### MOST HARMONIC NUMBERS (MHN) a.k.a. Pythagorean Temperament

The list of frequencies in this document is based on the "Most Harmonic Numbers", brought to my attention by Jamie Buturff

The "Most Harmonic Numbers Scientific Concert Pitch" (MHN SCP) concept is better knows as the "PYTHAGOREAN TEMPERAMENT" and works for instruments that use micro-tuning, as well as fret-less string instruments and trombone (a brass/wind instrument without buttons/tone-holes/pistons).

IMPORTANT NOTE: if you think about using the MHN SCP concept / Pythagorean Temperament, then I would like to suggest using a concept developed by Maria Renold, based upon it. Why? Well, the Pythagorean Temperament (and thus the MHN SCP concept) is "flawed" by design: when stacking 12 perfect Fifths you would "overshoot" the Circle of Fifths with about a quartertone, the circle does not close. Maria Renold developed a concept that fixed the issues concerning the "wolf interval", the "false Fifth". And thus I would recommend using it instead ...

The "Most Harmonic Numbers Tonic Concert Pitch" variation is my own interpretation and implementation of the 432-Tuning concept for 12-TET instruments. I have used it with instruments that can not be micro-tuned, but can change the concert pitch. These instruments are (for example) fretted string instruments and keyboards / synthesizers. This concept can also be implemented in post production when working with instruments that can not change temperament.

### MHN RELATIVE TO TWELVE-TONE EQUAL TEMPERAMENT:

The intervals between the tones when calculated using the Pythagorean Perfect Fifth (ratio 3:2) and those calculated using the 12-TET method are rather different. Below a table that displays the modifications required when using the Most Harmonic Numbers in comparison with 12-TET at Concert Pitch A4=440Hz.

 TONES HMN FREQUENCY (Hz) CENT CORRECTION DO C 512 -37.6312241210787 SI / TI B 486 -27.855595894044 Bb / A# 461.320 -18.0837059915098 LA A 432 -31.7666536334292 Ab / G# 410.063 -21.9907990325915 SOL G 384 -35.6747298650793 Gb / F# 361.500 -25.8996731339595 FA F 345.990 -16.1266721682703 MI E 324 -29.8139799005726 Eb / D# 307.547 -20.0360353716339 RE D 288 -33.7230226885739 Db / C# 273.375 -23.948955024014 DO C 256 -37.6345327289834

In the table above we find the following columns:

1. Solfeggio Syllables (not the "Ancient Solfeggio Frequencies").
2. Tone names.
3. Most Harmonic Frequency (between C4 and C5).
4. Cent correction (in comparison with 12-TET under concert pitch A4=440Hz).
5. A4 Concert Pitch for 12-TET instruments when micro-tuning is not possible, to adjust the first degree (prime / tonic) to it's most harmonic frequency.