Fendo family

The parent comma of the fendo family is 40/39, which is tempered out in the rank-2 fendo (2.3.13/5) and rank-3 unintendo temperaments.

This article shows rank-2 extensions of fendo.

Fendo

Subgroup: 2.3.13/5

Comma list: 40/39

Edo join: 5 & 7

Mapping: [⟨1 1 2], ⟨0 1 -1]]

CWE tuning: ~2 = 1200 cents, ~3/2 = 710.153 cents

Badness: 0.111

Main extensions:

The concept of "fendo" tuning is well-represented by EDOs 8, 13, and 18. Each pair of these produces a rank-2 temperament in 2.3.5.13 separating 13/8 and 5/4 by 3/2 (octave-reduced), or equivalently tempering together 13/8 and 5/3. They differ from unintendo in that 5 is found somewhere along the chain of fifths rather than being its own generator.

Fendo-18

Subgroup: 2.3.5.13

Comma list: 40/39, 832/729

Edo join: 8 & 13

Mapping: [⟨1 1 -2 0], ⟨0 1 7 6]]

CWE tuning: ~2 = 1200 cents, ~3/2 = 742.362 cents

Badness: 2.330

To reach 5/4 you stack 7 generators and to reach 13/8 you stack six. 5/4 is tuned sharp; 13/8 is also sharp.

Fendo-13

Subgroup: 2.3.5.13

Comma list: 40/39, 260/243

Edo join: 8 & 18

Mapping: [⟨2 2 1 5], ⟨0 1 3 2]]

CWE tuning: ~27/20 = 600 cents, ~3/2 = 726.538 cents

Badness: 1.064

To reach 5/4, you stack 3 generators; but note that the period is 600 cents. To reach 13/8, you stack 2 generators. 5/4 is flat; 13/8 is sharp.

Fendo-8

Subgroup: 2.3.5.13

Comma list: 40/39, 6656/6075

Edo join: 13 & 18

Mapping: [⟨1 1 6 8], ⟨0 1 -6 -7]]

CWE tuning: ~2 = 1200 cents, ~3/2 = 732.153 cents

Badness: 2.103

To reach 5/4, you go down 6 generators. To reach 13/8, you go down 7 generators. Both 5 and 13 are sharp.