Post by Joey GoldsteinThanks guys, but I don't think that either of you are really getting the
scale or the chord I'm talking about.
You're both talking as if the ratio of the m7 on the V7 chord can be
adjusted to some other ratio than 16:9.
There seems to be a variation on this interval. In addition, one gets
different Just major and minor scales:
If the Just major scale is constructed by requiring that the tonic,
dominant and subdominant major thirds are Just 4:5:6, then one of the
minor triad get out of tune. In C major, it is Dm, I recall. Similarly,
one can construct a Just minor scale by requiring its minor triads being
1:6/5:3/2 = 10:12:15, but then one of the major triads go out of tune.
This is described on the WP pages "Just intonation" and "Five-limit tuning".
Post by Joey GoldsteinIn the major key just intonation I'm talking about the b7 of V7 is
determined by the pitch of the root of IV, and the pitch of the root of
IV is determined by the root of IV being in a 3:2 relationship with the
tonic of the key.
Let's compute this interval. Just major is in key C
Key C D E F G A B C'
Degree 1 2 3 4 5 6 7 8
Just major T t s T t T s
where
s = mj = 16/15
t = Mj = 10/9
T = MJ = 9/8
All major triads are in tune, so G, to D is 3/2. D to F is s = 16/15 and
t = 10/9, which is 32/27. And this latter and 3/2 is 16/9.
Post by Joey GoldsteinMy understanding, and I guess I could be wrong about this, is that
tuning the root of IV this way will automatically result in the ratio of
the m7 within the V7 chord being 16:9 and only 16:9.
Please correct me if that's wrong and let me know what the correct ratio
of this interval is.
So in this position this is so - and it is the only dominant 7th within
the Just major scale.
Post by Joey GoldsteinMy own calculations on this are based on a JI for A major.
With A110 as the tonic, I calculate the freq of D as being D146.668.
(110 div by 3 = 36.666666. 36.667 X 4 = 146.668.)
E will, of course, be tuned to E165.
(110 X 3 = 330. 330 div by 2 = 165.)
So the m7 above E165 will be D293.336.
(146.668 X 2 = 293.336.)
So the two pitches involved in this interval are
E165 and D293.336.
And that interval is at a freq ratio of 16:9.
(16 div by 9 = 1.778. 165 X 1.778 = 293.37.)
Given a JI such as this, is there a 4-digit ratio that can describe the
intervals in *this* entire dom7 chord?
I.e. We know that its root 3rd and 5th are at 6:5:4.
When we add the b7, which is at 16:9 in relation to the chord's root,
what is the 4-digit ratio that correctly describes the entire chord?
It's not clear to me (see below) if Hans meant to answer this with
"36:45:54:64", or not.
Yes.
Post by Joey GoldsteinAgain, folks, as always, my math skills are *really* minimal, limited to
basic arithmetic I'm afraid.
So please keep any maths in your posts to a minimum.
It looks like Hans was talking about ways to multiply ratios, and even
that is beyond my math skills, unless it is explained to me.
So take this really easy. Somehow we got 36:45:54:64. Take away 64,
which is 36:45:54. Divide all numbers by their greatest common divisor
(gcd), which is 9, to get 4:5:6. So this part is the Just major chord.
And 64/36 = 16/9, so this means that the 64 in relation to the first 36
is the Just minor 7th under discussion.
Now compute frequencies. Use either of the forms 36:45:54:64 or
1:5/4:3/2:16/9. The latter is in fact easier.
If A2 = 110 Hz, then E3 = 110*3/2 = 165 Hz, so E2 = 165/2 = 82.5 Hz.
Then D3 = 82.5*16/9 = 146.667 Hz, which is what you got above, except
for the rounding.
Alternatively, one can use a table:
P1 M1 M2 P4 P5 M6 M7 P8
1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
A B C'# D' E' F' G' A'
So if A2 = 110 Hz, D3 = 110*4/3 = 146.667 Hz.
