JUST TUNINGS

Just intonation employs rational intervals between pitches. This tends to reduce or eliminate beating when several notes are played at the same time. Brass instruments and some vocal harmonies use just intonation. As there are theoretically an infinite number of just intervals (with a practical limit that can vary with the acoustical properties of the source of the sound), parameters can be chosen to reduce the number of intervals in a scale. These can include

• establishing an upper limit on the size of the factors,
• establishing an upper limit on the size of the prime factors,
• exclusion of factors based on acoustical choice,
• establishing a minimum space between pitches,
• etc.

Most just tunings use several of these parameters. For example,

• Pythagorean tuning limits the prime factors to (2, 3);
• major pentatonic (5-tone) tuning (typical of Andean music and much Chinese music) limits the prime factors to 2 and 3 and the minimum interval in the scale to 1/6 of diapason, or a whole tone (200 cents);
• heptatonic (7-tone) modal tuning (of Gregorian chants) limits the minimum interval to 3/40 of a diapason, or a halftone (90 cents).
• 5-limit tuning limits the prime factors to (2, 3, 5);
• just microtonal music generally chooses a prime limit of 13 or 19, although the individual composer has the freedom to make other rules.

The just scales listed below employ specific sets of prime factors, as labeled.

• 1-LIMIT
  • 0-factor
    • 1 (Monotone)
• 2-LIMIT
  • 1-factor
    • 2 (Octaves)
• 3-LIMIT
  • 1-factor
    • 3
  • 2-factor
    • 2, 3--Pythagorean intonation
• 5-LIMIT
  • 1-factor
    • 5
  • 2-factor
    • 2, 5
    • 3, 5
  • 3-factor
    • 2, 3, 5
• 7-LIMIT
  • 1-factor
    • 7
  • 2-factor
    • 2, 7
    • 3, 7
    • 5, 7
  • 3-factor
    • 2, 3, 7
    • 2, 5, 7
    • 3, 5, 7
  • 4-factor
    • 2, 3, 5, 7
• 11-LIMIT
  • 1-factor
    • 11
  • 2-factor
    • 2, 11
    • 3, 11
    • 5, 11
    • 7, 11
  • 3-factor
    • 2, 3, 11
    • 2, 5, 11
    • 2, 7, 11
    • 3, 5, 11
    • 3, 7, 11
    • 5, 7, 11
  • 4-factor
    • 2, 3, 5, 11
    • 2, 3, 7, 11
    • 2, 5, 7, 11
    • 3, 5, 7, 11
  • 5-factor
    • 2, 3, 5, 7, 11
• 13-LIMIT
  • 1-factor
    • 13
  • 2-factor
    • 2, 13
    • 3, 13
    • 5, 13
    • 7, 13
    • 11, 13
  • 3-factor
    • 2, 3, 13
    • 2, 5, 13
    • 2, 7, 13
    • 2, 11, 13
    • 3, 5, 13
    • 3, 7, 13
    • 3, 11, 13
    • 5, 7, 13
    • 5, 11, 13
    • 7, 11, 13
  • 4-factor
    • 2, 3, 5, 13
    • 2, 3, 7, 13
    • 2, 3, 11, 13
    • 2, 5, 7, 13
    • 2, 5, 11, 13
    • 2, 7, 11, 13
    • 3, 5, 7, 13
    • 3, 5, 11, 13
    • 3, 7, 11, 13
    • 5, 7, 11, 13
  • 5-factor
    • 2, 3, 5, 7, 13
    • 2, 3, 5, 11, 13
    • 2, 3, 7, 11, 13
    • 2, 5, 7, 11, 13
    • 3, 5, 7, 11, 13
  • 6-factor
    • 2, 3, 5, 7, 11, 13
• 17-LIMIT
  • 1-factor
    • 17
  • 2-factor
    • 2, 17
    • 3, 17
    • 5, 17
    • 7, 17
    • 11, 17
    • 13, 17
  • 3-factor
    • 2, 3, 17
    • 2, 5, 17
    • 2, 7, 17
    • 2, 11, 17
    • 2, 13, 17
    • 3, 5, 17
    • 3, 7, 17
    • 3, 11, 17
    • 3, 13, 17
    • 5, 7, 17
    • 5, 11, 17
    • 5, 13, 17
    • 7, 11, 17
    • 7, 13, 17
    • 11, 13, 17
  • 4-factor
    • 2, 3, 5, 17
    • 2, 3, 7, 17
    • 2, 3, 11, 17
    • 2, 3, 13, 17
    • 2, 5, 7, 17
    • 2, 5, 11, 17
    • 2, 5, 13, 17
    • 2, 7, 11, 17
    • 2, 7, 13, 17
    • 2, 11, 13, 17
    • 3, 5, 7, 17
    • 3, 5, 11, 17
    • 3, 5, 13, 17
    • 3, 7, 11, 17
    • 3, 7, 13, 17
    • 3, 11, 13, 17
    • 5, 7, 11, 17
    • 5, 7, 13, 17
