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Scales, Chords, Frequencies.
Hi Everyone here at iBreatheMusic.
I've been teaching myself music production for a while now, mostly with the use of a MIDI keyboard. That made it really easy for me. It didn't matter that I didn't know how to play keyboard, it was more important to me to train my ears and learn what a compresser is and how to EQ. My piano skills stayed at the very basic level because MIDI makes it easy. I can see what note I'm playing and correct any wrong ones after.
I wanted a musical form of expression though and MIDI wasn't cutting it so I picked up my old guitar. I found that it forced me to learn a lot more music theory a lot quicker as it isn't all spelled out nicely for you like a keyboard and all the frets look essentially the same, whereas on a keyboard you can see what a 'C' looks like because of the black notes around it.
So I learnt a bit more and decided to transpose my "G" chord up the fretboard to hear it higher up and found that the two chords sounded nearly identical. This really annoyed me because logically there is no way the chords should have been the same.
I thought about it and thought they might average out to be the same. I checked the frequency values of each of the notes in the two chords and tried to work out the averages. Nothing added up and the averages didn't correlate. There were no similarities.
So I looked at the two chords again, "G1=B1=D2=G2=B2=G3" and "G2=B2=D2=G2=B2=G4" And I saw that there are two G2's and two B2's. It seemed sensible to remove the duplicates. I added up the 6 frequencies from the first chord and the 4 unique ones from the second. At this point I was just trying to find a point of correlation.
I divided them both by 6, and found that done like this the first averaged at 100.27Hz and the second at 114.34Hz.
This seemed like it was close enough to explain why the two chords were more or less the same.
However I have only found a point of correlation. This doesn't satisfy me. I'm sure I have made mistakes somewhere, or have oversimplified things. It seems like there is something correct in there, and there's more than just chance at work, with music's mathmatical basis and the close link between notes and frequencies (G1 being half the frequency of G2). At this point I felt like I had to ask for help, so signed up here to see if someone can shed some light on it for me..
I lack the musical knowledge to explain to myself what happened and my maths is really rusty, but I think I sort of see why. I'm sure I explained this horribly, so thankyou if you've got this far. What feels most incomplete about my calculations is the dropping of the two duplicate notes in the second chord. Everything sort of made sense mathamatically and musically up until that point.
I gotta admit, It's the most fun I have ever had doing maths
A couple of things...what are the exact voicings you're comparing? I take it the first is an open G (320003, where the numbers indicate the fret played on each of the strings low-high)?
Second, by G1 G2 etc. I assume you're talking about Scientific Pitch Notation...in which case I'll note there are no G1 (49Hz), B1 (62Hz) or D2 (73Hz) notes playable on a guitar in standard tuning; the lowest note (open low E) is E2 (82Hz).
(If you're coming from a piano background, keep in mind standard notation for guitar is written an octave higher than it actually sounds (for readability purposes, keeps everything on the treble clef) - which may possibly be throwing you off?)
So, a 320003 G chord is G2 B2 D3 G3 B3 G4
Let me know the voicing you're playing for the second chord (what fret for each string) and I'll have a look from there...
Last edited by walternewton; 09-21-2010 at 07:20 PM.
You are quite correct about the octave difference - that the frequency doubles every octave.
Originally Posted by AlexQuatermain
So bottom G on guitar is 98 Hz, the open G string is 196, and 3rd fret on top E is 392.
Same applies to the B's and D's.
But something's gone wrong with your maths somewhere. I don't really know why you want to "average" the pitches. And in any case you are not comparing like with like if you include all 6 pitches of the first chord and only 4 from the second. (The fact that 2 pitches overlap is not relevant.)
So you have come up with an answer that doesn't refer to anything. (Those average frequencies are meaningless.)
If you were to add ALL the frequencies of the first chord, and ALL the frequencies of the chord an octave up, the latter would be double the first. That might seem too obvious (because the difference is obviously an octave). Of course, if you then wanted to divide the totals by 6, the higher answer would still be double the lower one.
