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5. The Constituent Level: Basic and Emergent Dimensions

Let’s start with the primary elements and look at the screws and planks in the furniture. The three basic dimensions are as mentioned Pitch (Frequency), Dynamic (Amplitude) and Time. None of them can exist on their own, but they are independent dimensions when a musical event is maximally reduced. If we take Time to be just enough to produce an audible event, a “now,” this event will always have a dynamic (amplitude), but its vibration frequencies will either be oriented around a singular pitch or pitches, or not. Sounds that are not, are sounds we hear in the world all the time whenever objects collide with an amount of force within proximity of our hearing. Even speech is not based around singular pitches, rather it goes up and down in a highly refined and intricate manner. Song on the other hand is produced when we keep only one pitch at a time when pronouncing words. We also use other things to make sounds based around singular pitches with instruments: the plucking, hitting or stroking of strings, or the blowing of air cylinders.

Pitch

The sounding of a Note of one pitch is actually a very complex composite of a Fundamental or Root frequency and its Overtones derived from the Harmonic series. To understand this, we need to look at the actual sound sources. With technology it’s possible to produce a sine wave that contains no harmonics, and which has a very clear, almost clinical tone. That is the equivalent of a string vibrating with only the root frequency of the whole string. But mathematically a string can vibrate in many more ways, starting with the division into two halves around the middle point, which each vibrates up and down, and continuing with a division of the string in thirds, fourths, fifths etc.

These different ways of vibrating are the Overtones or Harmonic fractals, and when a string is hit it in a certain way (which musical instruments are built around), the string vibrates in all different ways at once. However, different instruments have different composites and balance of overtones. This is called Timbre, and it’s why two different instruments playing the same pitch still sound different. For example strings and reed instruments have a “richer” sound comprising more overtones, and flutes and brass instead have a “purer” sound with less. (The attack of the notes is also a signifier for different instruments; it’s harder to identify them if it is removed).

When we talk about a Note, we actually mean a collection of frequencies with a clearly discernable root frequency. Human hearing can perceive frequencies in the range of around 20 – 20.000 Hz, with the upper end of this range declining significantly with age. If we add the discussion of overtone series in this, it means that we hear much less overtones for notes in the high register. It doesn’t matter as much which instrument plays a C in the 7th octave; it will sound very direct anyway. And correspondingly we hear a lot of overtones in the low register. The whole system of pitch, frequencies and Hertz are mapped on a logarithmic scale, due to the nature of the underlying mathematics or physics. It makes it harder to grasp because we are not made to think logarithmically. The piano with its big range of register takes all notes and arranges them in row that looks linear, but the frequencies changes logarithmically.

Dynamic

All notes, regardless of timbral quality, comes with a dynamic that is the amplitude of the frequencies in question. This dimension is simpler than pitch: it goes from soft to loud, however also in a logarithmic scale. In music we use basically six or eight discrete degrees ranging from pianissimo or ppp to fortissimo or fff. But to make things slightly more complicated our ears are equipped with a kind of natural compressor that attunes our perception to the overall sound level. If it’s really quiet around us, we are able to hear soft sounds. If it’s louder, our hearing contract so the loudness doesn’t damage our ears, and we are no longer able to hear softer sounds. This makes the dynamic dimension somewhat relative.

Time

Now, let’s consider the Time dimension in itself fully. Technically, pitch frequencies are only a measure of vibrations per time unit (Hertz = vibrations per second) and could be further reduced to that. However, pitch is an emergent property from these vibrations because we cannot perceive time in as short instances as fractions of a second. Pitch is what we hear when a recurring event reaches the lower end of our frequency perception range – if something sounds 20 times per second we cannot count those 20 times, but rather we hear a very low note. So in this regard, when we talk about time in music, we mean it on the level of human perception of time, like what we can understand as something occurring before and after something else. Pitches, although technically only vibrations in time, are heard as an immediate musical event, and how long it is sustained or stopped is what we perceive as musical time.

The time dimension consists of three parts or subdimensions: Rhythm, Tempo and Meter (or beat). Rhythm is the relationship between two notes and can be thought of as a vector scalable to any Tempo, which is connecting it to real time. Meter is an emergent subdimension out of several rhythms in a tempo: a super-rhythm of regular (or irregular) beats emerges in a meter after several rhythms has been sounded. There is also a secondary degree of the Tempo dimension, Tempo squared if you will, which gives rise to the secondary degree features of Accelerando and Ritardando, gradual increase and decrease in tempo. Finally, the special temporal feature of Rubato is changing the tempo back and forth within rhythms.

