Part 3 - the Diatonic Set (cont)

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.blogspot.com. Sorry for the wait, I have been in transit all over the place, sick, and torn between timezones. This entry was composed in no less than 3 separate cities!

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

We call this a major chord.

2) First interval=m3, second interval=M3, outside interval=P5

We call this a minor chord.

3) First interval=m3, second interval=m3, outside interval=TT

We call this a diminished chord.

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!

Music Theory, part 2

Welcome to part 2, typed up LIVE from terminal 5 at JFK.

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!

Music Theory, part 1

I think theory is a really important thing to learn, and while the study of theory comes with a ton of caveats and exceptions, I do believe that it helps you expand your knowledge of what is possible to achieve through music, and more importantly, shows you how to achieve it. There is more to say regarding the value of music theory, but I will make references it as time goes on rather than try to put it all in one big paragraph.

So let's get started.

Lesson 1 - fundamentals
1.1 The Harmonic Series
I am sure you all have heard the terms Perfect 5th, Major 7th, etc a lot of times before. It's going to be necessary to understand these intervals backwards and forwards in order to grasp some of the more advanced concepts we will come across. But instead of just teaching them by rote memorization, I would like to discuss WHERE the intervals come from and why they form the basis of Western harmony.
All music comes about via sound, which is a vibrating air column, which can be described mathematically as a wave. This is most obvious on a stringed instrument like a guitar, where you can physically see the shape of a wave form when you pluck it. Note that the two ends of the wave need to be clamped down, or zeroed, in order for it to vibrate; if one end was just hanging you wouldn't get sound. Those points where the amplitude is zero and the string is not vibrating are called nodes.
Now, what is not immediately apparent is that on any vibrating object, there are actually multiple waves superimposed upon each other. Essentially, as long as the vibrating string has nodes at both ends, any waveform can happen in between the two. In the below picture you will see many different possibilities of waves that can (and do) exist simultaneously on a vibrating guitar string.

Each one of these different vibration possibilities after the is called a harmonic, or partial. While the fundamental or first harmonic (top wave) is the loudest, the relative loudness of the higher harmonics is what gives every instrument its distinctive tone color. So while a guitar and a trumpet might be playing the same note, the higher-order harmonics are different and that's why they sound like different instruments.

Let's listen to what those individual harmonics actually sound like. To do that, we will artificially CREATE a node (say, by lightly touching the string with our finger) that silences the harmonics below (in the picture, those above) the one we want to hear. So if we touching the vibrating string at exactly one half the length, it will silence the fundamental but not the second harmonic, since at that point the fundamental has a nonzero amplitude, but the second harmonic is already at zero so it doesn't matter. What note do we hear? An octave above the fundamental. K, let's try isolating the third harmonic, at 1/3 the length of the string. Silences both the fundamental and the second harmonic, and we hear a fifth above the note that was the second harmonic. Now touch at 1/4 the length... we hear a note that's a perfect fourth above the third harmonic. It goes up and up until they are barely audible. This series of notes you hear is called the harmonic series.

1.2 Generating Intervals

But let's not yet assign the intervals between successive notes labels like "fourth" and "fifth." Instead I will refer to them as such: the interval between two notes is the ratio of their multiplicative factors relative to the fundamental. That is, the interval between harmonics 2 and 3 is, (2x as fast) to (3x as fast), or 2/3. [I am allowed to say that the second harmonic is twice the speed as the fundamental because the speed of vibration, or frequency, is inversely proportional to its wavelength, as long as the speed of sound remains constant (which it does).] Essentially, what we are saying is that the second harmonic is vibrating at 2/3 the speed as the third harmonic. Or: the third is vibrating at 3/2 the speed of the second, which is the convention I will be using from here on out.

K, so we will call the interval between the fundamental and second harmonic = 2/1. Between the second and third harmonics = 3/2. Third and fourth = 4/3. Do you see a pattern? We are adding a whole number to both the numerator and denominator for each successive interval. Now that we have a set of intervals that we can describe mathematically, let's see what we can do with them. How about multiplying them together? If I know that multiplying one frequency by say, 4/3, gets me a note a certain interval above the original, can I multiply AGAIN and get the same thing but above my new note? (Yes.) So finding the frequency of a note TWO "4/3 intervals" above the original would be 4/3 * 4/3 = 16/9 times the frequency of the original.

As you no doubt can predict, the more we multiply intervals (which are improper fractions) together, the larger and larger the ratios become. Is there any way to simplify them? Here is where I admit our first non-derived assumption: two frequencies that vibrate a power of 2 apart from each other (ie 2/1, 4/1, 8/1) are the same pitch. Essentially what I am saying here is that notes an octave (or multiple octaves) apart are still considered the same note. So, if the value of a ratio exceeds 2 (numerator is more than twice the denominator) then I know that interval is larger than an octave. And so if I divide it by 2 (or multiply by 1/2) until the numerator is no longer more than twice the denominator, I know that that interval is smaller than an octave. If the ratio between two pitches equals a power of 2, then they are the same note.

So armed with all these tools, I am going to try using any of these ratios as a Generating Interval to see how many distinct pitches I can get until I hit a power of 2, which I have assumed means that I have reached the 'same note' as the starting pitch. Let's play.... first up: 5/4. I don't know what interval this is, but how many times can I repeat it until I get back to the octave? 5/4 x 5/4 x 5/4 = 125/64, which is roughly 128/64 = 2 (a power of 2). So it looks like we could only use 3 of those intervals until we hit the first octave. OK let's try the next one, 6/5. I multiply this four times, and hit 1296/625 which again roughly equals 2. Damn, can't I break past 2? Maybe I can try using a simpler ratio, like 4/3.

