The Forge

The Forge is a collection of temperaments of temperaments, including subgroup temperaments, that all temper out the forge comma – the unnoticeable comma with the ratio 1640558367/1638400000, and a monzo of [-19 14 -5 3⟩. This collection of mostly rank-3 temperaments provides the foundation for multiple 11-limit and 13-limit temperament families.

Rank-3 temperaments

Forge

The parent temperament of this whole collection is a rank-3 temperament.

Subgroup: 2.3.5.7

Comma list: 1640558367/1638400000

Mapping: [⟨1 0 1 8], ⟨0 1 1 -3], ⟨0 0 -3 -5]]

Mapping generators: ~2, ~3, ~1701/1600

Tuning ranges:

Optimal ET sequence: 9c, 12, 34, 46, 58, 104c, 113, 125, 159, 171, 742, 913, 1084, 1255, 2681d, 2864, 3035, 3206, 3377, 4290d, 4461bd

Badness: 0.718 × 10^-3

Eleven limit children

The next comma of the normal comma list defines which 11-limit family member we are looking at. Some of the options are:

• 385/384, leading to the 113 & 125 & 159 temperament
• 441/440, leading to the 12 & 46 & 113 temperament
• 2835/2816, leading to the 12 & 34p & 113 temperament
• 6250/6237, leading to the 12 & 34p & 159 temperament- named "Ring"
• 8019/8000, leading to the 46 & 58 & 125 temperament- named "Trimmer"