User:FloraC/Quick reference

Taxonomy of tuning approaches

• Tuning rationalism
  • JI purism: this school recognizes that the acoustic quality of JI is of top importance. Some consider music as a platonic ideal object that cannot be approximated at all. Meanwhile, its weaker version is characterized by being maximally strict about JI approximation.
    • Primodality: I don't feel entitled to define this.
    • Stacking based a.k.a. lattice based: a more traditional approach to JI. They recognize both the acoustic quality and the algebraic structure of JI.
  • JI approximabilism: this school recognizes that the acoustic quality of JI and the algebraic structure of tuning systems are similarly important, and therefore accepts a tradeoff.
    • RTT: this school encompasses stacking based JI and applies approximation for custom structures.
  • JI agnosticism: this school suspends the question whether the acoustic quality of JI is of importance. It tends to focus on algebraic structures such as mos scales and generalizations.
  • JI indifferentism: this school does not believe the acoustic quality of JI is of importance. Practice in this school is orthogonal to the influence of JI.
• Tuning empiricism
• Tuning stochasticism

Important prime limits

2-limit (rank-1)

• Essential equivalence
• Completes the harmonic series for the first octave

3-limit (rank-2)

• Essential interval functions
• Completes the harmonic series for the first 2 octaves
• Rank is a highly composite number

5-limit (rank-3)

• Completes the harmonic series for the first 2 octaves and a fifth

7-limit (rank-4)

• Tonality: tonal
• Categorical characteristics: pivotal and semiambitonal
• Completes the harmonic series for the first 3 octaves
• Rank is a highly composite number

11-limit (rank-5)

• Completes the harmonic series for the first 3 octaves and a fifth

13-limit (rank-6)

• Essential interval colors
• Tonality: microtonal
• Categorical characteristics: ambitonal and semiambitonal
• Completes the harmonic series for the first 4 octaves
• Rank is a highly composite number

23-limit (rank-9)

• Limit of classical functional harmony
• Limit of classical concordance
• Tonality: pseudotonal and pseudomicrotonal
• Categorical characteristics: pseudoambitonal
• Completes the harmonic series for the first 4 octaves and a fifth
• Followed by a record prime gap

31-limit (rank-11)

• Completes the harmonic series for the first 5 octaves

37-limit (rank-12)

• Rank is a highly composite number

47-limit (rank-15)

• Completes the harmonic series for the first 5 octaves and a fifth

61-limit (rank-18)

• Completes the harmonic series for the first 6 octaves

89-limit (rank-24)

• Completes the harmonic series for the first 6 octaves and a fifth
• Followed by a record prime gap
• Rank is a highly composite number

Edo sizes

• Exo: 0–9
• Small: 10–38
  • Semitonic: 10–14
  • Subsemitonic: 15–26
  • Dietic: 27–38
• Medium: 39–79
  • Commatic: 39–67
  • Subcommatic: 68–79
• Large: 80–190
  • Hemicommatic: 80–132
  • Codettic: 133–137
  • Kleismatic: 138–190
• Mega: 191+
  • Subkleismatic
  • Hemikleismatic
  • …