Index
Chapter 7

Annex 1

8. Morphogenesis of chords and scales

This annex is almost identical to the chapter entitled «Morfogčnesi dels acords i de les escales musicals» in my book La convergčncia harmňnica, published in 1994.

At the end of this annex we have written up harmonic tables where are represented all (!) the chords and scales that could be formed with the 12-tone equal tempered system, gruped according the number of notes they contain (eliminating repeated notes in other octaves) and according to their harmonic structure. That is, we have represented all 'chord-classes' and all 'scale-classes'.

What appears in the tables are chords and scales as representantives of a wide family of musical statements, what is known as 'chord classes' or 'scale classes' (defined in 8.1 and 8.2). Depending on the horizontal or vertical musical thought each group of notes could be considered as chords (even if they are written as scales) or arranged as scales (even if they are written as chords). It is only for reasons of clearity that the groups of up to six notes are printed as chords while those of seven note or more appear as scales.

We will see that at certain equivalence level between inversions and modes the number of chords and scales that could be formed is the same so they could have the same harmonic representative which is the one that appears in the chord/scale class tables.

8.1 Equivalence level between chords

In musical theory or analysis, to say that two chords or two scales are the same could have many different interpretations. Normally, one trusts the reader's own musical judgement to deduce the correct equivalence level of the two chords. On a more precise level, two chords are the same only if they are in the same octave, have the same notes and layout. On another level, two chords are equivalent if they have the same notes but this equivalence is only stated by the bass. Also, if one chord is the transposition of the other, if it is the inversion, and so forth.


The different equivalence levels and the number of (different) possible chords in each equivalent group (each level includes the previous equivalence level) are shown in Figure 73.

Fig. 73


When working with the 12 tone equal tempered gamma a first equivalence already assumed and not specified in figure 73 is the enharmonic equivalence. That doesn't mean indifference when representing the same note in a way or another, on the contrary, following the thesis of our research, the separation of chords into fundamentals implies a precise harmonic notation, although sometimes a same pitch can be written differently, according to the harmonic or melodic context.

On the N1 level, two chords are equivalent only if they are exactly the same; at this level, the total number of chords is not specified because it is almost unlimited.

On the N2 level, two chords are equivalent if they coincide with the removal of empty octaves or doing strict transpositions of octave (of the whole chord). The number of possible chords in this group is still very wide (thousands of billions).

On the N3 level, the previous equivalence is included and the duplicated upper notes can be eliminated or upper octaves of individual notes added without affecting the chord's identity. The number of chords in this case is more that a thousand million (1.302.061.344 chords).

On the N4 level, the previous equivalences are included and two chords can be identified if one is the strict transposition of the other. The number of possible chords in this group is still very high (108.505.112 chords).

On the N5 level, the previous equivalences are included and two chords can be considered equivalent with any internal order of notes except for the bass. In this group we could include the triadic chords of traditional harmony and their inversions (as non-equivalent chords). The number of possible chords in this level is considerably reduced (2048).1

On the N6 level, the previous equivalences are included and two chords can be identified if one is the inversion of another one. In fact, at this level, a chord (also known as 'chord class') can be considered as a non-ordered collection of notes or, better to say as a collection of intervals (bearing in mind the N4 equivalence). The number of different chords is only 351 which are the ones represented on the Chord/scale classe tables. They would be the chords in any arrangement/order and in any inversion. It is curious that the numbering is symmetrical with respect to the number of notes they contain, the chords of 6 notes as central axis and more numerous (80). Sixty for the 5 and 7 notes, 43 for the 4 and 8 notes, etc. When we talk about 'chord class' we are referring to this chord group at this equivalence level.

Moreover, a N7 level could be considered, made up of the "chords" that constitute the fundamentals. The number of functional chords is not precise because on a certain level of complexity the chords can be decomposed into different convergent chords (fundamentals).


1 Are those that appear numerically in Table 5.


In general, we could classify chords into eight large functional families (see 3.1) and the symbology to represent them can be summarized in the formula XYZ (see 3.1 and 8.3), where X, Y and Z represent the fundamentals.

8.2 Equivalence level between scales

Taking into account that if we put the notes of a chord in horizontal form, we obtain a scale, the equivalence levels that we have seen in 8.1 also apply to scales and modes.

There is an equivalence relation between the concepts 'inversion of a chord' with the 'mode of a scale'.

We have 351 'chord classes' and 351 'scale classes'. If in a scale we fix a start note, we get one mode of this scale (just like if we change the bass of a chord, we get the different inversions of this chord). For example, the white keys of a piano (and their transpositions) form a 'scale class', if to this 'scale class' we fix different starting notes, we obtain different modes of this scale (known as medieval modes). A scale class has as many different modes as different notes have the scale (with the exception of cyclical scales, see Annex 3). A 7-note scale class has 7 modes. As there are 66 different kinds of 7-note scale class we will have a total of 462 different modes of 7 notes (66 x 7). Regardless of the number of notes, we have a total of 2 048 different possible modes, which coincides with the different possible chords in the equivalence level N6, formed by 'chord classes'.

