2. The harmonics of a sound as the basis of musical perception

2.1 Significant harmonics for the human auditory system

If we play C₂ on the piano (corresponding to a frequency of 131 Hz), the acoustic spectrum is approximately as follows:

Fig. 11

The vertical lines on the graph show the intensity of each of the harmonics of the note played.¹ This sound structure, a combination of simultaneous sounds with the same basic timbre (sinusoidal waves) but of differing intensity, enters the cochlea and excites a number of hair cells, each of which is sensitive to certain different frequencies. These frequencies travel along nervous fibres to the auditive cortex of the brain. This is where we identify the combination of sounds in figure 11 as "one" note on the "piano". That is, each sound is the sum of a collection of sounds of elemental timbre. This elemental timbre is always the same for all sounds; is the sound corresponding to a sinusoidal wave. ²

Although we have a collection of frequencies, the brain summarizes them all in one (which has the greatest common divisor) and is a frequency that may not exist in the collection of sinusoidal frequencies we have (!) (later on we will see the importance of this when we study the virtual fundamental). The information we 

¹ H.F.Olson (1967). This spectrum changes with time, from the moment the key is pressed until silence returns.

² According to the Fourier's theorem, any periodic wave (sound is a periodic wave) can be decomposed as the sum of sinusoidal waves.

receive from the "volume" of each harmonic the brain makes it a concept we know as a timbre and it is very useful to recognize what produces the sound and distinguish, for example, between the voices of friends or different musical instruments. The fundamental note, as shown in figure 11, does not have to be the most powerful one, even, as we say, it may not exist. What matters is the pattern, the relation that the frequencies of the harmonics establish with each other.

From birth, the brain is familiar with these frequency relations of the harmonics that form a sound, which are "intervallic" relations that always follow the same pattern and at first sight is very simple: if f is the frequency of the key of the piano (the fundamental sound), the harmonics series have frequencies 2f, 3f, 4f, 5f, 6f, etc. (the double, triple, quadruple, quintuple and so on of frequency f).

But translate this pattern of frequencies into musical notes is not so simple. In figure 12 we have represented, in arpeggio form (from a C₀), this pattern-chord translated to musical notes —with its divergence with respect to the three main tuning systems—. The curved drawing of figure 12, instead of the straight line to be expected from the function nf, is due to Weber-Fechner's law which states that sensations are in logarithmic relation to the stimuli. Thus, for example, a frequency which is 4 times higher (4f) than another (f) does not involve musically 4 octaves but only 2. The same thing happens with the sound intensity (thanks to that in the orchestras there can be many more strings than other instruments). A negative number indicates that the pitch of the harmonic is lower than the pitch of the note in the tuning system. All numbers are rounded to integers.

Fig. 12

Of these collections of notes and intervals, which are those that the brain «takes into account» to construct «musical laws»: scales, consonance, chords, sensations of tension and relaxion, tonality, etc.?

This is a question that does not have a unanimous answer. And it goes from considering the first three harmonics to considering the first 16.

In this chapter we will see that all harmonics can be considered, but at the same time they can be summarized in the 2f, 3f, 5f and very slightly the 7f (it is to say, the first 4 prime numbers [after f] of the harmonic series) and each one of them serves to establish different psycho-acoustic processes that generate the perception that we humans have of the musical tensions.

But before doing the demonstration let us first do a historical review:

Before the phenomenon of harmonics was known (discovered between the 17th and 18th centuries)³ theorists considered musical intervals as relations between natural numbers (2, 3, 4, 5 ...). For example, the octave as the 2/1 ratio, the fifth as the 3/2 ratio, the fourth as the 4/3 ratio, etc. Since, for example, using a monochord or a violin string, this was the ratio of the position of the fingers to the notes of the scales they used (figure 13).

Fig. 13

But note that to consider these vibrational relations between natural numbers in a string is the same thing as considering the harmonics of a sound. Taking a look at figures 12 and 13 and applying simple mathematical concepts it is seen that the relation 2/1 is the same as the relation 2f/1f, the 3/2 that the 3f/2f, the 4/3 that the 4f/3f, etc. That is to say, for example, the frequency corresponding to an upper fifth of a fundamental note is obtained by multiplying its frequency by 3/2 (the interval between 2f and 3f ), the fourth by 4/3 (the interval between 3f and 4f ), etc.

