Part 4 - fundamentals done, beginnings of Diatonicism

Hello again, everyone! For once I am not typing this from an airport because my flight was cancelled due to glorious amounts of snow (which I have yet to see). I hope everybody had nice holidays and and the upcoming New Year's celebrations will be great! I personally will actually not be experiencing New Year's because I will be flying at the time, never encountering 12:00 AM due to time zone trickery.

Today we are going to wrap up the fundamentals and then move on to discussing diatonicism, which includes chord function, progressions, etc. But first, let's review.

Pitch class space, or the total universe of notes in Western music, is composed of 12 equally spaced pitches. Within those 12, the optimally arranged set of 7 notes is known as the diatonic set. While the specific placement of notes in relation to each other is fixed, the set can be based on any of the 12 pitches, allowing us create 12 unique instances of the diatonic set, called keys. A scale is the set of notes formed by starting on any one note of the diatonic set and naming each successive note. A chord is similar, except we skip every other note. One type of chord is a triad, which is just the first 3 notes of a chord. Within the diatonic set we find that there are only 3 different types of triads - major, minor, and diminished. The difference between these is simply determined by what type of intervals compose the triad, and in which order. (Another type of triad is augmented, although this does not appear within the diatonic set.)

1.5 (cont) Seventh Chords

Alright, so this should be pretty conceptually simple. If a triad is the first 3 notes of a full chord, what kind of chord do we get if we also include the 4th note? The answer is a seventh chord. I have listed below all the seventh chords in the key of F.

Root #	Note (in F)	Triad quality	Seventh Chord	Outside interval	Classification
I	F	Major	F-A-C-E	Major 7th	Major 7
II	G	Minor	G-Bb-D-F	Minor 7th	Minor 7
III	A	Minor	A-C-E-G	Minor 7th	Minor 7
IV	Bb	Major	Bb-D-F-A	Major 7th	Major 7
V	C	Major	C-E-G-Bb	Minor 7th	Dominant 7
VI	D	Minor	D-F-A-C	Minor 7th	Minor 7
VII	E	Diminished	E-G-Bb-D	Minor 7th	Minor 7 b5

Note the different combinations of triad quality with the outside interval, aka the interval formed by the 1st and 4th note of the chord. (For ease of understanding, the 4th note of the chord, which is the 7th note of the corresponding scale, will be referred to as 'the 7th.' In fact, all notes in a chord will be referred to by their placement in the corresponding scale.) We have four that appear; I've listed them below with their common names. Note that the nomenclature doesn't appear consistent with the outside interval. Instead, the "7" in the chord name simply refers to the fact that we added the seventh note of that triad's associated diatonic scale.

Major triad + Major seventh = "Major 7"

Minor triad + Minor seventh = "Minor 7"

Major triad + Minor seventh = "Dominant 7"

Diminished triad + Minor seventh = "Minor 7 flat 5" (Less commonly called half-diminished 7)

So there are 4 distinct types of seventh-chords, whereas there were only 3 distinct types of diatonic triads. Try playing around with seventh chords and hear how they sound compared to triads. You'll feel that they add more 'color' to the sound, perhaps too much in some cases, perhaps only subtly in others. I encourage you to become very familiar with the sounds of the seventh chords, as they can be really beautiful when used in the right way. They are also extremely important in learning theory - you'll see how later.

Can you spell the following chords?

C major 7

E minor 7

F# minor 7 b5

F dominant 7

1.6 The Modes

When we were creating triads, we found that the quality of the triad depended on our starting note. That is, in the key of C, with C = note #1, creating a triad starting on the 2nd note (D) we get a minor triad, but if we start on the 4th note (F), we get a major triad. Out of 7 possible triads, there were only 3 distinct types formed. Does the same property apply to scales?

Scales, much like chords, can be major, minor, etc. Essentially, the scale that corresponds to a chord takes on its quality - so in the key of C, we would call the scale starting on D a 'minor scale,' since the D triad is minor. An F scale would be major, because the F triad is major. Let's analyze all the different scales we can get. For the sake of not getting you confused with things you may have learned elsewhere, I have chosen to use the key of Bb to demonstrate this, instead of C (most everybody uses the key of C to teach this stuff). I've also highlighted in red the 1st, 3rd, and 5th note of each scale to show the quality of the triad, and hence scale.

