Introduction
Welcome to my fifth article in the series on music discovery! My name is Beymer, and this and the previous articles chronical my journey in music composition. Please see my previous article “Music Discovery: Article 4 - Part 1 for my motivation for these articles and one of the best disclaimers I’ve ever written … LOL. Please subscribe!
In the last article, I defined the notion of a neighborhood of a musical point. Essentially, a neighborhood of a reference musical point was defined as a collection of musical points that had the same identical rhythm. By rhythm, I mean the neighbor points had the same note/rest duration sequence as the reference musical point. This also means the same time signature and the same number of notes and rests (at the same positions in the bars). Subsets of the neighborhood were defined based on the collection of pitches used by the musical points in the neighborhood. In this article, additional subsets will be defined again based off the collection of pitches used by a neighbor. Specifically, Quadratonic, Pentatonic, Hexatonic, Heptatonic, and Octatonic scales will be considered.
Scales and Modes
There are lots of books in libraries and bookstores, and there are lots of articles/videos on the Internet that talk about this subject of scales and modes. They do a much better job explaining them than I do in this article, so go there if what I write doesn’t do it for you. In what follows, I define these terms based off my understanding. I got all my information below from Wikipedia.
Let’s define a scale as a finite, ordered list of pure sounds. By pure, I mean that each sound in the list has its own unique frequency, and the associated electrical waveform can be described mathematically as follows:
V(Time) = Vp sin( 2 * PI * Frequency * Time)
where :
V(Time) is a sinusoidal waveform as a function of time.
Vp is the peak voltage.
PI is defined as 3.14159….
Frequency is how often the waveform repeats during an interval of one second (Hertz).
By ordered, I mean the sounds can be arranged in the list in ascending order of increasing frequency. By finite, I mean there are a countable number of these sounds. That is, it’s not infinite! Note that there are an infinite number of sounds as defined above, so the question is which frequencies should you choose for your finite list of sounds.
Let me start by discussing the Chromatic Scale. In particular, let me discuss the 12-tone equal-temperament chromatic scale. In this scale, 12 different pitches exist, with the highest pitch having a frequency that is twice the frequency of the lowest pitch. An octave is defined as a range of frequencies where the highest frequency is twice that of the lowest frequency. The chromatic scale and all the scales derived from it identify the octaves by whole numbers starting at 0 continuing up through 10; that is, assuming an audible range of 20 - 20,000 Hertz. C0 represents the frequency 16.35 Hertz and C4 represents the frequency 261.63 Hertz. The frequencies between C4 and C5 can be found by using the formula:
Fi = 2 ^ (i / 12) * 261.63 Hz
The symbol ‘^’ means to raise to a power of 2 . The variable ‘i’ is a whole number between 0 and 11. For example, the next frequency would correspond to i = 1, which would give 1.0595 * 261.63 = 277.18 Hertz. For i = 11, you would have 1.8877 * 261.63 or 493.88 Hertz.
The same formula is used (except you use the first frequency of that particular octave instead of 261.63, the first frequency of the 4th octave) for calculating the frequencies in all the other octaves. Also note that A4, the pitch just above C4, is set equal to exactly 440 Hertz, with all other pitches adjusted accordingly. With this approach, each pitch will be defined to be separated by a semi-tone or 100 cents. This means there will be a total of 1200 cents for an interval of one octave. The letters of the alphabet from A to G will be used to label these pitches, but since there are twelve pitches and the letters A to G only includes seven letters, you will have to make some new symbols to cover the remaining letters. This is done by adding the ‘#” symbol (sharp) to some of the letters. This means to add a semi-tone or 100 cents the pitch associated with a letter, which has the sharp symbol attached. Note that the assignment of letters to the pitches is such that none the white keys on a piano have a sharp symbol, only a letter. The black keys on the piano all have the sharp symbol attached to their letter.