Pythagoras and Plato⁴ believed that all intervallic relations in music (and in other

³ Foster attributes it to Marin Mersenne (1588-1648) (Foster 2010) although Rameau did not speak about it until 1750 (Rameau 1750).

⁴ In the Timaeus (Dialogs) (35b-36b) Plato relates the creation of the "soul of the world" according to the relations of numbers 2 and 3 (he establishes the Pythagorean semitone [limma] with total accuracy [256/243]).

natural principles) came from the relationship between numbers 2 and 3. That is, they had enough with the octave and the fifth —the first two harmonics— to build all the musical scales.

Years later Aristoxenus doubted about this and introduced, among others, the 5/4 relationship (harmonically the interval between 4f and 5f) within complex divisions of the tetrachord, although he said that the true musical scales were not mathematical but "felt".⁵

During the Middle Ages the Pythagorean school was prevailing although in England there was a style of composition that used the natural third (5/4) as a consonance.

In the 16th century Zarlino introduced the senario, composed of numbers 1, 2, 3, 4, 5 and 6 as generators of all intervals and musical laws. In fact, to establish the senario is the same to consider the relations until the number 5 since 6 = 3 x 2. Scientists and musical theorists like Kepler⁶, Descartes or Rameau⁷ agreed to consider only the relations between numbers 2, 3 and 5 as generators of musical scales and intervals and in fact this relation is the one that is still explained nowadays in many treatises of harmony.

Some theorists also introduce or share the ratio 7/4 (4f-7f interval) as the minor seventh, although this interval is more accepted as two fourths (4/3 x 4/3) from the fundamental (or two lower fifths). This is the case of Leibniz⁸ , Tartini⁹ , Euler¹⁰, Kirnberger and Vogel¹¹. Even the father of functional harmony, Hugo Riemann, although he favored the senario ratios for tonal relations, admitted that the interval of the minor seventh was an interval "given directly by nature" (Elementar-Musiklehre, Hamburg, 1883).

⁵ See: The measurement of Aristoxenus's Division of the Tetrachord, by Joe Monzo, http://www.tonalsoft.com/monzo/aristoxenus/aristoxenus.aspx.

⁶ See Cohen, 1984.

⁷ In the seventeenth century Descartes, in the Compendium musicae (1618) writes: "All the variety of sounds, relative to the treble and the bass, is born in the music only of these numbers: 2, 3 and 5". In his book Demonstration du principe de l'harmonie (1750) Rameau explains the C diatonic scale taking the first 5 harmonics of F, C, and G.

⁸ In a letter to Christian Goldbach (Luppi, 1989) Leibniz writes: "Nam our Intervalle vstinata omnia sunt rationum compositarum ex rationibus inter binos ex numeris primitivis 1,2,3,5. Si paulo plus nobis subtitiltatis daretur, possemus procedere ad numerum primitinum 7").

⁹ Tartini (1754 and 1767).

¹⁰ Partch (1974).

¹¹ Vogel (1975).

In fact, in the daily practice of music, with our equal-tempered system these ratios never occur in an exact way, but studies like those of Fransson, Sundberg & Tjernlund (1974) seem to show that considerable variations can occur in the tuning of a piece of music without it meaning that the notion of tonal and harmonic structure determined by these numerical relations is lost.

Thus, throughout history, musical theorists, in order to demonstrate musical laws, considered mainly the relations between numbers 2 and 3; also many of them incorporated the 5, and some few the 7 (that is, the relations corresponding to the harmonics 2, 3, 5 and 7: C, G, E, B¬).

Note that in the harmonic series, as we move forward, the notes are getting closer and closer together. Thus, for example, between harmonics 1 and 4 there have two octaves, but instead, between harmonics 24 and 30 we have 7 notes within a third, that is, approximately a quarter tone of difference between them; between harmonics 32 and 36 we have only one tone of difference, and so on. They are approaching logarithmically.

When sounds are very close in frequency, it operates what in acoustics is known as "auditory masking": when a set of sounds sound simultaneously at very close frequencies, the auditory system is unable to appreciate all the frequencies of the sounds; and the most potent ones completely nullify the perception of the weakest, making themselves unintelligible ¹²

This is what begins to happen from the harmonic 16, and from the 24 the sounds are already masked and indistinguishable between them.

Our thesis is that we can consider a sufficiently high number of the harmonic series (for example, up to harmonic 24) without this being an obstacle so that we can determine that the harmonics really significant for the auditory system can be reduced to the prime ones within the first seven (2f, 3f, 5f and 7f).