Post by Joey GoldsteinActually, looking at Hans' explanation of how he arrived at 36:45:54:64
is beginning to make sense to me, but I can't understand how he
calculated the "least common denominators".
It should have been the greatest common multiple, I think.
Post by Joey GoldsteinWhat does the symbol "^" mean?
Exponentiation. 2^3 = 2 three times = 2*2*2 = 8. 3^2 = 3 two times = 3*3
= 9.
Post by Joey Goldstein[I can't even find it on my keyboard and had to paste it in from Hans'
post. lol]
It is ASCII. On a US keyboard, it is <shift>-6.
Post by Joey Goldstein2^2 = 4.
But then he wrote 3^2 = 9 which doesn't make sense.
??
Multiply two times 3*3 = 9.
Post by Joey GoldsteinAnd again, my other question was about the min 3rds contained in this chord.
They appear to be of different sizes from each other to me.
Yes they are, 6/5 and the same value reduce by a syntonic comma 81/80.
Just major is uneven, due to two differently sized major seconds.
Post by Joey GoldsteinThe ratio of the m3 between the chord's 3rd and its 5th, by definition,
will be 6:5.
But what is the ratio between this chord's 5th and it min 7th?
Based on Hans' 4-digit ratio of the entire chord, would it be 64:54?
That reduces to 32:27, but no further, right?
Right: 32 = 2^5 and 27 = 3^3, so they have no common prime number
factors. But here, ":" is not a ratio, but indicates a list of numbers
that can be altered by multiplication by the same number.
Post by Joey GoldsteinAnd I wasn't even thinking about the tritone of this chord, but based on
Hans' ratio it would be 64:45, which doesn't reduce at all, right?
Right: 64 = 2^6, and 45 = 3^2*5, which has no common prime number factors.
Post by Joey GoldsteinThe rest of Hans' 4-digit ratio seems totally plausible to me.
45:36 reduces to 5:4.
54:36 reduces to 3:2.
And 54:45 reduces to 6:5.
The common denominator of all three digits (36, 45, and 54) being 9.
And 64:36 reduces to 16:9 (common denominator being 2).
So, it looks like Hans' ratio is correct, even though I don't completely
understand how he calculated it.
Thanks Hans!
You are welcome.
Post by Joey GoldsteinSo, in a major key JI the V7 chord is several times more
complex/dissonant than the I IV or V triads.
Right?
You get beats among the partials which are a bit off low numbered
harmonic ratios. The 16/9 is off 7/4, so the 7th partial of the lower
note will produce beats with the 4th partial of the upper note. The
Upper minor triad is off 6/5, so it will produce beats between the 6
partial of the lower note and the 5th of the upper note. And so on. It
gets quite complex. A pair of ears might be helpful. :-)
Post by Joey GoldsteinMight this partially account for why it is that early CPP composers,
operating in just intonations and exploring the maj/min key system (in
contrast to the earlier modal system) for the first time, treated dom7
chords as being so dissonant as to *always* require resolution?
Yes, I think so. Likewise, the +6 in quarter-comma meantone or E31 is so
stable that the only chord they can resolve to is a major triad. So
therefore, since +6 is out of the diatonic scale, one of the few uses is
as a predominant.
Post by Joey GoldsteinIt also seems to me that equal temperament has actually softened the
dissonance of the dom7 chord somewhat, which may have helped pave the
way to things like jazz musicians treating dom7 chords as consonances.
Does that make any sense?
Or am I totally off-base and making assumptions that are simply incorrect?
In E12, the M3 is off 5/4 by 13.686 cents and the m3 is off 6/5 by
15.641 cents (and the P5 is off 3/2 by 1.955 cents). So this makes the
E12 major and minor a bit unstable.
But if one can accept this instability in these chords to resolve
towards, the dominant should be more stable than the Just one using m7 16/9.
So early CPP composers may have resisted both the dominant 7th and
tempered tunings because of that instability. But as one accepts some
instability in the minor and minor chords, there is some improvement in
the dominant 7th.
So that seems a possible explanation.
Hans