Pitches in Time

When we add time to the equation, pitch and dynamic can vary and progress separately. If the pitch changes in discrete steps, we get Melody, that can be said to be an emergent dimension. If it changes gradually, we get Glissando (and if we have so much glissando that we lose track of any singular pitch, we start to leave the domain of music). If it changes back and forth a small amount, we get Vibrato. If the dynamic changes, we get a dynamic progression either discretely more or less Subito, or gradually Crescendo and Diminuendo. On a closer inspection of notes in time, we also find the area of Articulation. A note typically consists of two parts: an attack and the sustained note. Staccato gives primacy to the attack and leaves out any sustain in a short note. Tenuto has some sustain but specifies some rests of silence between several consecutive notes, whereas Legato specifies no rests of silence between them. Portato is a special case of a shorter amount of silence between notes. Accents either specifies a loud attack followed by a soft body or treats the whole note as louder, depending on the context.

Intervals

Interesting things start to happen when we combine several pitches simultaneously. Eventually we will end up with a system of harmony, but let’s start by combining only two pitches together. When taken together they form an Interval, which is more or less Consonant or Dissonant depending on the mathematical relations between their frequencies. We perceive intervals with close mathematical relations as more consonant, and the Octave is the most consonant with its perfect ratio of 2:1. The Fifth comes second with the ratio 3:2 (or 1,5:1), and then the Fourth with 4:3 (or 1,33:1). The fourth can also be seen as an inverted fifth, with the fifth as the root instead. Intervals with such pure mathematical relations are hence called Perfect consonances in the Western system. This also corresponds to the Harmonic series from before because they have some overlap with each other. The overlap of the octave is especially salient because it shares all overtones with the common denominator 2 (harmonics fractals no. 2, 4 and 8). This is why an octave sounds like it’s the same note only higher or lower, and this identity provides the grounds for all musical systems in the world. In our Western system we call pitches an octave apart the same letter note name, and the actual frequency is often specified only by context. After the perfect consonances comes the Imperfect consonances of major and minor thirds and sixths (with ratios 5:4, 6:5, 5:3 and 8:5), then Dissonances of seconds, seventh and the tritone.

The Chromatic scale

The system that governs all Western music is firstly the division of the octave into twelve Chromatic notes. The reason for this is that we need notes that have the ability to resonate with each other as fifths and fourths, as well as an equal distance between them in order to move freely within the system (i.e. the ability of transposition). Now, how many equal notes can we divide the octave in while also preserving intervals that most resemble the perfect fifth and fourth? We can investigate this mathematically by arranging perfect fifths on top of each other and see if they return to the same note as the root (given octave equivalence, which we have established). After twelve fifths we reach a note that is very close to the root seven octaves above it. It is not exactly the same however, and it turns out that we can never reach the exact same frequency (again with octave equivalence) because mathematically the equation

2^x = (3/2)^y

is never true for whole numbers. The higher the numbers the closer we come, and there is another point at 53 fifths where the fifths and the octaves “meet” up even closer and thus provides more fine-grained, even distances between all notes. However, having 53 notes within one octave is not practical for all sorts of reasons, not least that our hearing cannot differentiate between two consecutive notes in such a system. Twelve notes on the other hand are very practical and easy to distinguish between. The difference where twelve fifths and seven octaves meet is called a “Pythagorean comma” and is equalized among the twelve notes in “Equal temperament” tuning (“12-TET”), so that each interval is less “perfect” than it would be if the ratio were reached by mathematical relationships. It is a concession of accepting less consonance of intervals, but it is outweighed by the potential of the system. And for instruments or voices that doesn’t have fixed pitch values like the piano, they can choose to tune certain intervals either in equal temperament (if together with such fixed instruments) or to deviate momentarily from the system and tune intervals to their mathematically “pure” relationship.

Because of the fact that we used fifths to reach the number twelve, we know that every note in the system has a fifth both above and below it. In contrast, if we try to divide the octave into mathematically equal notes of another number, we will not get this property. A scale of ten or fourteen equal notes would potentially yield better other intervals for certain notes, but not fifths for all the notes.

The Diatonic scale

Twelve equal notes are great, but it does not have any stability if no notes are privileged over others (for example “12-tone” music that tries to abolish tonality makes very little musical sense). This is where we need Tonality, and we reach it by taking a subset of notes out of the chromatic scale and giving them a privileged position. This subset is called the Diatonic scale, and prevalence of it and the major/minor system that follows from it is the second governing feature of Western music. How we reach it is somewhat similar to the process of finding the chromatic notes. If we take the first seven of our previously stacked fifths, starting on an F, we get F-C-G-D-A-E-B. If we then rearrange them in the span of one octave starting on C we get the C major scale, or all the white keys on the piano. This is how the “octave” gets its name, it’s the eighth note after those seven.