The reason is that harmonics that are multiples of a prime harmonic have less importance to the auditory system than those who occupy prime position, since they are heard and understood as generated by them. Just as the fundamental is the synthesis of all its harmonics, the prime harmonics are the synthesis of the rest of

¹² Calvo-Manzano, 1991, and Moore, 1986: «The ability to hear frecuencies separately is known as frequency resolution or frequency selectivity. When signals are perceived as a combination tone, they are said to reside in the same critical bandwidth. This effect is thought to occur due to the filtering within the cochlea, the hearing organ in the inner ear. A complex sound is split into different frequency components and these components cause a peak in the pattern of vibration at a specific place on the cilia inside the basilar membrane within the cochlea. These componentes are then coded independently on the auditory nerve which transmits sound information to the brain. This individual coding only occurs if the frequency componentes are different enough in frequency, otherwise they are in the same critical band and are coded at the same place and are perceived as one sound instead of two».

non prime harmonics (their own harmonics). In the harmonic structure we could consider two levels of auditory perception (see figure 14):

-The fundamental as the perception of all the (prime) harmonics.

-The prime harmonics as perception of all the multiple harmonics.

Fig. 14

What really gives importance to each prime harmonic is the sum of its audible multiple harmonics, not masked and distinguishable by the auditory system.

Each prime harmonic p has its own harmonic series 2p, 3p, 4p, 5p, 6p ... (8th, 5th, 8th, major 3rd, 5th ...)

Within the first 24 harmonics we see that G (3f) has 8 harmonics, E (5f) has 4 and B (7f) has 3. They are the most important for the auditory system.

The next prime harmonic, the 11f, does not intervene at all in establishing harmonic relationships that mankind has used to make music (except for certain cases in electroacoustic music). The proof is that it falls almost exactly in the middle between F and F{ (on a equal-tempered piano, only one cent [one hundredth part of

semitone] in favor of F{). Something similar happens to the next prime harmonic, the 13f, which is placed between A¬ and A} (9 cents in favor of A¬).

It is not surprising that the situation of these two prime harmonics, equidistant between two notes of the equal-tempered scale, has produced and produces confusion. Thus, for example, in the harmony books of Schoenberg (1922) and Piston (1941) the harmonic 11 appears as F} instead of F{. In the treatise on harmony of Schenker (1906) and in the compositions of Albrecht and Scriabin based on the harmonic series appears the harmonic 13 as A} instead of A¬. In the treatises of Helmholtz (1863) and Piston both F} and A} appears, although Helmholtz already warns that they cannot properly be considered as musical notes. And this confusion with harmonics 11 and 13 continues today if one takes a look at the correspondences between notes and harmonics that can be seen on the internet. Prime harmonics 11 and 13 have had no influence on the establishment of musical laws, nor does it make any sense to assign them a musical note belonging to any of the known tuning systems.

This way so, in agreement with the history of musical theory from the Hellenic culture and from what we have seen on the prime harmonics, it is sufficiently demonstrated that, of all the harmonics of a sound, the only ones that have had a functional meaning for the the human auditory system have been 2f (C), 3f (G), 5f (E) and, very slightly, 7f (B¬).

The results shown in figure 14 reflect the history of harmonic theory in regarding the first five to seven harmonics to be those that are functionally perceived by the auditory system.

It remains to be seen whether in the future, perhaps by means of musical robots, the use of the following prime harmonics 11 and 13 through a coherent creative construction and microtonally tuned interpretation will be able to produce in our brain a new level of harmonic perception. Although I am rather skeptical in this regard. I have tested how a small composition might sound by using these harmonics (see https://www.youtube.com/watch?v=aHb2hovf95o).

2.2 Virtual fundamental
Before continuing to study the relationship of harmonics with musical harmony we will explain a psychoacoustic phenomenon of great importance in music.

It is the virtual or missing fundamental (known as `virtual pitch' or `missing fundamental' in acoustics): a frequency that does not exist in a sound, but that our ear tells us that it is precisely the frequency, the height, the pitch we hear. This occurs if we filter (we remove) the first harmonic(s) of a sound. In this case we continue hearing the fundamental frequency.

