Before we begin, let's review what we talked about yesterday. We looked at the harmonic series, a series of pitches derived from the naturally occurring harmonics of a vibrating string. We defined intervals, or the relationship between successive pitches in the harmonic series, as the ratio of two frequencies. And we found that multiplying a frequency by one of those ratios gives us the pitch of the other note in that interval; multiplying multiple ratios together had the same effect as adding successive intervals. Assuming that any ratio equal to a power of 2 represented the same note, just an octave higher, we were able to find out how many distinct pitches we could create using multiple iterations of any single interval before hitting the octave. The maximum number of equally-spaced, distinct pitches within the octave that we could create, using a generating interval from the harmonic series, was 12.
Today's essay is a continuation of Lesson 1- the fundamentals, and is titled The Roots of Tonality.
1.3 Pitch Class Space
is a fancy phrase for "the notes." The above diagram shows the 12 equally spaced, distinct pitches with whom we are all so familiar. Remember how we used a single generating interval to see how many distinct notes we could get before hitting the octave? This circular array of notes now lets us re-create that experiment using visually apparent intervals, as opposed to mathematical ratios. Although this is redundant because we've already derived the 12 notes through identical means, so we're of course going to prove ourselves right, it lets us visualize collections of notes in a way that using ratios won't.
Quickly, I would like to establish naming conventions. At the beginning I will refer to all intervals by the number of steps, aka 'semitones,' between them. So "interval 1" would simply be the interval between two adjacent semitones, 2 would be the interval formed by skipping one in between, etc. Eventually I will start blending in the more common terms - thirds, fourths, etc; but it should be noted that 'third,' 'fourth' etc have NOTHING to do with the number of semitones that compose that interval!
Say my generating interval is 3. So let's start at circle #0, and fill in every 3rd circle. You'll see that we only manage to get 4 of them before we end up where we started. Try the same thing with interval 4 - now you'll only get 3 unique notes. But if we try with interval 5, we go around and around until every circle is filled. (Colorful one below)
Note that the intervals which are factors of 12, that is to say numbers 1 2 3 4 and 6, create bounded sets - they never exceed the octave before hitting their starting note. Also importantly, the sets that they create are equally-spaced and symmetric. Draw a line bisecting the circle anywhere and both sides demonstrate rotational (and sometimes reflective) symmetry. And while this may be obvious, note that the product of 1) the interval # and 2) the number of notes created by that interval always equals 12.
So if I were to ask you to fill up the 12 spaces with 4 equally spaced, symmetrical notes, you would use interval 3 for that (since 12/4 = 3). But what if I asked you to fill up the 12 spaces with FIVE equally spaced, symmetric notes? 5 doesn't evenly fit into 12, so we will need to compromise somewhere and make some intervals larger/smaller than others. See if you can create such an arrangement - there are several, but no matter what you'll see that you need 3x interval 2 and 2x interval 3. And there is one arrangement in particular that is special in that it 1) is symmetric and 2) makes the abnormally large intervals as far away from each other as possible. See below left. (Orange = line of symmetry; Red = denotes large interval; Green = shows that they're as far apart as possible)
It also turns out that the first 5 notes created by using interval 5 are exactly the same (above right). KABOOM! Interval 5 (a perfect fourth, or P4) automatically creates the most efficient arrangement of notes! It's also known as the pentatonic set, more commonly referred to as the pentatonic scale. This arrangement represents the maximum number of notes that can be put into a 12-note system without creating any instances of interval 1. We generally think that interval 1, (the minor second, or m2) is jarring and must be treated very carefully in order to not sound too discordant. Since the pentatonic scale contains no instances of interval 1, any of its notes can be sounded in any combination and still sound "pleasing." This is why most wind chimes are tuned to the 5 pitches of the pentatonic scale, and why this performance was so easy to orchestrate.
To summarize, the pentatonic scale is the collection of notes formed by the first 5 instances of interval 5 (P4). And remember that interval 5 is special in that it's the only interval capable of creating all 12 pitches.
1.4 The Diatonic Set
We have so far accomplished inserting, in an equal and symmetric manner, 1, 2, 3, 4, 5, or 6 notes into a 12-note system. Our next challenge is fitting SEVEN notes into the 12-note system. Let's just forge ahead and assume that by using Interval 5, we can do this. Below left:
Note that the 2 areas that now contain a minor second (interval 1) were previously the abnormally large ones when we were only trying to squeeze in 5 notes. Essentially, this set of notes represents the maximum number of notes that can be fit into the 12-note system without having two adjacent instances of interval 1. We call it the diatonic set, or as many of you may know it, a "key." If we assign note names to each pitch (above right), the colored-in circles (in order of their derivation) are: B, E, A, D, G, C, F. Putting them in clockwise order, you get B C D E F G A; or, choosing to start on the C, it's CDEFGAB. Many of you will recognize that as the C 'Ionian' Major scale. (The fact that it's a C scale is arbitrary, I could have started this sequence on any note; I just wanted to use C in this example because it is simple and familiar.)
So to summarize, much like the pentatonic scale is composed of the first 5 notes derived via a perfect fourth (interval 5), the diatonic scale is composed of the first 7. The vast majority of the music we hear in Western culture is directly based on the diatonic scale; the major and minor chords/scales that compose most music are simply notes taken right out of this set.
Tomorrow, I will get into more detail about the major and minor scales and chords. I hope you enjoyed today's lesson!
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