Keys

[thebigcheese]

I think I can put it another way if that helps anyone understand... I've always been pretty good at math because they always give very logical reasons for everything. There's always some process that everything follows and they show you how this leads to this and is based on that. But none of the theory stuff is really giving me that logic, so I just get confused more...

 

[famous beagle]

Ok, this is a long one, but I'm really trying to help, so please read it!

Here are a list of facts that you may or may not already know.

1 A major scale has 7 (different) notes, and then it starts over again in the next octave.

2 You already know the intervallic pattern (W W H W W W H).

3 The notes of a scale are numered as degrees: the first is 1, or the root (or tonic). 2 is the 2nd, 3 is the 3rd, etc.

4 The interval from 1 to 3 in a scale is a 3rd, because three note names are in involved. In the case of C major, this would be C(1) D(2) E(3) --- C to E is a 3rd.

5 Consequently, D to F is also a 3rd: D(2) E(3) F(4) --- three note names

6 If you measured the intervallic distance from the root to any note in the scale, you'd get all kinds of intervals: C to D is a 2nd, C to E is a 3rd, C to F is a 4th, C to G is a 5th, C to A is a 6th, and C to B is a 7th.

So far, all this does is tell you the interval's quantity (i.e., the number of note names involved). An interval also has a quality. That's the major, minor, augmented, or diminished part of it.

Ok, now bear with me. Just read through this, and I promise that it's all aiming to answer your "why" question.

You need to learn this chart. It tells you how many half steps each interval is. It's given in reference to a C note, as an example, but any note can be used. Why do you need to learn it? Because understanding intervals is how you understand the way chords are built.

m2 (minor 2nd) 1 half step (C to Db)
M2 (major 2nd) 2 half steps (C to D)
m3 (minor 3rd) 3 half steps (C to Eb)
M3 (major 3rd) 4 half steps (C to E)
P4 (perfect 4th) 5 half steps (C to F)
aug4 (augmented 4th) 6 half steps (C to F#) or dim 5 (diminished 5th) (C to Gb)
P5 (perfect 5th) 7 half steps (C to G)
m6 (minor 6th) 8 half steps (C to Ab)
M6 (major 6th) 9 half steps (C to A)
m7 (minor 7th) 10 half steps (C to Bb)
M7 (major 7th) 11 half steps (C to B)
P8 (perfect octave) 12 half steps (C to C)

There are several things at work here that make learning it easier.

1 Major intervals are always one half step larger than minor ones.
2 Augmented intervals are one half step larger than perfect intervals (they can also be one half step larger than major intervals, but that's a bit advanced right now)
3 Diminished intervals are one half step smaller than perfect ones.

The augmented 4th and diminished 5th interval is the same note: F# = Gb. These notes are said to be enharmonic. (If you went to the piano, the same black key can be called F# or Gb.) The reason that C to F# is called some kind of 4th is because 4 note names are involved (C D E F). The reason that C to Gb is called some kind of 5th is because 5 note names are involved (C D E F G). The quality (augmented 4th or diminished 5th) is because 6 half steps are involved.

Augmented and diminished can also be used to describe other intervals; it's just not as common. For instance, the distance of 8 half steps from C could also be called an aug 5th (one half step larger than a P5th). This would be spelled C to G# and would sound the same as C to Ab. Again though, C to G# is called an aug 5th because 5 note names are involved, and C to Ab is a m6 because 6 note names are involved.

Again, this is all in relation to C. But if you understand what a half step is (one fret on a guitar or one key on a piano), you can relate it to any note. For example here's how this chart would look as measured from D:


D to Eb m2 (1 HS)
D to E M2 (2 HS)
D to F m3 (3 HS)
D to F# M3 (4 HS)
D to G P4 (5 HS)
D to G# A4 (6 HS) or D to Ab d5
D to A P5 (7 HS)
D to Bb m6 (8 HS)
D to B M6 (9 HS)
D to C m7 (10 HS)
D to C# M7 (11 HS)
D to D P8 (12 HS)


Ok, so why is this so important? Because understanding this shows you how chords are built. The triad is the basic chord. It has 3 different notes (hence the name). There are four different types: major, minor, augmented, and diminished.

Let's go back to our example with C.

