200-tone division of the octave

Need for equal temperament 200-tone division when using quarter-tones

In music theory and musical tuning, the equal temperament quarter point is a small musical interval of exactly 6 cents, equal to one step of the division of the octave (1200 cents) into 200 equal temperaments. It should not be confused with the Didymic or syntonic quarter comma which is used in meantone tunings and is only 5.375 cents.

The following postulates give rise to the desired optimal proportions:

• a whole tone contains 34 quarter commas,
• a chromatic semitone (addition of # or b) contains 19 quarter commas;
• a chromatic quarter tone (addition of < or >) contains 10 quarter commas.

Comparison with the 53-tone division

An octave consists of five whole tones and two diatonic semitones.

• In the 200-tone division, the five whole tones make up 85% of the entire octave, either 85/100 or 170/200 (5 whole tones of 34 quarter-commas)

• In the 53-tone division, the five whole tones provide 84.9% of the entire octave (90/106 or 45/53)

This division is thus a more refined extension of the 53-tone division of Nicholas Mercator and William Holder.

Table of the enharmonic quartons in the 200-tone system

For each note, the number of quarter commas (1/200th of an octave) above the root note Do is stated.

• 0 Do ______________ DO - C
• 4 Ti#
• 9 Do#> (Do#> and Do< form an enharmonic quarter-tone zone)
• 10 Do<
• 15 Reb (Reb and Do# form an enharmonic zone)
• 19 Do#
• 24 Re>
• 25 Reb<
• 34 Re ______________ RE - D
• 43 Re#>
• 44 Re<
• 49 Mib
• 53 Re#
• 58 Mi>
• 59 Mib<
• 64 Fab
• 68 Mi ______________ MI - E
• 73 Fa>
• 78 Mi<
• 83 Fa ______________ FA - F
• 87 Mi#
• 92 Fa#>
• 93 Fa<
• 98 Sob
• 102 Fa#
• 107 So>
• 108 Sob<
• 117 So ______________ SOL - G
• 126 So#>
• 127 So<
• 132 Lab
• 136 So#
• 141 La>
• 142 Lab<
• 151 La ______________ LA - A
• 160 La#>
• 161 La<
• 166 Tib
• 170 La#
• 175 Ti>
• 176 Tib<
• 181 Dob
• 185 Ti ______________ SI - H (or B)
• 190 Do>
• 195 Ti<
• 200 Do ______________ DO - C

The 1 quarter-comma interval between do#> and do< is called the Hersidian comma,

The interval of 4 quarter-commas between reb and do# is called the Didymic comma.

See also

Microtonal Music