art math music

When three different notes are sounding simultaneously they create the first instance of a chord. Chords can have many more than just three notes, but I would argue that they may have no less than three. Two notes is afterall an interval, is it not?

An interval can sometimes imply a chord, as when an artist creates the expectation musically of a particular chord but then only gives the two notes that define that chord most characteristically. The artist would be fulfilling most of the expectation of the listener, relying on cultural precedent to do the rest of the work filling in the missing tones.

So, there are lots chord combinations. To count them we have to think about the smallest interval, involving two different notes (the half-step). Then we need to count a half-step on top of another half-step as our first chord. A half-step with a whole-step on top of it would be our second chord. We’re “stacking” intervals here, to create chords. A half-step with a minor-third on top of it would be the fourth chord. But keep in mind that we have to exclude the unison and octave intervals as they are just a repeat of a note. Remember, a chord is defined as three different notes sounding simultaneously. That leaves eleven different intervals that can be stacked on top of eleven intervals, which makes 121 possible two interval chord combinations within each key. Multiplied by 12 keys, the number of possible two interval chord combinations jumps to 1,452. Remember also that chords can be composed of many more than three notes. For instance, if we consider all four-note combinations in the same manner there are 15,972 possibilities. Five note chords are very common. There are 175,692 possible combinations of 5 note chords.

Thank God there are a relatively small number of chords that are much more commonly used than the others. It helps also to name them in some systematic way. They’re much easier to reference in your memory once categorized.

There are four main types of triads. “Triad” is just another name for a three-note chord.

The four types are…

diminished

minor

major

augmented

A diminished chord is made by stacking two minor-third intervals. So diminished chords have the closest/smallest intervals of the four types.

An augmented chord is made of two major-thirds, stacked. It has the largest, or most largely separated, intervals of the four types.

Major and minor are easy to remember simply because a major-chord is a major-third with a minor-third stacked on top, conversely a minor-chord is a minor-third with a major-third stacked on top.

We’re only working with major and minor-thirds, stacking them in every permutation, and systematically naming discrete instances of chords when they appear.

There are only four possibilites for stacking major and minor-thirds.

minor-third, minor-third – diminished triad

minor-third, major-third – minor triad

major-third, minor-third – major triad

major-third, major-third – augmented triad

You could of course apply this process to all of types of intervals: seconds, fourths, fifths, sixths, sevenths. But seconds are too small, and sound very dissonant when played in triads…to most Western listeners. Fourths are pretty good, but a little too far apart. Chords in fourths are useful in certain musical situations, but used in excess fourths leave the music sounding empty. Our ears can discern notes at and above a fourth interval discretely a little too easily. Thirds are the happy medium.

So these triads in thirds are the basis for almost any chord you’ll see in common use.

For instance, an A-major triad is simply a major triad whose root note is an A. Likewise, an C-minor chord would be a minor triad whose root is C. In the triads we’ve been discussing thus far the root has always been the lowest note of the chord. However this will not always be the case, as when the intervals within a chord are inverted.

The root of the chord is the note in the chord against which all the others are compared. In theory, every note of a chord could be considered as the root, but the precedents of our musical culture create a situation where considering one chord tone over another as the root makes much more sense.

The notes comprising a chord are called “chord tones.” It is also common to distinguish these chord tones from one another by what interval separates each note in the chord from the root.

For instance, it’s common to talk about the 1, 3, and 5 of a chord. The 1 refers to the root. The first note in the chord, and a unison away from itself. The 3 is the second note in the chord, but it’s a third away from the 1. Similarly, the 5 is a fifth away from the root.

Intervals

An interval of space is a length. An interval of time is a duration. An interval of pitch in Western music theory is rather unimaginatively called an “interval.” Intervals are a way to compare the “space” between two notes.

Let’s say a person sings a note, and then another person sings a note such that they are sounding at once. Since two notes are sounding they can be quantifiably compared by using the taxonomy of musical intervals. Even if they are the same note, they can still be compared as being “the same note.” This type of interval is called a unison. A unison is when two instruments play the same note, in the same octave.

