Music Education: HALP

[Freeky]

So.  Intervals.  What are they, and what is the difference between M2, m2, M3, m3, P4, T, m6, M6, m7, M7, and P8?

[Xooxe]

My instructor said in the last fifty years, music has been tuned up to sound brighter, and her mother, who had been an opera singer with perfect pitch, had a really hard time not sounding flat.  So that's why things don't sound the same is because the tuning is different.

I didn't think of it like that, but it makes sense like drifting continents.

Separating the sharps on a piano makes sense since it helps you see which note is which. If it were all white keys it would be easy to get lost.

Yeah, I'm happy it's all uneven like that. I don't know whether I'm remembering correctly, but I thought the Greeks started out with seven notes, and then five more were tacked on at some point and slowly became tuned so that now they're equal to the others in terms of pitch spacing, but they're still sharps and flats in notation and theory. Really, the whole thing is slightly bizarre to me.

[Nephew Twiddleton]

Those are the spaces in between notes on a scale. Like if you start with c c# is the minor 2 (also the minor 9) d would be the major 2 and d#/e flat would be the minor third. So it goes root, minor(or flat) 2, maj2, min3, maj3, perfect fourth, dininished fifth (the devils note)perfect fifth augmented fifth sixth dominant seventh major seventh and then the octave which is the same note as the root but higher (this is why a second is also a ninth). This is actually how chords are also constructed. A major chord consists of a root a major third and a perfect fifth. A minor chord is the same but with a minor third. A major scale goes root maj2 maj3 perfect 4 perfect 5 sixth and major seventh. A minor scale is root major 2 min3 perf4 perf5 6 dom7

[LMNO]

Ok.

Music Theory:  The main reason for learning how to write music down is the same reason we write anything else down -- Communication, and Memory.  The fact that we can play a Bach piece hundreds of years after he thought it up is proof of it's efficacy.  Also, if two people know theory, it's much easier to tell someone that a melody you're playing is in [a minor] than it is to try and sing the scale to them.  Or to say, "could you harmonize that a fourth down?"

Because it's a set of patterns describing sound, it's also arbitrary.  To give a foundation, we tend to use A = 440Hz.  That frequency (440 Hertz) makes a sound we are declaring to be "A".  Ok, now it gets fun.

The Greeks (among others) understood that if you take a string and tighten it, it makes a certain sound.  Then they found that if you hold down that string in the middle, it makes a sound almost exactly the same, but higher.  We're calling that an "octave".  Would it surprise you to know that the A that is an octave above A=440 is A=880? 

The space between two notes is called an "interval".  So in this case, the interval between A=440 and A=880 is a "Perfect 8th" or "P8".

Ok, 440 times two is 880, and makes an octave.  What about 440 time three?  That's 1320, which is outside of our octave (440 to 880).  But if we bring that down by half, we get 660.

660 roughly¹ corresponds to E, which is a [Major 5] above E.  To an ear brought up on Western music, the octave is the most stable of the intervals.  The M5 is the second most stable.  By "stable", I mean it sounds complete; not dissonant; in harmony.

So, to get the fifth above E=660, we multiply by three again, and get 1980.  If we divide by 2 until it gets back down into our octave, we get 495, which roughly corresponds to B, which is a [Major 2] above E.  If you keep finding the fifth of each subsequent note, you'll end up with:

A=440                                                                                                     
E=660                                                                                                     
B=495                                                                                                     
F#= 742                                                                                                 
C#=556                                                                                                   
Ab=835                                                                                                 
Eb=626                     
Bb=469                                                                                                   
F=704                                                                                                     
C=528                                                                                                     
G=792                                                                                                     
D=594                                                                                                     
A=880                       

Which, when you rearrange it, become the 12 notes in the scale. 