I - Bb C D Eb F G A (Triad = Bb D F = Bb major)
II - C D Eb F G A Bb (Triad = C Eb G = C minor)
III - D Eb F G A Bb C (Triad = D F A = D minor)
IV - Eb F G A Bb C D (Triad = Eb G Bb = Eb major)
V - F G A Bb C D Eb (Triad = F A C = F major)
VI - G A Bb C D Eb F (Trad = G Bb D = G minor)
VII - A Bb C D Eb F G (Triad = A C Eb = A dinimished)

If we compare two minor triads, for example C minor and D minor, we see that the interval content of each is identical. (Both of them have a m3 between the first two notes, then a M3). But if we look at the SCALES that correspond to those two triads, we will see that they are different! Below, I have spelled out a C minor and a D minor scale (in the key of Bb). Underneath the note names I have labeled the scale degree*, and I also labeled what interval is between each note; minor seconds are red, major seconds are blue.

* Note that scale degree is different than the 'note number' in the key. For example, the note C is the 2nd note in the key of Bb, but it is the 1st note of the C minor scale. Moving forward, 'note number in the key' will usually be denoted by roman numerals, so usually when you see Arabic numerals associated with notes, they are referring to scale degrees.

You'll see that the intervals between the 1st and 2nd, the 2nd and 3rd, the 5th and 6th, and the 6th and 7th degrees of either scale differ. Therefore, although they are both considered minor scales, they're undeniably different. And so while there is only one type of minor triad, there are multiple types of minor scales. Although we have not done this, you will also find that there are 3 different types of major scales as well. Compare Bb major, Eb major, and F major (all in the key of Bb). Can you find which intervals differ from scale to scale?

To differentiate between the different types of scales, we give them particular names. Unfortunately these are names that you have to memorize, because I will be referring to them often. The column "distinctive feature" refers to a particular and prominent aspect of that mode's interval content, which might come in handy if you are trying to identify the modality of a particular scale. "Interval sequence" describes the intervals between successive degrees; M refers to Major 2nd, m refers to minor second.

#	Triad quality	Scale name	Distinctive Feature	Interval Sequence
I	Major	Ionian	m2 between 3 and 4	MMmMMM
II	minor	Dorian	M2 between 5 and 6	MmMMMm
III	minor	Phrygian	m2 between 1 and 2	mMMMmM
IV	Major	Lydian	TT between 1 and 4	MMMmMM
V	Major	Mixolydian	TT between 3 and 7	MMmMMm
VI	minor	Aeolian	m2 between 5 and 6	MmMMmM
VII	diminished	Locrian	TT between 1 and 5	mMMmMM

So, our naming convention for modes is as follows. If I say "E Dorian," I mean a scale whose
1) 1st, 3rd, and 5th degrees form the E minor triad
2) 2nd degree is a M2 above the 1st
3) 4th degree is a M2 above the 3rd
4) 6th degree is a M2 above the 5th (**distinctive feature of Dorian**)
5) 7th degree is a M2 below the 1st (or, m2 above the 6th, same thing).

Therefore, E F# G A B C# D.

Using the Interval Sequence column in the chart above, can you spell out the following scales? Refer to the original circle diagram if you are unsure of what two notes form a m2 or M2. If you have an instrument handy, I encourage you to play them!

A phrygian
B locrian
F# mixolydian
F ionian
C# dorian
C# aeolian
D lydian

Congratulations! Let's move on to ...
Chapter 2 - Diatonicism
Roughly, western (which I will controversially refer to as tonal) music is created out of chord progressions. That is to say, most tonal music is based on chords (either explicitly played or implied) which change over time, usually with some regularity or pattern. Diatonicism refers to the degree to which any given chord or chord progression 'belongs' to the key.

One thing I should mention is that, for now, when I say 'the key of C' I really mean the key of C ionian. This means that my designated 1st note is C and that the scale built on C is an ionian scale. This may seem obvious, but later we will talk about being in the key of, say, A minor, which actually has all the same notes as the key of C but the designated first note is now A.

2.1 Chord Spelling
For now, we are going to focus only on the diatonic chords, that is, the chords that can be formed out of the notes of the key and nothing else. As explained before, each of the diatonic triads corresponds to a certain modal scale. We call this 'taking' a scale: In the key of C, the C triad is major and 'takes' the Ionian scale; the D triad is minor and 'takes' the Dorian scale, etc. But if we don't know what key we're in, it will be impossible to say what scale is appropriate for a given triad. For example, the F major triad exists in three different keys - C, F, and Bb, and it takes a different scale in each (Lydian, Ionian, and Mixolydian respectively).
Therefore we tend to refer to chords not by their actual names, but by their position in the key. This is what was earlier referred to as "note number in the key," and we use roman numerals for it. The numeral I refers to the chord starting on the 1st note of the key; the II refers to the chord starting on the 2nd note, etc. Therefore, I could simply say "the IV chord in C," and I already know a huge amount of information-
1) The chord root (F) Because F is the 4th note in the key: C D E F G A B
2) The triad quality (Major)
3) The mode of the corresponding scale (Lydian)
4) The function (subdominant) will get to this later