To prove it I have done a video taking the harmonics of C. In this video we may see that if we delete the frequency corresponding to C₂ and even the first seven harmonics, we still hear the note C₂ (!). If we take only the harmonics corresponding to the notes E₄, G₄ and B¬₄, we continue to hear C. The reason is that we eliminate the first harmonics, but not their multiple harmonics, that is, the harmonic structure remains the same, that of figure 14. This video can be seen in: https://www.youtube.com/watch?v=0Y4-NQQ6hAY

To see a mathematical proof of this phenomenon see also:
https://www.youtube.com/watch?v=p3iWLkXAePM6
If instead of sinusoidal sounds we use timbred notes, for example on the piano, the effect is greatly diluted, but it still works (!). For example, let us go to the piano and play the note C₂ and immediately the chord of figure 15 several times. We will hear that when playing the chord, if we make the effort of remembering the C₂ that we have played before, this note magically appears as if it were actually sounding, it is psychologically perceptible (!) but does not actually exist in the chord.

Fig. 15

This phenomenon —in the history of musical theory (although surely ignoring its scientific basis)— has also taken part when constructing various harmonic laws; from concepts such as the basse fondamentale by Rameau, the terzo suono of Tartini (Tartini regards B¬ as the virtual fundamental of the diminished fifth D-A¬)¹³ or the consideration of the diminished triad chord EGB¬ (or, as Tartini, simply the diminished fifth: EB¬)¹⁴ as a dominant chord, usable to make cadences, that is, a chord with C as virtual fundamental (C as dominant of tonic F). Fact formally consolidated by Riemann by coding the diminished triad chord with the symbol \Dµ (D of Dominant).

Although the missing fundamental phenomenon was already known in acoustics, Ernst Terhardt introduced the term virtual pitch in 1974 and exemplified it with the visual analogy shown in figure 15bis. In visual perception we see the white square,

¹³ Tartini, 1767, p. 85

¹⁴ According to Tartini, with the B¬somewhat low to fit with the «natural» 7th of C. Tartini could do this because he was a violinist.

although it does not really exist; or we see the relief of the letters by putting only a few lines. The virtual fundamental would be an equivalent effect —acoustically speaking— to the "finish" rendered by the brain of the contours of figure 15bis.

Fig. 15bis

This acoustic phenomenon will help us in knowing the true fundamentals of complex chords and confirms us that the «true» fundamental (virtual) of the EGB¬ chord is C, it is not E.

In the following sections we will explain, one by one, the effect that, in our opinion, have the first four prime harmonics in the human auditory system that have allowed to develop many of the musical laws. That is, we will study separately the prime harmonics 2f, 3f, 5f and 7f.

2.3 Harmonic 2f
It is the octave of the fundamental.

If we look at figure 14 and if we consider the harmonic 2f as a fundamental sound, we see that the harmonics of 2f coincide in 50% with the harmonics of the fundamental sound f. That is, the harmonic 2f has half the harmonics of f (in any given neighbourhood).

From what we have explained previously about the virtual fundamental —we have seen that we can eliminate harmonics of f without this affecting too much the feeling of being still listening f— we could say that the auditory system considers f and 2f as the "same" note but with a different "timbre", since they "almost" have the same harmonics.

The great amount of harmonics in common between a sound and its octave and the acoustic effect of the virtual fundamental have made so that in music (in the human brain) these sounds have been considered, indeed, the same note. This fact is very useful both for the brain and for music theory because it has allowed to simplify information drastically, reducing the whole range of notes on a diatonic scale of only 7 notes (and their modes) (see in 2.6 the construction of the diatonic scale) and in a chromatic one of only 12 (once the equal-temperament is applied).

That is to say, the strong and powerful perception by the auditory system of harmonic 2f with its common harmonics associated with f has allowed to identify the sounds and their octaves as the "same" note and speak of notes and chords without continually specifying the actual pitch of them. However, it is the "same" note in quotation marks. We all know that, in musical practice, it is not the same to use a note or its lower or upper octave, especially if this changes the inversion of a chord, although "functionally" or "harmonically" it plays the same role.

Harmonic 2f is also the "culprit" that, functionally, the chords can be identified with their inversions.

In figure 16 we have a chord (it might be anyone) with its inversions. If we add the octaves to the chord notes (which really can be heard as harmonics), we see that the fundamental triadic position of the chord is repeated in the upper layers. The ratios between intervals remain the same and by means of the virtual fundamental effect the ear functionally matches the chords assigning to them the same fundamentals. However, sonance (consonance-dissonance) can be quite, if not very, different. But, as we said in the previous chapter, we should be able to distinguish the "sonance of the chord" from its harmonic function.

