In my studies, I was reflecting on this theory and just wanted your guys clarification..
A pitch in an octave is comprised of a fundamental frequency followed by its overtones. When we focus on an interval, its harmonic stability or instability is evidently stemming from the interaction of the two pitches harmonic series combined. As we add additional pitches to that interval, the relationships become furthermore complex due to additional overtones combined.
If I wanted to build a chord through selecting the intervals myself that would produce the desired level of consonance/dissonance, I would have to seek each pitches overtones in that chord and compare them to each other pitches overtones in that chord which would be rather tedious.
Is it because of this that the just intonation interval ranking was developed? Was it developed with the intention of making the stated above process easier by giving us a list of intervals and their ratios dictating the two pitches overtone relationships?
Is it also true that the manner in which the interval ranking functions in its process, is through providing a ratio for an interval with the ratio stating that in the case of the ratio 3:2 (Unison:Perfect 5th) the unison pitch has to perform 3 cycles and the perfect pitch 2 cycles for their wave crests to coincide, therefore determining their level of consonance/dissonance and this process in fact displays an accurate representation/translation of two pitches overtone interaction?
This would explain a lot for me. Thank you guys.
Here is the math formula provided by WillRoss1, simplified. Here you go Albrecht Hügli.
"Formula for producing a chord ratio by any infinte amount of interval ratios."
In this example we will utilize the Major 7th Chord, with the intervals Major 3rd (4:5), Perfect 5th (2:3) and Major 7th (8:15). This formula below applies to infinite amount of intervals you want to combine by following the exact process below.
Step 1. Mutipy the first interval ratio (4:5) by the result of multiplying all the other interval ratios first numbers by each other (2 and 8). 2 x 8 = 16.
16 x 4 = 64 / 16 x 5 = 80.
Step 2. Multiply the second interval ratio (2:3) by the result of multiplying all the other ratios first numbers by each other (4 and 8). 4 x 8 = 32.
32 x 2 = 64 / 32 x 3 = 96.
Step 3. Multiply the third interval ratio (8:15) by the result of multiplying all the other ratios first numbers by each other (4 and 2). 4 x 2 = 8.
8 x 8 = 64 / 15 x 8 = 120.
Your Major 7th Chord ratio is 64:80:64:96:64:120.
Step 4. Since there are duplicate numbers (64) in the chord ratio, we can turn the ratio instead into 64:80:96:120.
Step 5. To further summarize this chord ratio, we find the common factor (a single number that can produce each of the four numbers) for 64,80,96 and 120 which would be 8.
Step 6. Divide 64, 80, 96 and 120 each by 8.
64 ÷ 8 = 8
80 ÷ 8 = 10
96 ÷ 8 = 12
120 ÷ 8 = 15.
Your Major 7th Chord ratio is now 8:10:12:15.