Chapter 4: Harmonic Resonance
There is a universality to music cognition before the cultural, psychological, and memory influence kicks in.
However, there are no accepted standards for how to physically measure even the most basic musical qualities of consonance, dissonance, tension or resolution.
If there was a physical/mathematical model of music cognizance, we would probably have a contender for the theory of everything!
One theory of harmony ranks the intervals (from consonance to dissonance) as follows:
- Unison, octave, 5th, 4th, maj6, maj3, min3, min6, min7, maj2, maj7, min2, tritone
- 1/1, 2/1, 3/2, 4/3, 5/3, 5/4, 6/5, 8/5, 9/5, 9/8, 15/8, 16/15, 45/32
In this scheme, it is useful to think of these intervals as polyrhythms of varying 'composite cycles'. The higher LCM (least common multiple) of the numerator 'n' and denominator 'd' requires more time to 'resolve' and thus more dissonant.
However, it is more than just big numbers and small numbers.
For example, in 7 limit tuning, the tritone is 7/5 which has a relatively small LCM but is still more dissonant than say a min6 (8/5). Here the higher prime (7 vs 5) also contributes towards cognitive dissonance.
While, no perfect tuning system has ever been found, the wiggle room between perception of consonance and dissonance creates virtually infinite harmonic possibilities and blending it well is what makes music a great art form.
Inerval ratios can be visualized using circular geometry in many ways.
Play with the sliders below to create harmonic patterns for different ratios.
Fractal Harmony
Here, we can see the root note (C) set to 100 Hz and the 5-limit JI tuning used to generate the chromatic octave (100-200Hz)) The overtones (x2, x3, x4...) of each note are expanded to the right and many overtones produced are identical to the root overtones. In general, in an interval represented by n/d, every nth harmonic of the first note will be identical to every dth harmonic of the second no
12ET (equal temperament) tuning (the most popular tuning of all) divides the octave into 12 equal logarithmic steps. As a result, all the intervals are irrational (except the octave) and none of the overtones strictly overlap with each other.
However, many of these 12ET overtones are very close, as we can see these 'approximate matches' after applying an 'allowance' of ±15cents (100 cents = 1 semitone). The resultant chart of harmonic resonance is in fact very similar to that of the 5-limit JI system.
It is said that the human ear cannot detect pitch changes smaller than 8cents.
Comparison of 5-limit JI and 12-ET tunings (root =100Hz)
As you can see, all the differences are within the ±15cent range. This is one of the popular arguments for why there are 12 notes in an octave. However, a 19ET system also does a good job in approximating the simple ratios.
The 'uni-verse' is one-song, that is playing everywhere!
Can this system of harmonic resonance be applied to the whole universe?
If yes, music could be the unifying system that connects everything in a harmonic relationship, and makes up the unified whole!