Those of you from before, thanks for sticking with us! And those of you who now find yourselves receiving this email, welcome. You can read parts 1 and 2 below, or at http://thenoiseofmusic.
Today we're going to learn some more about the concept of Tonality, which forms the basis for (almost) all western music.
As always, let's recap what we learned last lesson:
Using the interval of a perfect fifth, which is the distance between two frequencies a 3:2 ratio apart, we were able to create 12 distinct and equally spaced notes within a single octave. We then found that the first 5 or 7 notes generated by that interval were the most equitably spaced and symmetrical arrangements possible within the 12-note system. Thus it was demonstrated that those two sets of notes, the pentatonic and diatonic sets, are directly derived from the natural physical properties of a waveform.
For your reference, the table below shows the relationship between interval, semitone, common nomenclature, and frequency ratio. Note that some of the intervals are in quotes - this is because intevals past the tritone are simply the smaller intervals in reverse. (for example, M2 and m7 span the same total distance in pitch space)
| Interval | Net distance, in semitones,from the originating note | Frequency Ratio (upper:lower) | Common name | Abbreviation |
| 0 | 0 | 1:1 | Unison | unis |
| 1 | 1 | 16:15 | Minor Second | m2 |
| 2 | 2 | 9:8 | Major Second | M2 |
| 3 | 3 | 6:5 | Minor Third | m3 |
| 4 | 4 | 5:4 | Major Third | M3 |
| 5 | 5 | 4:3 | Perfect Fourth | P4 |
| 6 | 6 | 45:32 | Tritone | TT |
| “7” | 5 | 3:2 | Perfect Fifth | P5 |
| “8” | 4 | 8:5 | Minor Sixth | m6 |
| “9” | 3 | 5:3 | Major Sixth | M6 |
| “10” | 2 | 9:5 | Minor Seventh | m7 |
| “11” | 1 | 15:8 | Major Seventh | M7 |
1.4 The Diatonic Set (continued)
Where I would like to start today is by establishing the idea of a key. Up til now I have used arbitrary starting points and absolute distances between intervals, in order to drive home the point that these rules of interval generation etc are universally true. But now I want to delve deeper into the concept of tonality, the first element of which is diatonicism. To do that we need to (temporarily) only focus on the 7 notes of the diatonic set.
First, let us analyze the number of each type of interval contained within the diatonic set. Looking at our circle diagram below, we can clearly see that there exist 2 instances of interval one (m2), 5 instances of interval two (M2) [below left], 4x interval 3 (m3) [below right], etc. Can you count up the rest of them?
What you’ll end up with is something called an interval vector, which simply describes the numbers of each interval type up to #6 (TT). The interval vector of the diatonic set is <'254361>. This means there’s 2x interval 1, 5x interval 2, 4x interval 3, 3x interval 4, 6x interval 5, and 1x interval 6.
Note that there is a unique number of each interval type. While we haven’t bothered experimenting with alternate arrangements of 7 notes within 12 spaces, the diatonic set is the only such arrangement that possesses this property. While this is not necessarily useful as a derivational tool, it does tell us a couple things - the main one being that not every interval or note is equal within the diatonic set.
To be specific, certain notes in the diatonic set are 1 semitone away from an adjacent note, whereas others are 2 semitones away from either of their adjacent neighbors. In terms of intervals, there exists only 1 tritone, whereas there are a whopping 6 fifths. It is the unequal nature of the diatonic set that gives rise to tonality. Tonality, essentially, is the idea that certain notes and intervals are special. You can already see this in the sense that, within the diatonic set, the tritone is the only interval of its kind.
1.5 Keys, scales, and chords
Earlier I stated that you could pick any arbitrary note to use as a starting point for generating the diatonic set. However, now we are starting to move into territory where we're going to need to differentiate keys. That is, we need a way to distinguish the diatonic set formed by starting on pitch #0 from the set formed on pitch #1. The absolute intervals are all the same, but the notes are almost completely different. Looking at the numbers from the circle diagram, going clockwise the set formed by starting on pitch #0 is [0,1,3,5,6,8,10] whereas starting on pitch #1, they're [1,2,4,6,7,9,11].