In Annex 2 we can see a detailed study of the 56 modes of the first eight 'scale classes' of seven notes, which are the ones most used in the musical history of all human cultures.

In Figure 74 we give an example of a 'chord/scale class' of seven notes with their different manifestations as scales, modes and chords (and inversions); all set of notes that appear in figure 74, seemingly distinct, correspond to a single 'chord/scale class' (the one corresponding to number 2 in the table of 7-note chord/scales classes), represented (in the tables) by the chord ~C~Bµ.

We could say that different note sets form part of the same 'chord class' (or 'scale class') if they can be ordered or reduced to the same intervallic pattern. All collections of notes belonging to the same 'chord class' can be decomposed into the same sub-chords, represented by the fundamentals, and in most chords up to six notes, are determined by their characteristic fundamental symbology (remember 3.1). It could also be said that a 'chord/scale class' is formed by a chord with all its inversions and transpositions in any vertical or horizontal combination. In figure


74 we have a lot of chords and scales but only one 'chord/scale class' (the number 2 of those with 7 notes) and can be decomposed into two dominant seventh chords at M2/m7 distance, that is, in two main fundamentals. Tonally one fundamental represents the function of dominant and the other one (a lower M2) the subdominant function, creating a clear tonal vector.

Fig. 74

The decomposition/separation of scales and chords into fundamentals (which basically represent the internal tensions of the 'quasi-fifths') gives us a lot of information on the resolutive properties of the chord or scale.


For example, as we have said, the scale in figure 75 (chord class 7-2) has a tendency to resolve towards A¬ or, in other words, A¬ is its tonic because the fundamentals of this chord/scale class are precisely the dominant and the subdominant of A¬ (enharmony C¬=B). But, at first sight, it may appear that the melody belongs to C minor. In fact, this tendency to C exists but it is only secondarily as a Phrygian homotonic resolution of the secondary fundamental (D¬µ|c), like the secondary resolution to G¬ (htonal homotonic relaxion from D¬) (D¬µ|g¬), even if it is not a note of the scale (Figure 15).

Fig. 75

A similar thing can be said about this scale in chord form. By its tonal tendency (and by the homotonic resolution of the main fundamental) this chord, as it appears in figure 76, would have a main resolution to a chord based on a fundamental E, but, as we have seen in previous chapters, would also have other harmonic relaxions between fundamentals. The precisest harmonic notation for this chord (isolated) would have to be with D{ instead of E¬ (I have put E¬ just to show how the convergent decomposition can help to find the melodic enharmonies). But this chord, for melodic reasons, can perfectly appear with E¬ when the resolution is due to the relaxion of the secondary fundamental on a chord based on D. For ease of understanding of Figure 76 the same notation has been used for all the examples, but in cases (a), (c), (d) and (e) it would be more correct to put D{.

Fig. 76


Obviously, the resolutions in figure 76 are not conclusive; they merely reveal a locally harmonic relaxion—in some of them, for example (e), there is sonance tension. I insist, once more, on the importance of distinguishing between harmonic (homotonic) local tensions, the tensions of sonance and the tonal tensions.

These examples, centered on this chord/scale class manifest us the utility of knowing the internal tensions that the fundamental symbology shows.

The term 'chord class' or 'scale class' should not be confused with the term PC-Set (Pitch Class Set), introduced by Milton Babbitt and developed by Allen Forte in his book The Structure of Atonal Music. Forte adds a new relation of equivalence between chords: he considers chords of the same group the symmetrical chords (the symmetrical inversion of chords), that is, for example, he considers within the same PC-Set the major triad and the minor triad or the dominant seventh chord and the Tristan chord. See more information on symmetrical chords and modes (and the relation with PC-Set) in Annex 4.

8.3 'Chords classes' and 'Scale classes' tables.

Once we have studied the meaning and utility of the 'chord/scale classes' we will explain the meaning of the chords and scales that appear in Table 4: Chord/Scale classe Tables.

On these tables there are represented all the possible (different) chord/scale classes that could be formed with the equal tempered palette of 12 tones (that is, all possible chords —or scales— at the equivalence level N6). They are grouped according on the number of different notes in each chord/scale class. In total there are 351 classes3, but each group starts with an independent numbering.

The chord/scale classes up to six notes have been written vertically in chord form and from seven notes have been placed horizontally, in scale form.

The symbology chosen to represent the chords is the one already introduced in 3.1 and we call it fundamental symbology.

That is to say, the symbology takes the form XZY, where X, Y and Z are fundamentals of the chord (in capital letter if they are functional —the fundamental has its M3— or in lowercase if they are not). Y represents the possible fundamentals forming intervals of fifth, major third and minor seventh (and m2) with respect to X ('harmonic' fundamentals), and Z are the possible fundamentals forming intervals


2 Bear in mind that if we play the examples in Fig. 76 in a continuous way, we can create a tonal field towards E, which would distort the local homotonic sensation of relaxion that we want to show.