This is a long way of saying that there is a mathematical relationship between the intervals we hear as "stable" and the intervals we do not.  As a further example,  we've seen that the 3/2 ratio is stable (that is, A=440x3 is roughly the same as E=660x2, (i.e.1320)).  What about A=440x5?²  That's 2200, which is the same as 550x4, which is really close³ to C#=556.  Which is a [Major 3] interval above A.  And {A , C# , E} is a Major Triad, which is you basic Major chord.

Oh, math.  Is there anything you can't explain? 

Anyway, I wanted to start out by explaining that when we talk about "stable, unstable, dissonant, consonant, harsh, or pleasant" intervals, we're more or less talking about the harmonic relationships between notes.  While a lot of music is subjective, there is an underlying objectivity to it, as well.

Ok.  Next time we'll get into the arbitrary symbols and nomenclature of music theory.  Also, if this means little and explains less to you, let me know.  I wouldn't want to waste your time.















¹Due to equal tempered tuning.  Never mind that for now.
²440x4 is 1760, which is 880x2, so we're back to the octave on that one.
³See footnote 1.

[rong]

For me, music theory is easiest on a piano.  Pianos are kinda set up for C major.  If you play all the white keys, starting and ending on C, its the C major scale.  A is the relative minor of C major. That means if you play all the white keys (same notes, different order, as C major) it is the A minor scale.  This leads to modes.  All the white keys, starting on D is D dorian. Starting on E is phrygian.  I forget which ones are lydian, mixolydian, aeolian, etc.

I think this is a good place to start - I will post more when I have computer access instead of trying to do it on my phone.

[Roaring Biscuit!]

re LMNO:

equal tempered tuning is where someone realised that your example was really impractical because when you derive your intervals using 3/2 ratio the second octave doesn't match up to the first?  So today the interval between semi-tones is 2^1/12 (or the twelth root of 2), which is mathematically perfect, but doesn't subscribe to the same logic that Western music had been using previously, which was largely built off greek traditions (where the 3/2 ratio came from), which only used one octave, so never had that problem.

There was another "fixed" tuning that basically applied a correction every octave, can't remember what it was called though, quite popular in the 15th century I believe

xx

[rong]

Speaking as a recovering mathematician, I agree that discussing music theory in terms of frequency is interesting, but not very practical for the aspiring trombonist.

After some consideration, I think the next logical step would be to illustrate that the written musical staff also correlates to the piano.  The spaces and lines directly represent the white keys. (unless there's a key signature indicated). The default key is C major.

[LMNO]

RB: yes. Equal temperament solved some of the weirdness. However, I wanted to start with a pure example, because everything after that is arbitrary.

[Freeky]

LMNO, I wish I could say that your explanation was insightful and helpful.  I really, really do, because you are awesome and I rarely have a chance to interact with you in a meaningful way. 

But all it made me do was go "duuuuuuh @____@"

Sorry about that. 

But hey, let's talk about nomenclature and symbols.  That sounds interesting and important.

[Triple Zero]

Well, FWIW, I found it really interesting (so much that I'm going to have to re-read it later on), so it wasn't entirely for nothing

Cause for me it's kind of the other way around, I tried to get a better understanding of music theory, read a couple of things, but the mathy bits were in fact the only parts I could make sense of So I know:

- that A is defined at 440Hz
- that frequency ratios made of small fractions such as 1:2, 2:3 and 5:3 sound pleasing to the human ear
- music theory calls them "intervals" instead of "ratios" and it is customary to put the big number first so 2:1, 3:2 and 5:3
- the 1:2 interval is called an "octave" because "oct" means eight and on a piano there's 8 white notes in an octave IF you count the C twice: C,D,E,F,G,A,B,C¹
- According to this list (sorry it's the Dutch one but the same page on English wikipedia also lists all sorts of really obscure intervals and is therefore quite hard to read), the 3:2 is called "perfect fifth" (kwint), 5:3 is the "major sixth" (grote sext) and 5:4 is the "major third" (grote terts) and to finish off the list of "small integer frequency ratios" there's 4:3 the "perfect fourth" (kwart) and 5:2 the "major tenth" (grote decime).²
- and finally that the twelfth root of two ratio per half note, with 12 half-notes in an octave, gives frequency points that get pretty close (not really, but that's the theory) to all these ratios of small integers.