So we need to establish one of those 7 notes in the set as being the "main note," sometimes referred to as the tonic. I am going to ARBITRARILY say that the diatonic collection we generated will be called by the name of the adjacent note moving clockwise. The reason I do this is because the jargon of music theory (which comes later) is based on this initial arbitration. It makes no difference from a theoretical perspective. So let's say we our starting pitch for generating was pitch #0, well, we will call the resulting set by the name of the note 1 semitone above that. In more familiar terms, that means the set of notes formed by starting on, say, B: [BEADGCF, or going clockwise, BCDEFGA] will be called the key of C. Reproduced below:
Now I will loosely define two terms: scale and chord. A scale is the collection of notes formed by moving clockwise around the circle. A chord is the collection of notes* formed by moving clockwise around the scale, skipping every other note. Two examples of scales in the key of C might be [ABCDEFG] and [EFGABCD]. The full chords that would correspond to those are [ACEGBDF] and [EGBDFAC]. If you trace the path around the circle to create either a scale or chord, you'll see that the interval between adjacent notes in a scale is either a minor second or a major second. For a scale, the intervals within are either minor thirds or major thirds. But for all 4 of those example scales/chords above, the placement of those intervals changes. And the particular placement of intervals in a scale or chord determines its quality.
Let's analyze all the possible chords we can create out of the diatonic set. For now, we are only going to focus on the first three notes of each chord, also known as a triad. Later, we will include more notes in our analysis to further distinguish the chords from each other, but for now the first three notes will give us a general idea of how to categorize them. Essentially what we are going to do is find what kind of intervals compose the triad, and in which order they appear. Let's begin. Interval 1-2 (or 'first interval') is between the 1st and 2nd notes; interval 2-3 ('second interval') is between the 2nd and 3rd; and interval 1-3 ('outside interval') is between the 1st and 3rd.
| Starting # | Notes | Interval 1-2 | Int 2-3 | Int 1-3 |
| 1 | C-E-G | M3 | m3 | P5 |
| 2 | D-F-A | m3 | M3 | P5 |
| 3 | E-G-B | m3 | M3 | P5 |
| 4 | F-A-C | M3 | m3 | P5 |
| 5 | G-B-D | M3 | m3 | P5 |
| 6 | A-C-E | m3 | M3 | P5 |
| 7 | B-D-F | m3 | m3 | TT |
We got 3 types of triads.
1) First interval=M3, second interval=m3, outside interval=P5
You may be thinking, what if the two inside intervals are both M3, making the outside interval a m6? That would be called an augmented chord, which is our fourth type.
It's important to note that we have been doing this all in the key of C, which contains no sharps or flats, and is thus pretty simple and easy to work with. Let's look at all the triads in the key of F#.
| # | Notes | Quality |
| I | F#-A#-C# | Major |
| II | G#-B-D# | Minor |
| III | A#-C#-E# | Minor |
| IV | B-D#-F# | Major |
| V | C#-E#-G# | Major |
| VI | D#-F#-A# | Minor |
| VII | E#-G#-B | Diminished |
Note that I have chosen to use icky spellings like E# instead of F. THIS IS ACTUALLY REALLY IMPORTANT. It doesn't make any difference sonically, but when it comes time to alter the notes of chords and scales it is of paramount importance that we make sure that we preserve the alphabetical order of notes in a scale. In short, in a diatonic scale there can only be 1 of each note 'letter.' If we had simplified E# to F, we would have to spell a sample scale as [G#, A#, B, C#, D#, F, F#] - see how we have 2 kinds of F, and no Es? Unacceptable. No matter what we have to append to the end of the note (sharp, flat, double-sharp, double-flat), we need to preserve this property.
I know that this has been a lot of information! I really want to end the Fundamentals chapter here, but there is still one big thing to go over, the modes. Tomorrow (or whenever) we will do that real quickly, and then move on to our next chapter, diatonicism.
Thanks for sticking around. As always, suggestions/comments/requests for clarification are invited!
apologies for the tables in this entry looking so crappy!
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