Fig. 16

We must do a parenthesis and comment on the particular case of the 2nd inversion, with a sixth and a fourth from the bass: if we consider the following significant harmonics after the octave, which are the fifth and M3 and the trend of the notes in solving dissonances towards notes to a lower second, we see that the ear can also understand that the bass is the real fundamental (we hear its fifth and its M3 as harmonics) and can understand the 6th and 4th as notes close to the 5th and 3rd, notes that "collide" with the latter (and need a resolution) and consider them as appoggiaturas that must resolve a second lower —with increased effect if the bass is an important note in the tonality—. This is the case of the known cadential æ. The

second inversion of a major or minor triad can then be interpreted by the ear in two

different ways, or both at once, since we can understand the cadential æ as T(æ)-D-T or Dæ-£-T.

Thus, harmonic 2f is responsible for identifying the octaves as a `same' note and for the functional equivalence between the inversions of the chords.

2.4 Harmonic 3f
It is the fifth (+ octave) of the fundamental.

If we look again at figure 14 and if we consider the harmonic 3f as a fundamental sound, we see that the harmonics of 3f match on a 33% with the harmonics of the fundamental sound f. That is to say, the harmonic 3f has 1/3 of the harmonics of f (in any neighbourhood).

The amount of harmonics that we should remove to the C to become G would be too many and the auditory system can no longer consider the G as a "timbre variation" of C, as it did with the harmonic 2f, its octave. But note that the harmonics of G (3f) fit perfectly with those of C (f&2f). For this reason, if the two notes sound simultaneously, we say that they form a perfect consonance.

If instead of being a 13th (5th + 8th) is a perfect fifth, the harmonics do not fit so perfectly, but the difference is only of an 8th in the first harmonics. The notes corresponding to the harmonics are the same. Something similar happens if we lower a further 8th and get the 4th, which is the inversion of the interval of 5th. We are losing consonance, but the upper harmonics continue to fit.

In fact the fifth (P5) is the only perfect consonance between two different notes. The consonance par excellence.

If we hear an interval near the fifth such as the minor sixth or the diminished fifth, the ear hears it as a "quasi-fifth" or "false-fifth", thus creating a harmonic tension that is resolved if the "quasi-fifth" is "tuned" to a perfect fifth (in the case of m6/M3 the tension is lesser because it contrasts, as we will see, with the imperfect consonance justified by the harmonic 5f). See figure 17 (and also figures 1, 2 and 3 of the first chapter).

Fig. 17

These simple harmonic tension-relaxions of figure 17 (produced by the intervals close to P5 and their resolutions), although it may be hard to believe, are the basis of most harmonic tensions of tonal music as we have seen in the previous chapter.

The fifth, as we shall see in 2.6, also intervenes in the construction of the diatonic scale in the major mode.

This way so, we could say that the harmonic 3f is the principal causer and generator of the functional laws (cadences, tonality, secondary dominants, etc.) of harmony. The re-reading of the previous chapter and the reading of the rest of the book will reinforce this fact, which might be summarized in the importance that the tension of the "quasi-fifths" and the tendency of the notes towards their lower fifth have for harmonic laws.

2.5 Harmonic 5f
It is the major third (+ two octaves) of the fundamental.

If we look again at figure 14 and if we consider the harmonic 5f as a fundamental sound, we see that the harmonics of 5f coincide on a 20% with the harmonics of the fundamental sound f. That is, the harmonic 5f has 1/5 of the harmonics of f (in any neighbourhood).

It is a weak linkage with the fundamental, but still perfectly perceptible by the auditory system.

This setting helps the M3 interval/chord to be relatively stable despite having the upper harmonics forming minor sixths, an interval close to the attractive fifth, stability that is definitely reinforced if we we confirm the fundamental adding also its fifth (that is, forming the major triad chord); we still reinforce it more if we have instruments of variable tuning or voices that drop a bit this M3 (in the resolutive chords) to make it correspond with the natural tuning of the harmonic 5f, which is a bit lower than the tempered M3 of keyboard instruments (see figure 12).

Before the 15th century the M3 interval was considered dissonant —or imperfect consonance with tension to be solved¹⁵ —because it was usual the Pythagorean tuning and vertically (heard as chord) the Pythagorean ditone is really dissonant. Keep in mind that between the ditone and the natural 3rd (5f) there is almost a quarter tone difference (22 cents). In fact in classical music of India they are considered (or were considered) two different intervals. The 5f as a shruti named Raktikâ, which expresses pleasure, sensuality..., and the Pythagorean 3rd a shruti named Raudrî, which has an antonym meaning: warrior, terrible (Danielou 1943 and 1967).

