User:CritDeathX/Sam's Idea Of Consonance
Okay, so a while back, I had developed this method of finding how consonant a chord was. The basic idea was to find how many combination/difference tones lined up with a chord.
Linear Tones
Formula
In order to find what tones line up with the chord, the resulting tone must either be
- The number up/down whatever octaves [h*(2^x)]. (e.g., 1/4, 3/6, 26/13)
- A power of 2. (e.g., 2/6, 8/9, 16/7)
- Another note/interval within the chord. (e.g., in the case of 4:5:6 [which will be demonstrated soon], 10/4)
Thus, by these rules, it cannot line up if it results in any other harmonic/interval.
Demonstration
I'll show an example of what I mean using 4:5:6.
5-4 = 1; 1/4 = lines up with 4
6-4 = 2; 2/4 = lines up with 4
6-5 = 1; 1/5 = lines up with 5
5+4 = 9; 9/4 = doesn't line up
6+4 = 10; 10/4 = lines up with 4:5
6+5 = 11; 11/5 = doesn't line up
As you can see, 4 of these tones line up with the base chord. To add a further reference for consonance, I suggest comparing how many tones line up with a chord compared to a unique x-note chord built off of 4. So for 3 notes, its 4:5:6, 4 notes is 4:5:6:7, 5 notes is 4:5:6:7:9, etc. Here's a list of how many tones line up with a(n) x-note /4 chord:
4:5:6 = 4
4:5:6:7 = 8
4:5:6:7:9 = 15
4:5:6:7:9:11 = 25
4:5:6:7:9:11:13 = 37
4:5:6:7:9:11:13:15 = 51
4:5:6:7:9:11:13:15:17 = 67
4:5:6:7:9:11:13:15:17:19 = 85
4:5:6:7:9:11:13:15:17:19:21 = 105
...:23 = 127
...:25 = 151
...:27 = 177
...:29 = 205
...:31 = 235
etc...
An interesting thing to note is that after the 5-note chord, the amount of times that the linear tones line up rises linearly by 2x.
To show an example of this method with the added reference, I'll show how 9:11:13 works here.
11-9 = 2; 2/9 = lines up with 9
13-9 = 4; 4/9 = lines up with 9
13-11 = 2; 2/11 = lines up with 11
11+9 = 20; 20/9 = doesn't line up
13+9 = 22; 22/9 = lines up with 9:11
13+11 = 24; 24/11 = doesn't line up
We then compare it to the 3-note reference point, 4:5:6. 4:5:6's tones line up 4 times, and 9:11:13's tones line up 4 times as well. By this conclusion, 9:11:13 should be as consonant as 4:5:6. (its also proportional like 4:5:6, so take that as you will)
I should note that they may not sound consonant on first listen, but if you were to hear it for a long enough time, you'd notice how weirdly consonant they are. I think this might be a useful method for finding alien harmonies without sacrificing the idea of consonance entirely.
Harmonal Limits
This is a term that I'm coining to describe certain patterns or outliers within a series of chords or a scale(s) based off of their linear tones.
Chord Progressions
Here's an example progression using a Fibonnaci sequence from Erv Wilson's letter to McLaren. We can look at the linear tones of these chords and see whether we can notice anything interesting:
| 1st chord | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|
| 8 | 8 | 9 | 8 | 8 | 8 | 8 | 10 |
Although its hard to figure out how to represent harmonal limits, I think the best way to do it is to have ([lowest number of unique notes in a chord]-[highest]) on the denominator and have ([lowest linear tone]-[highest]) on the denominator. In this case, it would look like (8-10)/(3-4), which if we were to translate this into a proper number it would equal 2.
Singular Chords
When it comes to just one chord, though, it'd be best to rely on which note most of the linear tones are found. For example, 9:11:13 in the last headliner had most of its linear tones occur on 9, so since 3 linear tones appeared on 9, we can say that the harmonal limit for this chord is 3.