3 Counting also the groups of 1, 11 and 12 notes, not represented in the tables.


of lower minor third and tritone with respect to X (fundamentals in the same tonal axis).

Almost all chords up to six notes can be expressed with this symbology, but not all.

When the fundamental has its minor seventh in the chord, we put a 7 above (except when Y is precisely the minor seventh of X or Z). A functional fundamental (uppercase) with its minor seventh has a significant tonal force in the chord or scale since it implies that it contains the 7M3 structure. Complex chords, or with many notes, can sometimes be symbolized (applying XZY) in different ways, but what is really important is to find the significant functional fundamental(s) of the chord (such as those with their minor seventh).

When the fundamental has its (upper) fifth in the chord, we put a point above it (in the tables). With two exceptions: one, when Y is precisely the fifth of X and second, when Z is the lower minor third of X and X is in uppercase, in this case we will not put a point in Z because this configuration always implies that Z has its fifth and will save us work by writing the symbology of minor chords. We put a point here in these tables but it is optional in the analyzes.

When the chord has other functional fundamentals that are not reflected in the main symbol, we put them below (in lower rows if they are several). This usually happens from 5-note chords.

As representative of each chord class we have placed the one that has tone C as central fundamental (the X). Being central does not mean that it is the most significant fundamental of the chord since Y or Z can have the structure 7M3 and X does not, but in general it is so.

From 7 notes we have put the group in scale form. We have not look for functional fundamentals since they grow exponentially in number and complexity (except in the first eight of 7 notes).

Finding the functional fundamentals of a scale consist to look for the M3 and tritone intervals in it. Only in the first eight scales —which with their respective modes have been those most used in the history of music (see Annex 2)— we have specified the functional fundamentals they contain; in addition, in the inferior part, we have placed the characteristic tonal cadence of the scale towards its main tonic, using all its notes.

In the tables we have not shown groups of 1, 11 and 12 notes since there is only one chord class for these groups or, in other words, all possible "chords" (or scales) of one, eleven or twelve notes are the "same" at the equivalence level N6.

In order not to create confusion I use the same numbering that appears in my book written in 1994, although I would now use another numbering because the


first one was based on the logic of two fundamentals (in the first book I only used binary symbology type XY and XZ) and now I use the form XZY in order to be able to code more chords, but, as I have said, the symbols used are of no more importance if the functional fundamentals of the chords are determined. For most chords of three and four notes (and even of five) the 'fundamental symbology' coincides in the two books.

8.4 How to find the fundamentals or the fundamental symbology of the chords

If there is a set of notes in the form of a chord or scale and we want to make a homotonic functional analysis with respect to the groups of notes that precede and follow it, then, the first step is to analyze their fundamentals and, therefore, know the internal tensions of the chord and its tonal influence.

The most practical way to do this is by directly analyzing the interval relationships between notes. It will be sufficient to look for the intervals of M3 and tritone. Indeed, in many cases it will be sufficient to find the M3. Once these intervals are found it is easy to deduce their functional fundamentals.

For example, if we have the chord EB¬C{F{A (Figure 77(a)): M3 intervals. (including enharmonic changes) are: A-C{ and F{-A{(B¬). Indeed, having A and F{ as a fundamentals, the chord is separated into two convergent (harmonic) chords. The fundamental symboly will, then, be: AF{7

Fig. 77

Another example, if we take the chord: CGC{G{ (Figure 77b), it could be thought that, as the chord contains two P5, the fundamentals would be C and C{. Actually this example is a trap that hides the true fundamentals since, as it was stated on 1.3, the strength of fifths to support a fundamental is very weak. In this chord there is only a M3 A¬(G{)-C, the rest of the function is given by the tritone G-C{(D¬) in its convergent direction towards E¬ (because E¬ is the fifth of A¬). In this way, the two fundamentals are A¬ and E¬ —this latter one virtual— and the fundamental symbology: A¬\E¬µ.

Only if the intervalic analysis is complicated or one is not sure of the results (or, why not, for curiosity) it is possible to look up the numerical tables (Table 5) at the


end of this appendix where for every group of notes it will be easy to find their representative in the chord/scale class tables and deduct their fundamentals.

Taking as an example the same chords as before (Figure 78):

Fig.78

First of all it is necessary to put the chord in scale form (into an octave according to the order of the chromatic scale, if there are any repeated notes, they are eliminated). It is possible to start with any note of the chord (here, we show the bass as first degree of the scale to be clearer). Then one looks at the number of semitones between the different degrees in the scale just formed; this gives us one digit numbers which are considered as digits of one whole number (in this case 2313). Beside number 2313 in this Table 5 number 61 is found. This number indicates that the representative of the chord is num. 61 in the Chord/scale class tables (from the five notes group). Indeed, you just have to do a major sixth transposition of the representative in the tables to get the notes of our chord. The process for the other chord is similar, in this case we get number 26 and looking at the representative in the tables we deduce the fundamentals. A plugin for the Sibelius program that does this and other analyzes is also available.4


4 The interested reader can request it free of charge to lbalsach@gmail.com

Chord/Scale class Tables
Annex2