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

INTRODUCTION:
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). 
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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 ...
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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.
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MOST HARMONIC NUMBERS & TWELVE TONE EQUAL TEMPERAMENT

The intervals between the tones when calculated using the Pythagorean Interval (perfect fifth) 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 12-TET A4 FREQ.
DO  C 512  -37.6312241210787  430.5 
SI / TI B  486  -27.855595894044  433
  Bb / A#  461.320
-18.0837059915098   435
LA A  432  -31.7666536334292  432
  Ab / G# 410.063
-21.9907990325915   434
SOL G  384 -35.6747298650793   431
  Gb / F#  361.500
 -25.8996731339595  433.5
FA F  345.990  -16.1266721682703  436
MI E  324 -29.8139799005726   432.5
  Eb / D# 307.547
-20.0360353716339   435
RE D  288 -33.7230226885739   431.5
  Db / C#  273.375
-23.948955024014   434
DO C 256  -37.6345327289834   430.5

In the table above we find the following columns:

  1. Solfeggio Syllables.
  2. Tone names.
  3. Most Harmonic Frequency.
  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.

[1] SOLFEGGIO SYLLABLES

"In music, solfège (French pronunciation: [sɔl.fɛʒ], also called solfeggio, sol-fa, solfedge, or solfa) is a pedagogical solmization technique for the teaching of sight-singing in which each note of the score is sung to a special syllable, called a solfège syllable (or "sol-fa syllable"). The seven syllables commonly used for this practice in English-speaking countries are: do (or doh in tonic sol-fa),[1] re, mi, fa, sol (so intonic sol-fa), la, and ti/si ..."

IMPORTANT! When we speak about the "Solfeggio Syllables", we are NOT talking about "The Ancient Solfeggio Frequencies" as proclaimed in the tuning-theory by Dr. Joseph Puleo and Leonard G. Horowitz. Their theory has no relationship with 432-related tuning.

"In the eleventh century, the music theorist Guido of Arezzo developed a six-note ascending scale that went as follows: ut, re, mi, fa, sol, and la. A seventh note, "si" was added shortly after. The names were taken from the first verse of the Latin hymn Ut queant laxis, where the syllables fall on their corresponding scale degree."

There are two main types of Solfège (Wikipedia):

  1. Movable do, or solfa, in which each syllable corresponds to a scale degree. This is analogous to the Guidonian practice of giving each degree of the hexachord a solfège name, and is mostly used in Germanic countries.
  2. Fixed do, in which each syllable corresponds to the name of a note. This is analogous to the Romance system naming pitches after the solfège syllables, and is used in Romance and Slavic countries, among others.

 

[2] TONE NAMES

Solfeggio Syllable tone connection ("Fixed Do" system):

  • C (Ut / Do)
  • D (Re)
  • E (Mi)
  • F (Fa)
  • G (Sol)
  • A (La)
  • B (Si / Ti)

For more information, continue reading at: http://en.wikipedia.org/wiki/Solf%C3%A8ge

[3] MOST HARMONIC NUMBERS / FREQUENCIES

Not all "Most Harmonic Numbers" are located between C4 and C5 (the middle register on the piano). I have brought all 12-tones back within the middle register and marked the tones located outside the middle register with "[ ]"