Things that confuse me here:

¹ this one is not that confusing, as long as you know it is completely arbitrary, and just is that way because that's how it is. Problem I had with a lot of introductions to music theory texts is that they explain everything as if it's completely logical and obviously follows from the theory discussed so far. I've no problem with arbitrary, but tell me when it's arbitrary and tell me when it's actually based on previous (possibly arbitrary) assumptions.
That's what I liked so much about Dion Fortune's The Mystical Qabalah, it was really clear about when it laid completely arbitrary foundations ("the Sephiroth looks like <this> graph and its vertices and edges tell a symbolic story that roughly goes like <this>") and when it applied (some sort of) logical reasoning to build on these foundations via associations etc.

² for example, I just explained why (I think) the 2:1 ratio is called oct-ave ("eighth"), but that doesn't really give me any clue why the other "ratio of small integer" intervals got theirs? In the same way perhaps?
Let's see, the 3:2 is the "perfect fifth", here's a table of the half notes (semitones?) in the octave:

C   ▇  1.00000
C^#  ▇  1.05946
D   ▇  1.12246
D^#  ▇  1.18921
E   ▇  1.25992 ~ 1.25 = 5/4 major third
F   ▇  1.33484 ~ 1.333 = 4/3 perfect fourth
F^#  ▇  1.41421
G   ▇  1.49831 ~ 1.5 = 3/2 perfect fifth
G^#  ▇  1.58740
A   ▇  1.68179 ~ 1.667 = 5/3 major sixth
A^#  ▇  1.78180
B   ▇  1.88775
C   ▇  2.00000 = 2:1 octave

The black or white square shows what colour key it would be on a piano, and the number is the ratio according to the 12th-root-of-2 tuning

(to Freeky: the 12th-root-of-2 is the number by which we multiply the frequency every step, as you can see from the C->C# step in the table, that number is about 1.059. Because it's the twelfth root of two, repeatedly multiplying by that number twelve times, is the same as multiplying by two, as you can see from the table, the final C has twice as high frequency as the initial C.)

So 3:2, the "perfect fifth" has a ratio of 3/2 = 1.5, which is really close to the 1.49831 ratio of the C->G interval. And whoa! If you count the white keys, including both the first and the last of the interval just as with the octave, there's FIVE of them! ZOMG PINEAL 2:3 LAW OF PERFECT FIFTH

I don't trust it yet, let's check. 5:3, the "perfect major sixth" is 5/3 = 1.667, which is kinda-sorta close to the 1.68 between C->A. And that range on the piano indeed spans six white keys!

So what's major about it? According to Wikipedia, the "minor sixth" is 8:5 = 8/5 = 1.6, that's closest to the 1.587 of C->G#. Because it ends on a black key, it's just one step short of spanning six white keys. Is that what "minor" means?

So what's "perfect" mean and why is there a minor and a major sixth but not a minor and major fifth?

Or is that one of those arbitrary things? I see in this list
http://en.wikipedia.org/wiki/Interval_(music)#Main_intervals
that for the "major" intervals go one semitone down and you get the minor version, but with the "perfect" intervals you can go either one down or up, and get the "diminished" and "augmented" versions.
Sort of, kind of.
And then you get that the major third is the same as a diminished fourth, at least that's how it looks in the table. Exactly the same?
The perfect fourth 4:3 = 4/3 = 1.333 = C->F, add one semitone to get the augmented fourth, which is the same as the diminished fifth. Again, exactly the same? Or is there some slight difference in frequency that does not get expressed in this method of obtaining notes intervals and frequencies? I thought I read something along those lines, once. If that is so, then what is this frequency difference based on?

Okay, awesome, I figured out most of that whole naming system thing is based on counting the white keys on a piano. SWEET.