The harmonic 5f is the causer that the m6/M3 "dissonance" —due to its proximity to the P5/P4— is accepted by the ear as semi-consonance, something that does not

¹⁵ The regola delle terze e seste said that these imperfect consonances had to resolve its tension with a perfect consonance (Dalhaus, 1990) and Gafurius, in Practica musicae (1496), describes the thirds and sixths as irrational consonances (Tenney [1936]).

happen to the tritone that is also separated from the perfect consonance by half tone of difference.

The harmonic 5f establishes the imperfect consonance of the M3 (and by bounce that of the m3, since it is the interval that is formed between 5f and 3f). That is to say, it is a harmonic that establishes some laws of music sonance, but does not affect to the functional laws.

Broadly speaking, we could say that the harmonic 3f (the P5) is the causer of the functional laws of harmony and the harmonic 5f (the M3) to extend the perceptions of sonance.

Before continuing with the harmonic 7f, with these two harmonics we have studied (3f and 5f) we can already demonstrate how the diatonic scales have been formed from the perception of these harmonics.

2.6 Formation of music scales
The (major) diatonic scale can be formed in a very simple way considering only the harmonics 3f and 5f.

In fact, only with the 3f we can form the diatonic scale, but with Pythagorean tuning.

For example, let us take F and let us construct a scale formed by the 3f harmonics of the harmonics 3f, that is, the fifths of the fifths.

We will obtain F, C, G, D, A, E, B, that is, C-D-E-F-G-A-B.

There is another way to build this scale, adding the use of the 5f:

We take the harmonics 3f and 5f from the first two fifths (3f) of F (figure 18).

Fig. 18

We have not made a discovery. This second construction is the same that Rameau gave in his book Démonstration du principe d'harmonie (1750).¹⁶

¹⁶ In the summary of the Academie Royale des Sciences at the end of the book: «Ainsi l'Auteur exprime fa, ut, sol par les nombres 1, 3, 9, & la proportion qu'ils forment, est ce que M. Rameau appelle Basse fondamentale d'ut en proportion triple, ou simplement Basse fondamentale. Les trois sons qui forment cette Basse, & les harmoniques de chacun de ces trois sons, composent ce qu'on appelle le Mode major d'ut».

Which of the two constructions from the harmonics is the one that the auditory system really considers when listening to this scale? Most probably, both at the same time (the Pythagorean and the so-called «natural») although the tuning is not the same; in fact equal-tempered system comes to make an intermediate tuning of the two (though closer to the Pythagorean).

The reader might have noticed that the two constructions start from the tone F. So, if F generates the seven notes of the scale, would not it be more logical to put F at the beginning instead of C and become the scale of F rather than C?

The answer is that these seven notes create a tonal vector towards C and not to F and therefore they rest in C. Why do they rest in C and not in F? Due to the laws of "quasi-fifths" we have seen in 1.1 and 2.4, that is, the tensions that the intervals of M3 (GB) and tritone (BF) create and which form the 7M3 structure that we have seen in 1.4 (in the case of C major, the structure formed by notes GBF). A 7M3 structure creates a powerful tonal vector. The major mode has a unique 7M3 structure that determines which of the seven notes is the "tonic", the note that provides more rest.
Briefly explained, before the 16th century, and more concretely before the formal change of modal theory in Glareanus' Dodecachordon (1547), the theoretical modes, in order to avoid tritone, were based on the hexachord (cantus naturalis) of Guido de Arezzo (11th century), that is, the diatonic scale without B: Do (Ut), Re, Mi, Fa, Sol, La. The hexachord does not have the 7M3 structure and the resting note is not so clear and depends on the melodic drawing. Therefore, in those times, music was easily modal because, apart from being basically horizontal, many notes could be the finalis note. According to the theory of the time, these final notes could be Re (D), Mi (E), Fa (F) or Sol (G). Although the modal theory is more complex since —in a same chant— hexachords could also be used from the notes F (using B¬) and G (as a kind of transport or modal modulation). If the notes F and B} were used in the same melody, they were notes that were not close (in time); if they appeared, between them it was customary to put the finalis note, but, for example, F-G-A-B} was almost never used, in these cases B¬ was used.