Again we have the property of perfect fifths and fourths within the scale, but the distances between the notes are not equal, as with the chromatic notes. Between them we have five whole-note steps and two half-note steps in chromatic terms. Mathematically, this configuration of seven notes is the one where those intervals are maximally separated; any other collection of seven notes will have different intervals at more uneven positions. But still, because of the unevenness we get different scales when we transpose by changing the starting note of the scale. This is what causes the different “Modes” of the diatonic scale. Major is the mode starting on C and Minor is the mode starting on A, with this set of “white keys” notes. The other modes are as follows, with the set of white keys and the starting note:

• C: Major (also Ionian)

• D: Dorian

• E: Phrygian

• F: Lydian

• G: Mixolydian

• A: Natural Minor (also Aeolian)

• B: Locrian

These modes were more common in earlier Western classical music – Medieval, Renaissance and Baroque to some extent – before equal temperament came into full effect. They have different properties that follows from which intervals occurs on which positions in the scale. For example, the Locrian mode is not popular because it is the only one that does not “catch” a perfect fifth on the root, instead B-F is the very dissonant tritone. The prevalence of the major/minor system comes from these properties and especially how triadic harmony fits onto them.

These modes were more common in earlier Western classical music – Medieval, Renaissance and Baroque to some extent – before equal temperament came into full effect. They have different properties that follows from which intervals occurs on which positions in the scale. For example, the Locrian mode is not popular because it is the only one that does not “catch” a perfect fifth on the root, instead B-F is the very dissonant tritone. The prevalence of the major/minor system comes from these properties and especially how triadic harmony fits onto them.

Harmony

Two pitches at the same time creates an interval, as we have seen. Now, if we add a third note to an interval, we form a Triad or a Chord, and we have proper Harmony. If we take three pitches that are not the same, the most consonant triad will be the major chord with a perfect fifth as well as the major third in between. The major third has a consonant relation to the root (it is also the 5th harmonic of the root), and the minor third interval that is created with the fifth is also an imperfect consonance. The second consonant triad is the minor chord with those intervals reversed, so in effect a less consonant interval between the root and third, but more consonant between the third and fifth. From another perspective it turns out that if we want to take three notes within the diatonic scale, the major chord is the way to maximally separate those notes by spreading them out onto the 1-3-5 scale notes in diatonic terms (while still keeping the perfect fifth). If we now take the modes again and apply the pattern of 1-3-5 on each scale note, we will see how we ultimately end up with the major/minor system. Starting on C, the seven triads are then C major, D minor, E minor, F major, G major, A minor and then a special B diminished triad consisting of two minor thirds (and no perfect fifth). How these chords are distributed in the different modes is what gives them their identity. The triads that appear on the three consonant scale notes 1, 4 and 5 are the most important, and hence we get the following results.

• The Major mode emerges as the most consonant with major chords on all three important scale notes 1, 4 and 5

• The Minor mode emerges as the most minor with minor chords on 1, 4 and 5

• The Lydian and Mixolydian modes are more major but also mixed:

  • The Lydian has major chords on 1 & 5; diminished triad on 4

  • The Mixolydian has major chords on 1 & 4; minor chord on 5

• The Dorian and the Phrygian mode are more minor but also mixed:

  • The Dorian has minor chords on 1 & 5; major chord on 4

  • The Phrygian has minor chords on 1 & 4; diminished triad on 5

• The Locrian mode is the most dissonant with the diminished triad on 1; minor chord on 4; major chord on 5

Now, the area of harmony expands very quickly when we start to consider other triad formations than 1-3-5, chords consisting of four notes (Tetrads) or more, chords that uses chromatic notes not in the diatonic scale (grounds for Modulation), chord Extensions (like 7, 9, 11, 13), chord Alterations (like sus4, +5, -5), and endless combinations of all of the above. It’s a field of study on its own and too vast to consider in this framework.

Emergent Dimensions

Technically, if we go back to the constituent dimensions of music harmony can always be reduced to several individual pitches played simultaneously. But there are important properties that are only emergent on the level of harmony, which is why we need to treat Harmony as its own, albeit emergent, dimension. When we consider Harmony in Time, we get to the even higher level of Harmonic progression, and we have finally arrived at the full scope of tonality and Functional harmony, which governs the relations between different harmonic entities over time (somewhat differently depending on genre and style). When we put everything in this chapter together, we have a full picture of voices forming melodic lines in the horizontal direction with time, and harmony in the vertical direction. The configuration of the different patterns of voice here is called Texture, and the relative dynamic among then Balance.

This is also where Notation comes in as a technological innovation in music history that propels it forward. It is only by writing down specific prescriptions for which notes different voices should play or sing, that any proper polyphony or harmony is possible in practice, since two voices need to be in constant agreement for making such music. This happened in Western music in in the Medieval and Renaissance periods; in the Baroque period it had evolved to more advanced Counterpoint and in the Classical period the musical texture got further divided into melody and accompanying harmony. For a good overview of notational features and how they define different music cultures around the world, see Nicholas Cook’s Music – A Very Short Introduction (1998, chapters 4 and 5).

To sum up, when we consider how the three basic dimensions work in relation to each other, there is a plethora of features and new dimensions emerging (with the help of music notation). Harmony is the most important of these, so important that we count it as a basic dimension within the Western system. Other important emergent features are Melody, Rubato, Texture, Balance and finally Harmonic progression. It is along these dimensions that the musical units on the Perceptual level are formed rather than from only the basic dimensions, which we will see in the next chapter.