*Musings on this*, 12 years ago, in my demoscene days, I wrote some software synthesizer code. And because my specialty was 4kb demos, I had to make those reeeeeeally tiny. And for that reason I was quite happy that I just needed to store the numbers 440 and 1.05946, repeatedly multiply the first with the second and you got a nice list of all the semitones you could possibly need. (actually I started 3 octaves lower at 440/8=55, because of, well, BASS ).

And just sort of like the MIDI spec, I could code the index of the semitone into one byte (0-255) and span more than enough octaves. So a melody of, say, 8/4 beats, would code into just 8 bytes. Not bad. (not counting the timing data and of course the 303-inspired lowpass-resonance filtersweep cutoff envelope)

But now I wonder, isn't this sort of throwing out the baby with the bathwater? If all you actually want is frequency ratios of small integer fractions, and you're not tied to piano keys because this is your homebrewn softsynth and it'll do whatever the fuck I tell it to, couldn't I just code the melody as frequency ratios instead? It would be more accurate than this 12th-root-of-2 stuff too!

Because instead of one number between 0-255 (8 bits) you can also fit two numbers between 0-15 (2x4 bits) into one byte. That seems to be enough to encode most small integer fractions you could need, right? And your intervals would be bloody spot-on.

Would that be awesome or would that sound really weird? I recall reading somewhere that since most musical instruments--especially in mainstream popmusic--are tuned with the 12th-root-of-2 scale, our ears are so used to it that hearing the actual proper intervals the way they were intended would actually sound out-of-tune to us? Or am I confusing some things now, it could have also been some medieval scale tuned even more different.

Ok this is turning into a long rambling post, but I'm kinda happy to find that I actually know more about this stuff than I thought Not bad given my musical abilities are limited to playing the first phrase of Yankee-Doodle on a keyboard, slowly, after five failed attempts, one-finger hunt-n-peck style like a very bad typist

Finally, at the top of this post I said I was gonna read LMNO's post again later. Because I didn't immediately understand some bits but then I wrote all of the above and read back your post and I got it (it was the "A=440x3" bit that was kind of ambiguous, better write "A@440Hz x 3" or "A(440) * 3"--take this advice on formula-text notation from a seasoned programmer: binary operators in infix notation shall be surrounded with a space on either side for readability )

Well there's about 99999 things I want to try out and discuss, but I'll save that for another post.

[Nephew Twiddleton]

I got kinda lost there trip but my guess is that the fourth and the fifth are called perfect because they harmonize well (your standard rock guitar chord or power chord consists of a root its fifth and sometimes the octave) as for the reason for diminished and augmented im not sure. And ive only really heard of the minor sixth refered to as augmented fifth but some of the nomenclature can get shifted depending on the root note. An a minor sixth chorf is identical in spelling to an f major seventh but the root note is different. Also fun fact the diminished fifth is in the middle so that b flat is the dimished fifth of e and e is the diminished fifth of b flat.

[Freeky]

(to Freeky: the 12th-root-of-2 is the number by which we multiply the frequency every step, as you can see from the C->C# step in the table, that number is about 1.059. Because it's the twelfth root of two, repeatedly multiplying by that number twelve times, is the same as multiplying by two, as you can see from the table, the final C has twice as high frequency as the initial C.)

I feel like I ought to get this, but using words to describe the math is throwing me off. 

Could you rewrite this as an equation?  Is this a reasonable thing, by which I mean would I understand better what you're saying, given the discussion?  I don't even know.

[rong]

A = 440 Hz
Bb = A# = 440*2^1/12 Hz
B = A## = 440*2^1/12*2^1/12 Hz = 440*2^2/12 Hz
.
.
.
A (one octave higher) = 440*2^12/12 Hz = 440*2 Hz = 880 Hz

anyhow- getting back to written music and the white keys.  once you start thinking of the notes in terms of their number, instead of their letter, you can write music in any key just as easily as you can write it in C. you just have to put the appropriate key signature (sharps and flats at the beginning of the staff) and realize that the first note, or the "1" or "I" or "i" starts on the corresponding line or space.