When the palette of notes gradually passed from hexachord to heptachord (the diatonic scale), the Do (C) and La (A) notes were added as finalis notes (Glareanus, 1547), which ended up being the predominant modes because then it is easier to incorporate the tritone; and the heptachord already has the 7M3 structure (GBF) that places tone C as finalis, as the most "acoustically" resolutive note. And the eolian mode (finalis A) was becoming our current minor mode when, little by little, the leading-tone G{ was joining into the mode, since, in this way, the 7M3 structure (EG{D) were included in it.

2.7 Minor chord and minor mode

Is there an explanation of the minor chord from the harmonics? There have been (and there are) many discussions about it. For me, the explanation of the consonance of the minor chord is quite simple: the major chord and the minor chord are the two single chords formed by the intervals that are created with the notes corresponding to the harmonics 2f, 3f and 5f (and their octaves). If we take a look at figure 14, we will see that the intervals created between the notes are: 5th (CG), 4th (GC), major 3rd (CE), minor 6th (EC), minor 3rd (EG) and major 6th (GE). There are only two chords that, with their octaves, have ALL and ONLY these intervals (figure 19): they are the major and minor triads. In fact, as "interval classes", they are only combinations of P5, M3 and m3.

Fig. 19

This is why these two chords are the most consonant of all those who can be formed with three different notes.

Next a question would come with which music theorists do not seem to agree on the answer: what is the fundamental of the minor chord? (fundamental in the sense of functional representative of the chord).

Conventional musical theory often fails to make an adequate distinction between two very different concepts —(con)sonance of a chord and a chord's function as a generator of harmonic tension. (Con)Sonance refers to the greater or lesser sensation of smoothness or roughness we experience on hearing the chord, while the function of a chord refers to the harmonic tension which that chord creates.

For example, the two chords shown in figure 20 share the same harmonic function as they both create a tonal vector towards C. Their degree of sonance, however, is totally different. In this example the root is the same whether we define it as the bass of the "root state" (ordering the notes in thirds), or as the representative of the function of the chord. But the bass of a chord ordered in thirds is not always the same as the root defined representative of the function of the chord. This is true in the case of minor chords.

Fig. 20

The chord ACE probably has its greatest consonance when A is the bass note (and is the bass ordering the notes in thirds). For conventional theory A is therefore the root of the chord. But students of harmony soon learn that they cannot use this chord as the dominant of D and so A, which is the dominant of D, cannot represent the function of the chord. For functional harmony A is not the fundamental of the chord. This confusion concerning the root of the minor chord is due to the fact that we are talking about two different ideas —on one hand the (con)sonance of a chord and on the other its function.

In the 18th century Rameau had already defined the chord ACEG as having a double function with A or C as the "basse fondamentale", depending on the harmonic progression or the inversion (A as a "sixte ajoutée"). When Riemann classified the functions of all chords into just three categories, (subdominant, tonic and dominant)¹⁷ he made the D minor chord in the key of C major into a subdominant (he anotated it Sp) (fundamental F); the chord of E minor became a dominant (Dp) (fundamental G); the A minor chord became a tonic (Tp) (fundamental C), and the diminished chord BDF was a dominant with G as its virtual fundamental. Riemann was in agreement with Tartini, many years before this concept was defined acoustically by Terhardt.

All this tends to indicate that from the point of view of functional harmony, the fundamental of chord ACE is C.

In this way musical theory seems to give A as bass to underpin the most consonant chords and C to represent the function of the A minor chord.

This is in agreement with our separation of chords (scientists would use the word chromatography) according to their convergent structures that we explain in 1.7 and 3.1.

The ACE chord can be separated in its P5 (AE) (3f of A) and its M3 (CE) (5f of C) (figure 21).

The interval that creates tension in the chord is that of M3 (CE), which tends to resolve or continue locally relaxed with chords containing the fundamental F or B.

Fig. 21

¹⁷ Rameau had already said something similar in Génération harmonique (1737): «Il n'y a que trois Sons fondamentaux, la Tonique, sa Dominante, qui est sa Quinte au-dessus, & sa Sous-dominnate, qui est sa Quinte au-dessous, ou simplement sa Quarte».

Concerning the minor scale(s), as we know, in music practice, three different types of them coexist, the so-called natural, harmonic and melodic ones, which vary according to different combinations of VI and VII degrees (at the end of 2.6 we have already seen how the minor mode was formed from the eolian mode). In any case, it is possible to show, as with the major chord, its internal harmonic coherence:

Let's choose F again. Let us take the three upper fifths and the three lower fifths: F: C, G, D and A¬, E¬, B¬: F, that is, C-D-E¬-F-G-A¬-B¬, we obtain the natural minor scale.