Octave?

An octave is another type of interval. It’s sort of the inverse of a unison. In fact, a unison does invert to an octave and vice versa.

An octave is the doubling or halving of some given frequency. So if the given frequency was 100 Hz, then the first octave above that would be 200 Hz, the second 400 Hz, and so on. The first octave below 100 Hz is 50 Hz, the second 25 Hz, etc. There is a continuous set of frequencies between these octaves, but the Western musical tradition since the mid 1700s has been to divide the octave evenly into 12 pieces. Each of these pieces is called a half-step. We haven’t always divided the octave this way. J.S. Bach was a proponent of equal temperament however, and helped usher in the new tuning system by composing “The Well-Tempered Clavier.”

The idea here is to create a naming system able to describe all the possible relationships between two notes. Twelve was the number at which the Western tradition settled upon, but there are other systems in other cultures that divide the octave into more than twelve notes. These intervals would just sound out of tune to a Western listener.

I can build a list of all the possible intervals within an octave by comparing the first note in said octave to each subsequent note.

For example, if given a note we can compare that note to another note that is a minimum of one half-step above our initial note. We could then add a half-step and compare the note that is two half-steps above to our initial note. We could then compare the note three half-steps above the initial note, and so on. When we reach the twelth half-step, we’ll have reached the first octave above our initial note. All we have to do now is name all of those relationships and we can talk about the two notes in music.

A complete list of the possible intervals within the octave:

unison (0 half-steps apart) – the same note sounded by two separate sources simultaneously

minor-second (1 half-step apart) – also known as a half-step or a semi-tone

major-second (2 half-steps apart) – also known as a whole-step

minor-third (3 half-steps apart)

major-third (4 half-steps apart)

perfect-fourth (5 half-steps apart)

augmented-fourth (6 half-steps apart)

perfect-fifth (7 half-steps apart)

minor-sixth (8 half-steps apart)

major-sixth (9 half-steps apart)

minor-seventh (10 half-steps apart)

major-seventh (11 half-steps apart)

octave (12 half-steps apart)

There are few more little complications that must be addressed to fully understand the Western system of describing intervals. I have given what I believe to be the most commonly used names to the intervals, but to be comprehensive we’d have to give each interval a number of names. This is where the interval naming system can become pretty convoluted, so bear with me.

Seconds, thirds, sixths, and sevenths can be diminished, minor, major, or augmented.

Unisons, fourths, fifths, and octaves can be diminished, perfect, or augmented.

These terms are not to be confused with minor, major, augmented, or diminished chords. We’re talking about intervals here, which are composed of two notes only.

What?

Let’s look at an example. Take a major-third (the distance of four half-steps apart between two notes). If I were to add another half-step to the higher of the two pitches in the interval I would have the interval of a perfect fourth, but I could also call that interval an augmented-third. The augmented-third interval is said to be enharmonically equivalent to the perfect-fourth. They are indistinguishable audibly. The difference is purely nominal. Likewise, if I were to take away a half-step from the top note of the major-third interval I would produce the interval of a minor third, but a minor third could also be called an augmented-second. An augmented-fourth is enharmonically equivalent to a diminished-fifth. A doubly-augmented-fourth is the same as a perfect-fifth. A doubly-diminished-fifth is just the same as a perfect-fourth. A diminished-fourth is equal to a major-third. An augmented third is the same as a perfect-fourth.

Intervals can also be inverted. For instance, imagine two notes sounding at once. There is a higher and a lower night, unless they are the same note. Ok…imagine any two different notes sounding at once. In this instance, there is always a higher and lower note. If you take the bottom note up an octave, or take the top note down an octave you have “inverted” the interval. It’s like turning the interval inside-out. Conveniently major converts to minor, augmented to diminished, and perfect to perfect such that…

unison inverts to octave

minor-second inverts to major-seventh

major-second inverts to minor-seventh

minor-third inverts to major-sixth

major-third inverts to minor-sixth

perfect-fourth inverts to perfect-fifth

augmented-fourth inverts to diminished-fifth

perfect-fifth inverts to perfect-fourth

minor-sixth inverts to major-third

major-sixth inverts to ___________

_____________ inverts to minor-second

______ inverts to ______

If you can fill in the blanks, you’re starting to understand intervals.