[Triple Zero]

(to Freeky: the 12th-root-of-2 is the number by which we multiply the frequency every step, as you can see from the C->C# step in the table, that number is about 1.059. Because it's the twelfth root of two, repeatedly multiplying by that number twelve times, is the same as multiplying by two, as you can see from the table, the final C has twice as high frequency as the initial C.)

I feel like I ought to get this, but using words to describe the math is throwing me off. 

Could you rewrite this as an equation?  Is this a reasonable thing, by which I mean would I understand better what you're saying, given the discussion?  I don't even know.

Well, yes. By which I mean you'd understand it better (cause indeed I didn't take enough care to word it most clearly since the post was already getting much longer than I anticipated ) but given the discussion--well this is the sort of thing I like to think about, understanding it might not directly help you with what you asked in the OP ITT, but extra understanding certainly can't hurt.

Ok here's with formulas.

So if you have x² and you take the square root of that, you get x back right?

  square_root(x²) = x

And if you have x³ and you take the cube root, you again get x back, right? (your calculator might not have a cube root button. Windows Calculator switched to "scientific mode" does, though)

  cube_root(x³) = x

Now if you have x¹² and you'd take the 12th root, you'd also get x back. Remember that taking the square root of x is the same as doing x^1/2? and taking the cube root of x is equivalent to doing x^1/3, and indeed, taking the 12th root of something is like doing x^1/12. Guess what? That's what the x^1/y button on the scientific-mode Windows Calculator is for!

So

  twelfth_root(x¹²) = (x¹²)^1/12 = x

So what's that mean? Let's first calculate the twelfth root of two (you can check this with Windows Calc):

  twelfth_root(2) = 2^1/12 = 1.059463

Now remember the musical note A above middle C has a frequency of exactly 440 hertz? What frequencies would the next notes after that A have? So that's A, A#, B, C, C#, D, D#, E, F, F#, G, G#, A2. Spanning one octave in semitone steps. I call the final A "A2" because it's one octave higher (dunno what the proper musical notation is for that).

So we know that

A = 440Hz

We also know that

A2 = 880Hz

Because it is exactly one octave higher, so twice the frequency. But what about the semitones in between? This is where the twelfth root of two comes in handy! Let's call it R, okay? So

  R = 1.059463

Now check this, stepping through the octave, one semitone at a time:

A  = 440 * R⁰ = 440.00
A# = 440 * R¹ = 466.16
B  = 440 * R² = 493.88
C  = 440 * R³ = 523.25
C# = 440 * R⁴ = 554.37
D  = 440 * R⁵ = 587.33
D# = 440 * R⁶ = 622.25
E  = 440 * R⁷ = 659.26
F  = 440 * R⁸ = 698.46
F# = 440 * R⁹ = 739.99
G  = 440 * R¹⁰ = 783.99
G# = 440 * R¹¹ = 830.61
A  = 440 * R¹² = 880.00

At each step, we multiply the frequency of the previous semitone by R, and wonder of wonders, after doing that twelve times, we find that the frequency (880) has exactly doubled compared to the start (440)!

Because that's just how the twelfth-root-of-two rolls



HEY another question to people that might know this: Why is it that the octave generally starts at C and not A, like any normal human being would expect? Sure it's just modular arithmetic, but what if I told you that I developed a new decimal system, it's just the same as the normal one, except that, well, you know how everybody starts counting at zero or one like the fucking mainstream sellouts that they are? Yeah? That's right, this motherfucking decimal system starts counting at motherfucking three. Like a boss. Remember, we started counting at three, before everyone was doing it. Watch this space.

[Nephew Twiddleton]

My baseless guess is that because the key of c major is identical to the key of a  minor is that people based theyre music on more scales first. I submit gregorian chant as evidence since that was the first written music. And if i recall normal speech patterns are generally in c major.