As with the major diatonic scale, we can also construct it in another way: let us take the three minor chords formed on the first two fifths of F (FA¬C) (CE¬G) (GB¬D), we return to have the natural minor scale.

However, when chords are used, the natural minor mode has a problem (problem not so obvious when it is only melody or horizontal polyphony). This mode has the tonal tension of the 7M3 structure of another tone, in case of C natural minor would be the structure B¬DA¬ that causes that the rest note become E¬ instead of C (remember that E¬ major and C minor share the same key signature). So what composers do is try to change the 7M3 structure, at least in cadential processes. They change the structure B¬DA¬ by the GBF structure, that is, they simply use B} instead of B¬. The scale thus formed is precisely the C harmonic minor scale.

In order for the harmonic minor scale not to sound so "exotic" with the appeared augmented 2nd (A¬-B}), often for ascending melodies A} is used instead of A¬ (G-A}-B}-C) and for the descending ones B¬ is used to take advantage of the melodic phrygian resolution towards the dominant (C-B¬-A¬-G), the so-called basso di lamento (which coincides with the Greek Dorian tetrachord¹⁸ ). In this way we obtain the so-called melodic minor scale, consisting of an ascending htonal resolution (GB)|C and a descending phrygian resolution (A¬C)|G (figure 22).

Fig. 22

¹⁸ Plato considered the Dorian [diatonic] mode (the medieval Phrygian) as a model of order for the Republic (Levarie, 1992).

2.8 Harmonic 7f

It is a sound very close to the minor 7th as seventh degree of the diatonic scale or very close, in other words, of the tempered minor 7th of the piano.

As we see in figure 14 the weight of this prime harmonic is quite small due to the few and distant harmonics they share in common with the fundamental.

In 2.1 we have seen that some theorists, such as Leibniz, Tartini, Euler, Kirnberger and Vogel, considered this sound (at least) as Riemann considered it, that is, an interval «given directly by nature» although its natural tuning, due to the current use of the equal-tempered tuning is rarely put into practice.

The "blue notes" used in jazz surely have their origin in the perception of this seventh harmonic. For example, in the blues it softens the clash that occurs in a chord between the tempered major third of the tonic/root (E}) (piano or guitar) and the minor seventh of the subdominant F (E¬) (usually in the voice or instruments with variable tuning) as the distance between the E} and the E¬ (blue note [natural minor 7^th] in this case) is widened (50 to 81 cents) and the clash is more "sweet" when the E¬ sounds above the Cµ (E¬ sounds a bit lower than the temperate one and softens the dissonance E}-E¬). Also the barbershop quartets usually sing the harmonic seventh in 7^th chords.

I have done a test (https://www.youtube.com/watch?v=FU2ov_r9Li0) which consists in seeing the tuning difference of a dominant seventh chord according to the equal-tempered tuning and according to the harmonic tuning following the frequencies that correspond to G, E and B¬ according to the harmonics 3f, 5f and 7f (disregarding octaves), using sampled voices. In this example, we hear, alternately, the equal-tempered chord and the "harmonic" chord. First we hear the tuning difference of the two B¬, which is clear (the harmonic B¬ sounds lower than the tempered one). But when we hear the whole chords, this difference of tuning decreases and it is not clear which is the chord that sounds more "tuned". Perhaps our habit of listening to the equal-tempered system makes the tempered one, in the first instance, that seems more "tuned", but the second has a "harmonic sweetness" that does not have the first.
Note that although the harmonic 7f is weak, considering the first 4 prime harmonics of a sound (2f, 3f, 5f, 7f, which are the only really significant ones for the auditory system) we are hearing a chord very similar to that of dominant seventh of the fundamental, which helps that, hearing only a single note, we have the sense of resolution upon hearing a note a 5th lower or a minor 2nd lower (and so on) (htonal and phrygian resolutions).

The 7f harmonic will also help us to find the true fundamentals of chords from which we will have guidance on the real tensions and local relaxions between chords

(see 1.6, 1.7 and the next chapter) as it allows us to summarize the 7M3 structure (which coincides with the significant harmonics of a fundamental note) with a single symbol. For example, in the case of GB(D)F, using G⁷.

But, as in the case of the harmonic 5f, it has not influenced the construction of the relations and functional laws of harmony and tonality.

Chapter 3