A major triad contains a root, a major 3rd, and a perfect 5th. Looking at our interval chart with C as the root, you can see how to build a C major triad.

C is the root, E is the major 3rd, and G is the perfect 5th. C-E-G

A minor triad contains the root, a minor 3rd, and a perfect 5th. It's only one note different from a major triad. So, looking at the chart, we would adjust to spelling of C minor to this:

C is the root, Eb is the minor 3rd, and G is the perfect 5th. C-Eb-G

An augmented triad contains a root, a major 3rd, and an augmented 5th. Looking at our chart, we'd spell it this way:

C is the root, E is the major 3rd, and G# is the aug 5th. C-E-G#

And a diminished triad contains a root, a minor 3rd, and a diminished 5th. Our chart would show that it's spelled like this:

C is the root, Eb is the minor 3rd, and Gb is the diminished 5th. (Remember, diminished intervals are one half step smaller than perfect ones.)


Before I go on, let me know if this makes sense so far. If you have any questions, ask them specifically if you can, and I will try to answer them.

 

[thebigcheese]

That does help, actually. Thanks

The only question I have so far is why the 4 and 5 intervals get to be called perfect. I know that they're special, I'm just not really sure why. It helps when figuring out how to form and name a triad, though (because there's only one note that can be a minor or augmented).

(As a side note, I also pasted that all into a Word document so I can print it out and use it as reference later... very helpful so far)

 

[ggunn]

I think I can put it another way if that helps anyone understand... I've always been pretty good at math because they always give very logical reasons for everything. There's always some process that everything follows and they show you how this leads to this and is based on that. But none of the theory stuff is really giving me that logic, so I just get confused more...

Music is extremely mathematical; you just haven't made the connection yet.

Music theory (it's weird to me that they call it "theory", since how it all works is well known) is one of those things that seems like a bunch of disconnected facts until you begin to see the pattern behind it. It's like learning a language... strike that, it IS learning a language. That's really all it is, when you get down to it. Learning theory won't change the way music feels to you; it will open up possibilities for you to explore and allow you to communicate with other musicians more effectively.

EDIT: One more thing - nobody ever learns it all. It's something that you will continue to learn about as long as you play or even think about music. That's a good thing.

 

[thebigcheese]

Music is extremely mathematical; you just haven't made the connection yet.

I figured that there had to be some logic to it, but yeah... still trying to find that connection.

 

[ggunn]

I figured that there had to be some logic to it, but yeah... still trying to find that connection.

What I hear from you is frustration, which is normal in the steep parts of a learning curve. I remember when I went back to school after being out for ten years and had to take a course in differential equations, thinking "this doesn't make any sense at all; how can ANYBODY ever understand this crap???" I later came to realize that the frustration itself was an impediment to my understanding, and I was wallowing in it. It was only when I was able to rise above the frustration and look at it as an unnecessary indulgence that I began to "get it".

 

[muttley600]

Your comfortable with maths and explanations based on maths, that’s good. From what you are saying it sounds like you want an explanation of why certain intervals that make up a chord are important? That’s good too because the easiest way to understand that is with a little simple maths, some simple physics and our old friend the vibrating string. I’ll try and precise some of my teachin g notes to explain with maths a bit of physics and some perspective as to why the fifth and other intervals are important. This will also help you when you move onto understanding some basic things about tunings and acoustics but that can come later.

I’ll start right at the beginning and stop me if and when it starts to confuse. Hopefully it will explain how intervals are derived and why they are important.
We have a simple vibrating string, don’t worry about it’s length of tension or anything else, What we are interested in to start with is simply that it produces a note.

We shall call that strings length 1 as a mathematic unit or 1/1, same thing. It has a pitch or frequency. We know by shortening that string that we will raise the pitch if all other things remain the same.

Now suppose we fret the string exactly half way along its length. We have effectively divided the string in two and if we do that on a guitar it would be at the 12th fret and we sound the same note an octave higher.

As we have divided the string into two equal parts mathematical we have the simple task of calling it 2:1 as a ratio. Or two to one.

We’ll divide the string further in a minute but first a little basic physics or acoustics.