So, there you have it … the Equal-Tempered Chromatic Scale. The pitches in this scale start with or are calibrated to A4 (440.00 Hertz). Using the formula described above, you can generate more pitches below and above the calibrated A4 pitch. The pitches are organized into octave collections (12 pitches), which begin with the ‘0’ octave (starts with C0, which has a pitch of 16.35 Hertz) and increases to about the ‘10’ octave (starts with C10, which has a pitch of 16,744.04 Hertz). So, for the C4 octave, there would be C4, C4#, D4, D4#, E4, F4, F4#, G4, G4#, A4, A4#, B4, and then C5 of the next octave. There are 12 notes in this octave, C4 through B4.
The Octatonic, Heptatonic, Hexatonic, Pentatonic, and Quadratonic Scales are just subsets of the Chromatic Scale. That is, you just choose eight of the pitches from an octave for the Octatonic Scale: You choose seven of the pitches from an octave for the Heptatonic Scale: You choose six of the pitches from an octave for the Hexatonic Scale: You choose five of the pitches from an octave for the Pentatonic Scale: You choose four of the pitches from an octave for the Quadratonic Scale.
For a given set of ordered pitches, the beginning note of the scale is referred to as the tonic note. Starting with C4 gives you one mode. Starting with C4# gives you another mode and so on. The number of modes present in a scale depends upon scale. Refer to the Internet for much more detail about modes.
In the following sections, each of these scales will be used in turn as the allowed collection of pitches for a neighbor of a given reference rhythm point. The pitches will be randomly selected as described in the previous articles. The mode for each scale will simply be the one associated with the first note of the scale, which is C. The rhythm will be fixed for all generated riffs. The fixed or reference rhythm I used is a 4-bar riff that has syncopation on the third beats of a 4/4-time signature. The reference rhythm is shown below using the Diatonic Scale (Heptatonic Scale). See below:
Quadratonic Scale for the Collection of Pitches
With this scale, we need to select four out the possible twelve pitches contained in the Chromatic Scale for each octave. Let’s start with C4; go up 3 semitones to D4#; go up 4 semitones to G4; go up 2 semitones to A4; leaving 3 semitones until C5. See below:
Here are two samples using the Quadratonic Scale:
Pentatonic Scale for the Collection of Pitches
With this scale, we need to select five out of the possible twelve pitches contained in the Chromatic Scale for each octave. Let’s start with C4; go up 3 semitones to D4#; go up 2 semitones to F4; go up 3 semitones to G4#; go up 2 semitones to A4#; leaving 2 semitones until C5. See below:
Here are two samples using the Pentatonic Scale:
Hexatonic Scale for the Collection of Pitches
With this scale, we need to select six out of the possible twelve pitches contained in the Chromatic Scale for each octave. Let’s start with C4; go up 2 semitones to D4; go up 2 semitones to E4; go up 2 semitones to F4#; go up 2 semitones to G4#; go up 2 semitones to A4#; leaving 2 semitones until C5. See below:
Here are two samples using the Hexatonic Scale:
Heptatonic Scale for the Collection of Pitches
With this scale, we need to select seven out of the possible twelve pitches contained in the Chromatic Scale for each octave. We have been using the Diatonic Scale, which is one possible Heptatonic Scale, in all of the previous articles of the series. The Syncopated Starter riff shown above is an example.
Octatonic Scale for the Collection of Pitches
With this scale, we need to select eight out of the possible twelve pitches contained in the Chromatic Scale for each octave. Let’s start with C4; go up 1 semitone to C4#; go up 2 semitones to D4#; go up 1 semitone to E4; go up 2 semitones to F4#; go up 1 semitone to G4; go up 2 semitones to A4; go up 1 semitone to A4#; leaving 2 semitones until C5. See below:
Here are two samples using the Octatonic Scale:
Summary
Ok, in this article, I presented several different scales or pitch collections to use with neighbors of a reference musical point. I didn’t name these finite subsets of a neighborhood like I did in the previous article, but if I had, I would have named these subsets after their pitch collection or scale. Hope you enjoyed it.