    • 5, 11, 13, 17
    • 7, 11, 13, 17
  • 5-factor
    • 2, 3, 5, 7, 17
    • 2, 3, 5, 11, 17
    • 2, 3, 5, 13, 17
    • 2, 3, 7, 11, 17
    • 2, 3, 7, 13, 17
    • 2, 3, 11, 13, 17
    • 2, 5, 7, 11, 17
    • 2, 5, 7, 13, 17
    • 2, 5, 11, 13, 17
    • 2, 7, 11, 13, 17
    • 3, 5, 7, 11, 17
    • 3, 5, 7, 13, 17
    • 3, 5, 11, 13, 17
    • 3, 7, 11, 13, 17
    • 5, 7, 11, 13, 17
  • 6-factor
    • 2, 3, 5, 7, 11, 17
    • 2, 3, 5, 7, 13, 17
    • 2, 3, 5, 11, 13, 17
    • 2, 3, 7, 11, 13, 17
    • 2, 5, 7, 11, 13, 17
    • 3, 5, 7, 11, 13, 17
  • 7-factor
    • 2, 3, 5, 7, 11, 13, 17
• 19-LIMIT
  • 1-factor
    • 19
  • 2-factor
    • 2, 19
    • 3, 19
    • 5, 19
    • 7, 19
    • 11, 19
    • 13, 19
    • 17, 19
  • 3-factor
    • 2, 3, 19
    • 2, 5, 19
    • 2, 7, 19
    • 2, 11, 19
    • 2, 13, 19
    • 2, 17, 19
    • 3, 5, 19
    • 3, 7, 19
    • 3, 11, 19
    • 3, 13, 19
    • 3, 17, 19
    • 5, 7, 19
    • 5, 11, 19
    • 5, 13, 19
    • 5, 17, 19
    • 7, 11, 19
    • 7, 13, 19
    • 7, 17, 19
    • 11, 13, 19
    • 11, 17, 19
    • 13, 17, 19
  • 4-factor
    • 2, 3, 5, 19
    • 2, 3, 7, 19
    • 2, 3, 11, 19
    • 2, 3, 13, 19
    • 2, 3, 17, 19
    • 2, 5, 7, 19
    • 2, 5, 11, 19
    • 2, 5, 13, 19
    • 2, 5, 17, 19
    • 2, 7, 11, 19
    • 2, 7, 13, 19
    • 2, 7, 17, 19
    • 2, 11, 13, 19
    • 2, 11, 17, 19
    • 2, 13, 17, 19
    • 3, 5, 7, 19
    • 3, 5, 11, 19
    • 3, 5, 13, 19
    • 3, 5, 17, 19
    • 3, 7, 11, 19
    • 3, 7, 13, 19
    • 3, 7, 17, 19
    • 3, 11, 13, 19
    • 3, 11, 17, 19
    • 3, 13, 17, 19
    • 5, 7, 11, 19
    • 5, 7, 13, 19
    • 5, 7, 17, 19
    • 5, 11, 13, 19
    • 5, 11, 17, 19
    • 5, 13, 17, 19
    • 7, 11, 13, 19
    • 7, 11, 17, 19
    • 7, 13, 17, 19
    • 7, 11, 13, 19
  • 5-factor
    • 2, 3, 5, 7, 19
    • 2, 3, 5, 11, 19
    • 2, 3, 5, 13, 19
    • 2, 3, 5, 17, 19
    • 2, 3, 7, 11, 19
    • 2, 3, 7, 13, 19
    • 2, 3, 7, 17, 19
    • 2, 3, 11, 13, 19
    • 2, 3, 11, 17, 19
    • 2, 3, 13, 17, 19
    • 2, 5, 7, 11, 19
    • 2, 5, 7, 13, 19
    • 2, 5, 7, 17, 19
    • 2, 5, 11, 13, 19
    • 2, 5, 11, 17, 19
    • 2, 5, 13, 17, 19
    • 2, 7, 11, 13, 19
    • 2, 7, 11, 17, 19
    • 2, 7, 13, 17, 19
    • 2, 11, 13, 17, 19
    • 3, 5, 7, 11, 19
    • 3, 5, 7, 13, 19
    • 3, 5, 7, 17, 19
    • 3, 5, 11, 13, 19
    • 3, 5, 11, 17, 19
    • 3, 5, 13, 17, 19
    • 3, 7, 11, 13, 19
    • 3, 7, 11, 17, 19
    • 3, 7, 13, 17, 19
    • 3, 11, 13, 17, 19
    • 5, 7, 11, 13, 19
    • 5, 7, 11, 17, 19
    • 5, 7, 13, 17, 19
    • 5, 11, 13, 17, 19
    • 7, 11, 13, 17, 19
  • 6-factor
    • 2, 3, 5, 7, 11, 19
    • 2, 3, 5, 7, 13, 19
    • 2, 3, 5, 7, 17, 19
    • 2, 3, 5, 11, 13, 19
    • 2, 3, 5, 11, 17, 19
    • 2, 3, 5, 13, 17, 19
    • 2, 3, 7, 11, 13, 19
    • 2, 3, 7, 11, 17, 19
    • 2, 3, 7, 13, 17, 19
    • 2, 3, 11, 13, 17, 19
    • 2, 5, 7, 11, 13, 19
    • 2, 5, 7, 11, 17, 19
    • 2, 5, 7, 13, 17, 19
    • 2, 5, 11, 13, 17, 19
    • 2, 7, 11, 13, 17, 19
    • 3, 5, 7, 11, 13, 19
    • 3, 5, 7, 11, 17, 19
    • 3, 5, 7, 13, 17, 19
    • 3, 5, 11, 13, 17, 19
    • 3, 7, 11, 13, 17, 19
    • 5, 7, 11, 13, 17, 19
  • 7-factor
    • 2, 3, 5, 7, 11, 13, 19
    • 2, 3, 5, 7, 11, 17, 19
    • 2, 3, 5, 7, 13, 17, 19
    • 2, 3, 5, 11, 13, 17, 19
    • 2, 3, 7, 11, 13, 17, 19
    • 2, 5, 7, 11, 13, 17, 19
    • 3, 5, 7, 11, 13, 17, 19
  • 8-factor
    • 2, 3, 5, 7, 11, 13, 17, 19
• Prime-number-only
  • All-prime

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Last revised: 21 August 2008

visitors since 30 May 1999