When a string vibrates it doesn’t just vibrate back and forth between the two fixed points, like a skipping rope, although it is this “fundamental” motion that gives us the “fundamental” frequency or pitch. The strings sound is also made up of other notes which naturally occur in the string as it vibrates. These are called partials or overtones and their strength and relative amplitude go to make up the timbre of the note. What is vital to us here is that those vibrating “mini” skipping rope movements occur at very specific places and are predictable and measurable. They constitute what is known as the harmonic series. These are very important.

The other partials sounding from this string are higher frequencies They are specific pitches in an order representing whole number multiples, ( 2,3,4,5,6 etc......) of the fundamental vibration. The octave is special for a very good reason. All those partials that are vibrating do so physically and mathematically at the same places when the two note are sounded together.

When two notes are sounded together the closer those partials in the two notes match the more pleasing the interval sounds. Pythagoras discovered years ago that further dividing the string into a ratio of 1.5:1 gives a ratio that is very pleasing. That ratio or string division in whole numbers is 3:2 or a fifth.. The reason it sounds good is that the two harmonic series of vibrations in the 1;1 string and the 3:2 string divisions are very closely matched. More so than in other divisions.

Let me know if this is making sense and I’ll take it further or clarify anything that isn’t clear.

If it does make sense you are well on the way to understanding why a specific interval is harmonically important. We can then look at how other intervals occur naturally and how they relate to each other.

If you want a greater in depth explanation this link is very well written. Ignore the first few bits and jump straight to "Overtones, Ratios , and Intervals". If it isn't clear don't worry I'll explain it with less jargon.

 

[Oancient1]

Now hold the phone pardner! I'm a rocker, and according to my research, 63.1% of the songs I've written are in G!

(Of course, that's because I'm a big fan of the relative minor pentatonic scale, and I like to play that in...yep you guessed it...E.)

How fortunate we are for that! Likewise C and Am. For so many years I avoided centering a song around C only to discover that I could solo in that key using some of the same slop I used for the blues in A. Even if I program the MIDI to slip in an occasional B dim, the Am pentatonic still works. Get rid of the thirds entirely and I can even use B as a cjhorus ending chord and ride the Am pentatonic way up to B major.

I love rock music.

Tom

 

[Oancient1]

That does help, actually. Thanks

The only question I have so far is why the 4 and 5 intervals get to be called perfect.

I think it is because are the same in both the major and minor scales. But I only recently began paying attention to all this stuff.

Tom

 

[thebigcheese]

@muttley: I don't understand how you arrived at the ratios, but it sounds like basically you are saying that strings vibrate such a way that they create overtones at other frequencies than just the fundamental. Because of the specific frequencies that these overtones occur at, certain intervals are going to sound better when played together with the fundamental (or the root note in a key), and it just so happens that these intervals are the 4, 5, and 8 intervals. Right?

 

[muttley600]

@muttley: I don't understand how you arrived at the ratios, but it sounds like basically you are saying that strings vibrate such a way that they create overtones at other frequencies than just the fundamental. Because of the specific frequencies that these overtones occur at, certain intervals are going to sound better when played together with the fundamental (or the root note in a key), and it just so happens that these intervals are the 4, 5, and 8 intervals. Right?

In essence yes, if you can get a handle on this now it will be a real help to you later. Trust me.

If you have a string that vibrates. That note is the fundamental.(1:1) If you divide it in two that is the octave (2:1). If you divide it 1.5 to 1 you have the fifth (1.5:1 or 3:2 same thing). That interval or division of the string is important because it is pleasing acoustically, physically, mathematically and musically. More so than other intervals. The bit about the harmonic series and overtones is not vitally important for you to understand but it explains why that interval sounds "pure". In very simple terms the overtones or partials in that string are not clashing. None of this will help you understand keys or music theory but it explains why certain intervals are important which I think you asked about?

I'll try and find a drawing to illustrate the ratios but essentially it's simple math that you'll get as soon as you see it drawn out. The overtones are important because when they occur closely in two different pitches it provides a pleasing relationship. The fifth is the closest match after the octave and is termed "pure" as a result. When the strong overtones or partials are badly matched it results in audilble beats such as when you tune are guitar string to the harmonic.

