Definitions of tuning terms: tetramu, (c) 2003 by Joe Monzo

Definitions of tuning terms
© 1998 by Joseph L. Monzo
All definitions by Joe Monzo unless otherwise cited

tetramu, 4mu

A term I and a few other tuning theorists coined in July 2003, based on an idea by Aaron Hunt, to describe one a family of terms referring to units of resolution in MIDI tuning, in which the prefix specifies the exponent of 2 which describes the number of MIDI tuning units per semitone and the final "mu" is an acronym for "MIDI unit".

At the setting for tetramu pitch-bend resolution, a semitone is divided into 2⁴ = 16 pitch-bend units. Thus there are 16 * 12 = 192 tetramus in an "octave", so the tetramu measurement system may be thought of as 192-EDO tuning, with a tetramu being one degree of 192-EDO.

The internal data structure of the tetramu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or -) showing the direction of the pitch-bend up or down, and two other bits which are not used, as follows:


  let "d" designate the bits that cannot be used
  because it is reserved for the SysEx flag, to
  indicate that this is a byte of pitch-bend data.

  let "s" designate the bit that represents the
  sign of the pitch-bend data, + or - .

  let "x" designate unused bits
  

  the tetramu spec thus uses a total of 2+4 = 6 bits.

  thus, the maximum possible value is:
  
      dsxx 1111  [binary]

    =  +/-    F  [hex]

    =  +/-   15  [decimal]

    note that the first nibble can only indicate the sign + or - .

Therefore the tetramu gives a range of possible values from 0 to +/- 15 [decimal] = 0 to +/- F [hex]. Even for humans (at least, those familiar with hexadecimal numbering) this provides a convenient measurement as it only requires one hexadecimal digit to show deviation from the 12edo MIDI-note.

A tetramu is calculated as the 192nd root of 2, or 2^(1/192), with a ratio of approximately 1:1.003616666. It is an irrational number, but is extremely close to the 555:553 ratio (3¹ 5¹ 7^-1 37¹ 79^-1): a tetramu is only a little more than ¹/₁₂₀ of a tetradekamu (~ ¹/_20,000 of a cent) larger than 555:553, which for all intents and purposes makes it identical to that ratio.

A tetramu is

approximately ¹/₄ of a 53edo, syntonic, or Pythagorean comma,
exactly ³/₈ ( 0.375, ~ ¹/₃ ) of a 72edo-moria,
exactly 1 ⁹/₁₆ ( 1.5625, ~ 1 ¹/₂ ) 300edo-savarts,
exactly 3 ³/₁₆ ( 3.1875, ~ 3 ¹/₅ ) 612edo-schismas,
exactly 5 ⁵/₂₄ ( 5.208333..., ~ 5 ¹/₅ ) milli8ves,
exactly 6 ¹/₄ ( 6.25 ) cents,
exactly 55 ⁵/₂₄ ( 55.208333..., ~ 55 ¹/₅ ) türk-sents.

The formula for calculating the tetramu-value of any ratio is:

tetramus = log₁₀(ratio) * [ (2⁴ * 12) / log₁₀(2) ]

For practical use in tuning MIDI-files, an interval's semitone value must first be calculated. The nearest integer semitone is translated into a MIDI note-number (which can generally also be described by letter-name plus optional accidental: A, Bb, C#, etc., followed by an "octave" register-number). Then the remainder or deficit is converted into tetramus plus or minus, respectively.

The Pythagorean "whole-tone" (9:8) and "major-3rd" (81:64) are +1 tetramu larger than the nearest 12edo MIDI-note, 5:4 and 15:8 are -2, 7:4 and 7:6 are -5, and 11:8 and 11:6 are -8 or +8 depending on which of the two neighboring MIDI-notes is used.

Definitions of tuning terms © 1998 by Joseph L. Monzo All definitions by Joe Monzo unless otherwise cited

Definitions of tuning terms
© 1998 by Joseph L. Monzo
All definitions by Joe Monzo unless otherwise cited