Make a timbre fits 15-ET

This blog article is a snippet of my article, for more info, please visit: https://docs.google.com/document/d/1LGTcZN83AEFCCjSqfKJOCeIrQKVmdZ8BEUwbBSYOec4/edit?usp=sharing

The partial pattern of most strings and wind instruments is in the pattern of harmonic series. In the harmonic series, the 2nd and 3rd partials combine in the ratio of a perfect fifth, 3/2. Stacking the fifth 12 times roughly returns to the original tone in octaves. The 12-ET is the tool to solve the Pythagorean comma. That is the background of why 12-ET fits normal instruments.

The challenge right now is I need to build a partial pattern that fit 15-ET. Comparing 12-ET and 15-ET, intervals that line up with each other are major 3rd and octave. Current instruments with harmonic series partials sound really out of tune. The partials have to be rebuilt artificially. 

The process of building the partial pattern

First, calculate all the ratios of 15-ET.
15-ET
1
1.0473
1.0968
1.1487
1.2030
1.2599
1.3195
1.3819
1.4473
1.5157
1.5874
1.6625
1.7411
1.8234
1.9097
2

Second, get all the intervals from the natural harmonic series and the ratios, in the first 2 columns. Compare the ratios of 12-ET and 15-ET, and pair similar ratios together.

IntervalRatioFrom the harmonicsRatio: 15 ET
Octave21 and 22
Fifth1.52 and 31.515716567
Fourth1.33333 and 41.319507911
Major third1.254 and 51.25992105
minor third1.16676 and 71.148698355
Minor seventh1.754 and 71.741101127
major second1.14297 and 81.148698355
minor second1.090911 and 121.09682498

Third, rebuild the spectrum for 15-ET based on the pattern of natural harmonic series.
Harmonic series from 12-ETRatios for 15-ET spextrum
fundamental1
oct2
oct+5th3.031433133
2 oct4
oct+5th+4th+M35.0396842
oct+5th+4th+m76.964404506
3 oct8
oct+5th+4th+m7+M2+M29.624200289
oct+5th+4th+m7+M2+M2+M2+m212.12573253

Dissonance curve of the tailor-made 15-ET timbre
The result above demonstrated this timbre when two notes, and different intervals create different levels of dissonance and consonance. From the left, 1:1 creates a low level of dissonance; similar to ratios like 1:1.26, 1:1.32, 1:1.52, 1:1.66, 1:2 etc. The 15-ET can capture the ratios -- fits the timbre.

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