K, I'm multiply (4/3) by itself again and again and still not hitting anything that looks like a power of 2. By the 12th time, my ratio is 16,777,216 / 531,441. Let's divide ... 31.57 .... Oh yeah! That's so close to 32, or 2^5! It seems that after ascending the interval of 4/3 12 times, I have finally landed on a power of 2, that is, landing on the same note as I started.

Fine, so if changing our generating interval from 5/4 to 4/3 helped us get 12 (as opposed to 3) distinct pitches before coming full circle, maybe I can use an even simpler ratio and get even more pitches! Let's try 3/2. (3/2)^12 = 129.76 = 2^7. Still only 12 pitches came out of that effort. So 12 seems to be the maximum number of distinct pitches we can derive from the intervals found in the harmonic series, and since we ascended the same interval each time as we calculated those 12, it follows that each one of those 12 pitches are equally spaced.

And that is why Western music has 12 pitches. Yes Western. Other cultures have different scale-construction systems which I'm not going to get into here because frankly, I don't understand them. Tomorrow we will discuss the intervals as you know and love them, by their actual names! Yaaaaay!

Note: You saw that when I landed close to, but not on, a power of two, I deemed it 'good enough' to call it the same note. If we were to be sticklers and say that we only hit the same note once we straight-up get a real power of 2, then we would be waiting forever. As long as we're using a generating interval derived from the harmonic series (that is, a whole number ratio), there is no way to truly bound the number of notes produced within the octave. This is because the harmonic series diverges! It explains why notes can go infinitely high and why it's possible to create a smooth 'slide' between two pitches. There are ways to use non-harmonic generating intervals to create 21-, 23-, 27-note systems, or really whatever you want. But if we want stop the madness, base our note system in the harmonic series and keep the notes more-or-less equally spaced, we have to accept that 31.57 = 32. Attempting to reconcile the inherent mathematical imperfections with the idea that all notes should be equally spaced is what led to the development of 12-Tone Equal Temperament, which Bach demonstrated with the Well-Tempered Clavier.


Edit: I received a couple questions asking to clarify what makes an octave so special and to re-explain the ratios between different harmonics. As follows...

The fundamental frequency can be any arbitrary frequency. I intentionally wasn't using any specific one to define our fundamental, to drive home the point that these properties are universally true. However, the upper harmonics that can be generated from that wave are dependent on our choice of fundamental. That is because the frequencies of the harmonics are multiples of the fundamental frequency.

When we isolate the (naturally occuring) second harmonic, we are in essence muting the first harmonic and hearing all the harmonics that are left over. (Those harmonics were actually there the whole time, but the first harmonic is much louder.) The second harmonic has a wavelength half as long as the first harmonic, as you can see in the diagram. Since v=f*λ (and v=the speed of sound which is constant), that means that the second harmonic has a frequency double that of the first harmonic. Through some mix of pitch perception and cultural context, we have come to identify the interval formed between a note of frequency f and another with frequency 2f as "the same note" despite the fact that they are obviously different frequencies. That is why we have the concept of octaves - it's a way to express the difference between two notes that have the same same pitch class (i.e. C, Eb, F#, etc) but different frequency. Mathematically, we say that doubling a frequency has the effect of adding an octave.

You can demonstrate this by playing an open string and then touching it lightly at the 12th fret, which is exactly half the length of the string. You'll hear the octave come through - that is the second harmonic you're hearing. Note that you can do this on any of the strings, which shows that it doesn't matter what our fundamental frequency (can be E, A, D, G, B whatever) is, the second harmonic is still always an octave above the fundamental.

"so, does "original" mean the first of those two notes, and the note you get by multiplying by 4/3 is the second note, and so you have an interval between those two notes"

this is completely correct. But I will go over it again to make sure you understand.

Let's say our fundamental frequency (aka first harmonic) has the frequency f. We just determined that the second harmonic has frequency 2f.

If you look at the diagram, you will see that the third harmonic has a wavelength 1/3 the size of the fundamental. Therefore its frequency is 3f. Fourth = 4f, Fifth = 5f, etc.

So if multiplying the fundamental frequency by a whole number gets us the pitch of whichever upper harmonic corresponds to that number, then we can choose to define the interval between the two notes (in this case, the fundamental and the upper harmonic) by the factor by which we multiplied. That is, the interval between the 2nd harmonic and the fundamental can be called "2:1." Or the interval between the 3rd harmonic and the fundamental can be called "3:1." OK great - but what if we want to find the interval between harmonics 3 and 2? That would be the ratio of the frequencies 3f and 2f, otherwise known as 3:2. That's the interval we call a fifth. (On the guitar, it's the interval between the 12th fret harmonic and the 7th fret harmonic)

Therefore if you multiply any frequency by a factor of 3/2 you will get a second note that forms that interval (a fifth) with your starting note. Again, your starting note can be completely arbitrary. Let's think of it this way - I want to find the frequency of a note a fifth above frequency m. Via the above means, it would be 3m/2. But we could also say, if we call m our fundamental, the 3rd harmonic will sound like a fifth, just an octave too high. So we'll take that frequency, 3m, and then subtract an octave by dividing the frequency by two. Voila, 3m/2.

3/2 is the multiplicative factor for a fifth. The other intervals are represented by other multiplicative factors... the so-called "Perfect intervals" are whole number ratios like 5/4 and 7/6 etc. However if we use the interval of a fifth to GENERATE those other notes (ie- start on C, go up by 2 fifths and you'll get the note D, so you have created a major second by using a fifth to get there) the ratios will be slightly different, this is what I allude to in blue at the bottom of part 1.