Hi there Can someone please help me understand the following... I was going through a just intonation example in the textbook and am unsure how they came about it....the ratio is 4:5:6 in the key of C and they write the tonic chord is C,E,G. C=1 I get up to there...then it goes on ans says E=5/4 and G=6/4=3/2...How did they get E=5/4 is it just the fact that E is a major third above C hence 5/4...then for G=6/4=3/2...is it cos G is a major 5th above C-if so whats with the 6/4? Then it goes on to the dominant chord G,B,D in the ration 4:5:6 this determines B=5/4*3/2=15/8....How do we know what to multiply? I also don't get why they go D=3/2*3/2=9/4 can someone pleases explain that and also the reasons why it says the D is dropped into the same octave as C makes D=9/8 Then they do the subdominant chord F,A,C in ration 4:5:6. they write in the book they "work down" from C and obtain F=2divided by 3/2 and A=2divided by 6/5. why do they do it this-why can't I just use the intervals straight without dividing C=2 F=4/3 (cos it's a perfect 4th above C) and that A=5/3 cos its a major 6th above C... Someone please help me Thanks in advance physics_06er
you know added to the fact that I know nothing of this intonation example music concept, I like totally loose you on what is it exactly you are dealing with and what seems to be the problem?...
Ok, first of all there are twelve notes in an octave: C C# D D# E F F# G G# A A# B then C again, B again, etc. It repeats. Each of these notes is a semitone apart from the previous note. The interval of a fifth is 7 semitones (e.g. C to G, D to A, D# to A#). The interval of a major third is 4 semitones. Say you have a string that sounds C. If you half the length (double the frequency of the sound it produces) you'll get C an octave up. If you triple the frequency (use a string 1/3 of the length), you'll go up an octave and a fifth to G. If you go up an octave and a fifth (multiply frequency by 3) and down an octave (divide frequency by 2) you'll have used a ratio of 3/2 to go up a normal fifth to G. So 3/2 is the frequency ratio from C to G, or from D to A or G up to D, etc. Similarly, if you multiply by 5 you go up two octaves and a major third, so dividing by 4 to take you two octaves down, the ratio 5/4 goes up a major third. E.g C to E, D to F#, G to B. So now you can find the ratio from C to B: go up a fifth from C to G (ratio of 3/2) and a major third from G to B (ratio of 5/4). Thus the total ratio from C to B is (3/2)*(5/4) = (15/8). To get to D, go up two fifths from C: C to G, then G to D. This total ratio (3/2)*(3/2) = (9/4) takes you up an octave and a full tone. To go up just a full tone, you want to go down an octave so multiply frequency by 1/2: (9/4)*(1/2) = (9/8). Make sense?
In the harmonic or Pythagorean scale of G major, the frequencies of the notes are as follows: G 24hz A 27 B 30 C 32 D 36 E 40 F# 45 G 48 In any major scale these same ratios hold. It's just that if your tonic is C then you have to create F natural which will be 32*(32/24) which is not an integer. That's why we usually use G major as a demonstration key. You'll probably understand this better if you do the arithmetic for all the intervals yourself and see how they're derived. Just pretend the frequency of C is 24 and all the calculations will come out right. Notice that in a harmonic scale the frequency ratio of a whole tone is sometimes 9/8 and sometimes 10/9. However, the ratios of all the larger intervals are constant and they are "ratios of small integers," which is the Pythagorean definition of harmony. The only one that is not is C-F#, the minor fifth, which as we all know is a very "disharmonious" interval common in modalities that use unresolved progressions heavily like the blues. In the chromatic or scientific scale the ratios have all been normalized so that every semitone is 2**(1/12) and therefore every full tone is 2**(1/6). This allows the notes that do not actually occur in a scale to be interpolated and assigned standard frequencies. And you can't use the Pythagorean formulas any more. Somehow our ears work around that. A lot of study has been put into understanding this. Zephyr will straighten out my terminology. Please Register or Log in to view the hidden image!