Before we move on, see if you can figure out the following chords (and associated scales).
V chord in F
IV chord in G
II chord in A
II chord in F#
I chord in E
VI chord in G

2.2 Chord Roots
The root of a chord is its first note. Finding the root of a chord is simple if the chord is spelled like this: C-E-G. The root is C, and by looking at the other 2 notes inside we know that it's a C major triad. However, often we'll see chords spelled like this: E-G-C. Here, the first note is E... but it has the notes of a C major triad. It is, in fact, a C major triad, but we call it an inversion. No matter how a chord is inverted, its root remains the same. The original spelling, C-E-G, we call root position. The fact that a root-position chord is the same exact thing (from a theoretical standpoint) as any of its inversions is an extremely important concept to understand. In practice, we'll rarely see chords written strictly in root position, so we need to know how to look at a chord and determine its root. As you become more familiar seeing/hearing chords and their inversions, this will become more intuitive.

So how do we 1) tell if a chord is inverted, and 2) determine its root?
Remember much earlier when I described a chord as being the collection of notes formed by going up a scale, skipping every other note? That method will result in a root position chord; and it means that the interval between each successive note MUST be a third. If we look at the note collection G-C-E, we can see that the interval G-C is not a third; there are two notes skipped (A and B), making it a fourth. That's how I automatically know it's an inversion. To find out what this triad is in root position, rearrange the notes until both inside intervals are thirds (another trick is to rearrange the notes until the outside interval a fifth). Thus, C-E-G.
Can you rearrange the following inverted triads into root position?

C#-F#-A

Eb-Gb-Cb

C#-F#-A#

Fb-Ab-Db

2.3 Chord progression basics, and Strong vs. Weak Motion

Each instrument has a different way of 'playing' a chord. "Chordal" instruments like the guitar or piano are able to produce all the notes of a chord at once, while low instruments like bass or bari sax usually play the root of the chord in order to further reinforce the sound, and higher instruments (which can only play one note at a time) might play a sequence of notes, or melody, that emphasize the component notes of a chord.

Chord progression is simply the idea of having a series of chords played or implied in some sort of sequence. In the vast majority of music, the sequence of chords and the order in which they appear is predetermined.

The way that we describe a chord progression is by talking about its root motion. Essentially all this means is that we discuss the sequence of chords by the roman numerals of their roots. So, instead of saying "E major, then A major, then F# minor, then B major," we should refer to that same progression as "I - IV - II - V in the key of E." This makes it easier to analyze the chord progression, and if need be, transpose it to another key.

When one chord changes to another chord, some of the component notes may stay the same, while others may change. Let's take a couple examples, in the key of Ab:

I -> II: Ab-C-Eb -> Bb-Db-F (0 notes in common)

I -> III: Ab-C-Eb -> C-Eb-G (2 notes in common)

I -> IV: Ab-C-Eb -> Db-F-Ab (1 note in common)

I -> V: Ab-C-Eb -> Eb-G-Bb (1 note in common)

We hear the most distinction between chords that have fewer notes in common. That is, the I chord and the III chord have 2 out of 3 notes in common, so the two chords are considered more similar than, for example, the I chord and the V chord, which only have 1 note in common. Aurally, the fact that Ab major and C minor are different chords is obvious. However, when it comes to building chord progressions, they have a very similar effect, and are in fact often interchangeable. Thus we can describe two different types of 'motion' between chords: Strong and Weak. Essentially, Strong Root Motionis motion that results in 2 or more notes CHANGING (i.e. a maximum of 1 note staying the same). So, moving from I-II or I-IV would be examples of strong root motion. Weak Root Motion is the opposite, basically any motion that preserves at least 2 of the same notes. Therefore, root motion by a third (or sixth, which is essentially the same distance as a third) is called weak, whereas root motion by a second or fourth/fifth is called strong.

Those of you with instruments, pick any chord from any key. Improvise a melody over that single chord, and then change to another chord in the key via weak motion and sing the exact same melody. You'll see that many of the notes still 'feel' right. Now try moving to another chord via strong motion and doing the same, and listen to the results. While the effectiveness of the melody over the different chords is not a direct function of the type of chord motion, you should start to see that some (weakly related) chords are virtually interchangeable with each other in terms of supporting the melody. Examples: I and VI, II and IV, V and VII, etc.

Alright, this email has gotten long enough. Next time I promise to delve more into Chord Function and Resolution, basically the two factors governing how chord progressions work. Good work today and thanks for reading!

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!