 

[ggunn]

That does help, actually. Thanks

The only question I have so far is why the 4 and 5 intervals get to be called perfect. I know that they're special, I'm just not really sure why.

The 4 and 5 are the most harmonically closely related notes in the scale to the 1, and the 5th is the most nearly perfect interval in the tempered scale. Going up a 4th is the same as going down a 5th (ignoring the octave, and the octaves are "perfect", ignoring stretch tunings), so a 4th and a 5th are, in a sense, the same thing.

But the 4th and the 5th are NOT perfect in the tempered scale, only moreso than any other interval.

 

[JR#97]

The circle of fifths is the answer. So kudos who posted the chart. The rest of you have to stay after class and peel gum off of the bottom of the desks.

 

[ggunn]

The circle of fifths is the answer. So kudos who posted the chart. The rest of you have to stay after class and peel gum off of the bottom of the desks.

Yes. Shifting the frequency by a mutiplicative factor of 1.5 sends you up a fifth. From the circle of fifths, you know that doing this 12 times will land you on the note that you started from. Unfortunately the numbers don't work out quite that way, so each 5th has to be bent a little to tweak it in. Think of the circle of fifths as a lock washer that needs to be compressed to make the ends meet; the compression stress (tempering) is distributed equally all around the circle.

 

[muttley600]

The 4 and 5 are the most harmonically closely related notes in the scale to the 1, and the 5th is the most nearly perfect interval in the tempered scale. Going up a 4th is the same as going down a 5th (ignoring the octave, and the octaves are "perfect", ignoring stretch tunings), so a 4th and a 5th are, in a sense, the same thing.

But the 4th and the 5th are NOT perfect in the tempered scale, only moreso than any other interval.

I wouldn't confuse with the issue of temperament for now, we all know where that will lead..

thebigcheese, As promised a simple diagram of the string divisions and their ratios. From it you can see how the ratios are arrived at and how they directly relate to what is going on with a vibrating string. All these vibrations are present in any vibrating string and regardless of it's length they occur at the same relative positions. Those between the 1:1 (fundemental) and 2:1 (octave) are the partials and 3:2 (the fifth) is harmonically the most well matched to the other equivalent vibrations in the root string or note.

Again this doesn't help you understand keys but it explains why the fifth is so important.

 

[muttley600]

Yes. Shifting the frequency by a mutiplicative factor of 1.5 sends you up a fifth. From the circle of fifths, you know that doing this 12 times will land you on the note that you started from. Unfortunately the numbers don't work out quite that way, so each 5th has to be bent a little to tweak it in. Think of the circle of fifths as a lock washer that needs to be compressed to make the ends meet; the compression stress (tempering) is distributed equally all around the circle.

Here we go, syntonic and Pythagorean commas. I knew this was going to end in tears.

thebigcheese, ignore all this bit for now..

 

[JR#97]

Yes. Shifting the frequency by a mutiplicative factor of 1.5 sends you up a fifth. From the circle of fifths, you know that doing this 12 times will land you on the note that you started from. Unfortunately the numbers don't work out quite that way, so each 5th has to be bent a little to tweak it in. Think of the circle of fifths as a lock washer that needs to be compressed to make the ends meet; the compression stress (tempering) is distributed equally all around the circle.

Good for you if you want to get geeky about it. But the original post is about finding what key a song is in. Circle of fifths would be a more proper tool than analyzing frequencies, eh?

 

[muttley600]

Good for you if you want to get geeky about it. But the original post is about finding what key a song is in. Circle of fifths would be a more proper tool than analyzing frequencies, eh?

Not really no, not when the OP has asked for an explanation of why the fifth is such an important interval and how it is determined. How is the circle of fifths going to explain that?

 

[Moonrider]

Well, what I mean is we've only gone over major scales in class, so those are the only ones I'm familiar with. I've also seen the scales in the "Fretboard Logic" book, which I think are called pentatonic scales (maybe), so I know those, too.

Tom Serb's "Music Theory for Guitarists" is one of the very best entry level theory books out there. It'll help make sense of the questions you have.

 

[ggunn]

Not really no, not when the OP has asked for an explanation of why the fifth is such an important interval and how it is determined. How is the circle of fifths going to explain that?

Also, he indicated that he is interested in the math behind the theory.