30th-octave temperaments

Template:Fractional-octave navigation 30edo is not a harmonically notable temperament, but some of its multiples are. Due to plenty of divisors of number 30, these temperaments are also difficult to catalog due to their belonging in multiple temperament families, and this page is designed to accommodate that.

Zinc

Zinc temperament is named after the 30th element and in the 7-limit it can be described by tempering out the landscape comma and the neon comma. Tempering out the landscape comma divides the octave in 3, the neon comma in 10, and respectively zinc divides the octave into 3 x 10 = 30.

Zinc has mos scales of size 60 and 90. These scales are significantly distinct from either landscape rank-3 scales or the 10th-octave MOS scales that produce neon or its descendant temperaments like deca.

Subgroup: 2.3.5.7

Comma list: 250047/250000, [-45 -24 14 18⟩

Mapping: [⟨30 2 15 66], ⟨0 5 6 2]]

Optimal tuning (CTE): ~53747712/52521875 = 1\30, ~216/175 = 364.389

Optimal ET sequence: 270, 1380, 1650, 1920, 2190, 4650

Badness: 0.0742

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 250047/250000, 97869261875/97844723712

Mapping: [⟨30 2 15 66 122], ⟨0 5 6 2 -2]]

Optimal tuning (CTE): ~18865/18432 = 1\30, ~216/175 = 364.389

Optimal ET sequence: 270, 1110, 1380, 1650, 1920, 2190

Badness: 0.0220

13-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 123201/123200, 250047/250000, 1990656/1990625

Mapping: [⟨30 2 15 66 122 193], ⟨0 5 6 2 -2 -9]]

Optimal tuning (CTE): ~351/343 = 1\30, ~216/175 = 364.389

Optimal ET sequence: 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de

Badness: 0.0150