
21. In the following charts, the closest possible voice-leading for all possible relationship-types of each of these categories is given, and the available small intervals for close voice-leading are indicated. Also, each progression will be "scored" by the minimal amount of semitone displacement required to "pull it off" within a four-part texture (or within a three-part texture, if only three-note triads are involved.) Also note, that these voice-leading charts are mathematically "absolute" in nature, and do not necessarily reflect the common practice of any particular time or place. It should be noted, however, that the bias of these charts is to consider each two-chord relationship-type as if one of the chords is "tonic," and to consider the progression to be a "tonal" event. In an actual piece of music, a third chord in a succession may or may not sound like it is in the "same key" as the first chord. Thus, these charts are normalized to a "C-tonic" triad, and represent the pure elements of harmonic progression using only the two-chord vocabulary outlined within this article.
22. These charts do not insist on "spelling chords in 3rds," à la Rameau. Rather, (excepting the tonic triad, which must be spelled in 3rds) these charts do insist on spelling chords as they would be linearly voiced using exclusively pitches represented by some transposition (in this case, C) of the "Modal-Chromatic" Scale: 1 - b2, 2 - b3, 3 - 4, #4 - 5 - b6, 6 - b7, 7 - 1. This is the spelling of all twelve chromatic notes as they would be found in the six "white-key modes" from Phrygian up to Lydian (via the circle-of-fifths, but transposed to a common starting pitch). Or, as this author prefers using Moveable-Do Chromatic Solfeg Syllables over using scale-degree numbers, the scale can be thought of as Do - Ra, Re - Me, Mi - Fa, Fi - Sol - Le, La - Te, Ti - Do. The scale requires the tonally important perfect fifth of scale degrees 1 and 5 to remain locked as constants. The use of scale degrees b5 (as in the Locrian Mode) or b4 (as in the particular "Heptatonia Secunda" mode that can be described as Locrian with b4) will lie outside the scope of this article, as the modes which contain these scale degrees lack the perfect-5th above tonic that this author feels is the only true cause of the phenomenon of tonality. The use of raised scale degrees, as in traditional secondary leading tones, is not considered here, since a secondary leading tone (such as in a V/ii - ii progression within a major key) implies a temporary shift to a different tonic, and as such is merely a transposition of one of the unique elements that these charts set out to depict in their entirety. Additionally, raised scale degrees typically used in "micro-tonicization" (e.g., scale degrees #2-3 within a tonic Major Triad, or #1-2 within a V-chord) are also not considered here, as the scope of this article does not set out specifically to address the added complexities of proper spelling of non-chord-tones within the basic harmonic elements listed here.
23. A good "classical" example of the "Modal-Chromatic" Scale used "correctly" is found in the bar just before the recapitulation of the first movement (m.79) of W. A. Mozart's A-Minor Piano Sonata. The ascending chromatic scale, in A minor, within the span between scale degree 5 (E) and an octave higher, is spelled E, F, F#, G, G#, A, Bb, B-natural, C, C#, D, D#, E. If Mozart had thought the way most music instructors seem to think, that one must spell ascending chromatic scales "in sharps," then he would have used an A# rather than a Bb. Mozart's use of the Bb here seems to indicate he was still aware of its role as a scale degree b2 relative to A, even though the passage is played so quickly that some might not think "it mattered."
24. In any case, 2-chord progressions within the following charts will be spelled exclusively along these lines of reasoning, and the progressions will use the closest possible voice-leading as the "model" for each progression. For example, a progression from an E-major Triad to a tonic C-major Triad is spelled with the voice-leading fragments: (B - C), (E - E), and (Ab - G). It might seem strange to a pianist or a guitarist to spell an E-Major Triad (E, Ab, B), but when voiced for choir, or the instruments in some ensemble, this tonal-linear way of spelling makes even more sense than "spelling in thirds." The voice that has the Ab-G fragment, as opposed to a G#-G fragment, understands the sound of an half-step upper-neighbor to scale degree 5: Le - Sol ( = b6 - 5), but does not really know what to make of a displaced scale degree 5: Si -Sol ( = #5 - 5). An example of an A-Major-Minor-7th Chord progressing to a tonic C-Minor Triad is spelled with the voice-leading fragments: (Db - C), (E - Eb), (G - G), and (A - G). Once again, the spelling of the A-Major-Minor-7th Chord (A, Db, E, G) makes perfect sense from a tonal-linear perspective.
25. Since we have been examining six distinct chord-types, the number of families of possible associations between themselves and each other is, for n=6, (n2+n)/2, or 21 families. Here then is every harmonic relationship-type, by family, discussed within this article.
Click the family-type to see all of its possible relationship-types.
An X-X relationship-type refers to some intervallic relationship between two chords of the same type
(e.g., two major triads, with roots some distance apart).
An X-Y relationship-type refers to some intervallic relationship between two chords of different types
(e.g., a major triad and a minor triad, with roots some distance apart).
- For X-X harmonic relationship-types,
with X having a transpositional degree of symmetry of S, there are:
N = int(6 ÷ S) unique harmonic relationships.
- For X-Y harmonic relationship-types,
with X having a transpositional degree of symmetry of S1,
and Y having a transpositional degree of symmetry of S2,
and S1 and S2 having a greatest common factor, F, there are:
N = 12 ÷ [ ( S1 . S2 ) / F ] unique harmonic relationships.
Some familiar chords and pitch collections along with their transpositional degree of symmetry
major triad, minor triad, diminished triad, major-minor-7th, half-diminished-7th , major-7th, minor-7th, diatonic set, heptatonia secunda set, "black-key" pentatonic.
French augmented-6th, Z-cell, ocatonia secunda set.
augmented triad, hexatonic set.
fully-diminished-7th, octatonic set.
For example, if you would like to know the number of unique relationship-types that there are between a French augmented-6th chord and a "Z-cell" [0,1,6,7], then, since S1 = S2 = 2, and therefore F = 2, there are:
N = 12 ÷ [ ( 2 . 2 ) / 2 ] = 6 unique harmonic relationships between a French augmented-6th chord and a Z-cell.