# 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

1. The number up/down whatever octaves [h*(2^x)]. (e.g., 1/4, 3/6, 26/13)
2. A power of 2. (e.g., 2/6, 8/9, 16/7)
3. 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.