Chapter 7

This annex is almost identical to the
chapter entitled « 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.
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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 N On the N On the N On the N On the N On the N Moreover, a N | ||||

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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 X
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 N 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 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 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.
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 N 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 X | ||||

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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 X 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 N 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
X
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: A 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¬ 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. | |||||

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