So it looks like you've been looking for an answer in the wrong place - and in any case I don't really understand your question. I dont know why you think the chords ought to be more different than they are. In one respect (musical functionality) they are identical. They are both G chords in the same voicing. The only difference is that one is an octave higher than the other, so the only mathematical "explanation" necessary is that the frequencies have all doubled.
It is true that there can be some kind of psychoacoustic difference, due to our threshold of pitch perception. We don't hear frequencies below 20 Hz as pitches at all. More relevantly, we don't hear frequencies above around 4000 Hz as musical tones. (Healthy young ears will hear frequencies up to 20,000 Hz, but between 4K and 20K it's all timbre - harmonics of lower pitches - and non-pitched noises.) So the higher the chord, the more of its upper harmonics will disappear from human perception. The upper harmonics - although very faint - will give a chord more "colour". So (hypothetically at least) notes in higher registers ought to sound purer.
The other (more crucial) mathematical aspect is the relationship between scale notes, particularly chord tones. IOW, how B and D relate to G and to each other.
Our sense of "harmony" - of what notes blend smoothly when played simultaneously - derives from simple ratios between note frequencies. (This in turn relates back to the harmonic series, because notes in simple ratios will share some of their harmonics, and therefore have a perceptible acoustic affinity.)
G-B is a "major 3rd", and the frequencies relate in a 4:5 ratio;
B-D is a "minor 3rd, and the ratio is 5:6;
G-D is a "perfect 5th", the strongest interval in the chord, because it has the simplest ratio (after the 2:1 octave): 2:3.
Altogether, the 3-way ratio between a G-B-D triad is 4:5:6.
All 3 notes also relate back to a "virtual" low G of 98/4 = 24.5 - the hypothetical "1" in the 4:5:6 ratios.*
However, this is in an ideal "pure" world. The demand for 12 equal divisions of an octave mean that these ideal ratios have to be detuned slightly. This is known as "equal temperament", and is how pianos are tuned and how guitar frets are set. The mathematical figure there is the not-simple-at-all 12th root of 2: 1.0594631 (etc). If you multiply one pitch by that number, you get a semitone higher; do it 12 times and you end up at an octave higher.
Here are the two sets of pitches, based on the same G root, for comparison:
Pure G triad: G = 98.00; B = 122.50 (5/4xG); D = 147.00 (3/2xG)
Equal tempered G triad: G = 98.00; B = 123.47; D = 146.83.
Our ears have a tolerance that means we don't perceive the latter notes as being "out of tune" - although if we heard the first set, we'd probably say they made a "purer" chord.
A digital tuner would tell you that a B of 122.5 was 14 cents flat - that's 0.14 of a half-step. That's because they are programmed for equal temperament.
The pure D, meanwhile is only 2 cents out.
This is why - incidentally - rock guitarists use power chords. Distortion enhances all the harmonics, and brings the 14 cent clash between the 5th (5x) harmonic of the root and the octaves of the major 3rd into sharp relief. The musicians don't understand the math, but they hear that nasty muddy sound - and they fnd if they leave out the 3rd of the chord, they're left with a good strong perfect 5th (a negligible 2 cents out).
* If you want all the frequencies in a full 1:2:3:4:5:6 harmonic series of that low G, you come up with the following:
1 x 24.5 = G
2 x 24.5 = 49 = G
3 x 24.5 = 73.5 = D
4 x 24.5 = 98 = G
5 x 24.5 = 122.5 = B
6 x 24.5 = 147 = D
- so you can see how the 4th , 5th and 6th harmonics of one note will produce a major triad for that root (this is the case for any note you choose, of course). That's why major triads sound so "right", the basic building block of harmony.
Last edited by JonR; 09-21-2010 at 10:47 PM.
I have to agree, I am a bit amazed with all of these info you guys are sharing. But I do know one thing bout power chords. Basically it is a chord consisting of only the root note of the chord and fifth interval that is why power chords are also know as the Fifth Chord. They are the main elements for rock music.
Originally Posted by JonR
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