Alternatively, if this seems like gibberish to you then read it again! Understanding intervals is essential to grasping the concept of chords, which is the topic of the next installment of this blog.

A Single Note

What is a note?

A note is actually a multidimensional thing. It has both pitch and duration.

It’s like how a vector has direction, and magnitude. The magnitude might be zero, but it’s there.

In the case of a note, the pitch might be zero but the note still has duration as the vector maintains its direction.

The duration of a note might be forever, but it cannot be without duration at all. The instance of a note duration with zero pitch is called a rest-note, or simply a “rest.”

Sound, as we hear it, is about the exchange of kinetic energy. Some sound “source” jostles the air at a sufficient velocity and the jostling is detected by tiny hairs in our ears which transduces the kinetic energy into electrical impulse, interpreted via nerves by our cerebral cortex.

I believe the use of the word “note,” in any language, puts the sound in question squarely into a musical context. Described musically, the sound could quite possibly be considerted “out of tune.”

We tune, musically, to a thing called “A 440.” This is short for “440 Hertz(Hz), which is called ‘A.'” Our society has simply chosen to call the frequency of 440 Hz “A,” and everything else is tuned from that pitch.

What is pitch? What is Hertz? What is frequency?

These terms are related. When a sound source, such as a guitar string, vibrates it jostles the air around it. The rippling waves created by these jostlings propagate longitudinally through the air. If you can imagine a transverse wave vibrating and oscillating through the air in three dimensional spheres of undulating positive and negative pressure then you start to get the picture. Oh yeah, and the oscillations can happen many times per second. For human hearing we’re talking 20 vibrations per second minimum, to be heard as sound. We as humans can detect very many more vibrations per second, however. The top end of human hearing is 20,000 oscillations per second. I try to imagine visually the many wrigglings that the longitudinal wave worm would have to perform per second to create the effect of even 8,000 oscillations. It’s difficult.

Hertz was a guy. The measure of oscillations that pass through a given point in a second at the speed of sound is named after him. The more Hertz (Hz), the higher the pitch. The exact number of oscillations per second in Hertz is referred to as a sound’s frequency.

Waves can take on different forms, but there is a fundamental wave form against which all others are compared. It is called a sine wave.

This wave form is the building block.

A guitar string playing a note seems like a single sound, but it’s actually lots of sine waves at different frequencies sounding all at once.

Most sound sources are not capable of reproducing a perfect sine wave. Things like animals, cars, humans, instruments create what are called complex waveforms. They’re complex because they’re made up of multiple sine waves sounding at once. The waves interfere with one another both constructively and destructively to create a sound signature that we perceive in amalgam.

To imagine this, ask yourself “how can two people sing the same note, but sound different from one another at the same time?”

The answer is that a single note, unless it’s a sine wave, is not just a single not but the combination of many sine waves sounding simultaneously. When a person sings a note a whole series of notes sound in resonance with the lowest, fundamental note. The lowest, loudest sounding note is called the “fundamental” and is what we would call the pitch of the note being sung. However, every multiple of that fundamental frequency actually sounds, just much more quietly, simultaneously with the fundamental. Although the higher multiples of the fundamental frequencies are much quieter they still impact the overall sound enough to create an audible signature. This series sounding in harmony with fundamental is eponymously called the “harmonic series.” The relative volumes of the upper harmonics as compared to the fundamental is what gives a sound it’s recognizeable signature; it’s timbre.

First Prismatic Landscape Mural

Just completed a painting. It’s a mural about eleven feet tall and twenty something feet long. I haven’t measured the length. It just looks about twice as long as the height. It’s a landscape. Or it’s not. Whichever’s funnier.

Acrylic on drywall.