The Forge: Difference between revisions

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'''The Forge''' is a collection of temperaments of different ranks, including subgroup temperaments, that all temper out the [[forge comma]] – the unnoticeable comma with the ratio 1640558367/1638400000, and a monzo of {{monzo|-19 14 -5 3}}.  This collection of temperaments provides the foundation for multiple 11-limit and 13-limit temperament families.
{{Technical data page}}
'''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 {{monzo|-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 ==
== Rank-3 temperaments ==
The parent temperament of this whole collection is a rank-3 temperament.
 


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


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 14: Line 16:
Mapping generators: ~2, ~3, ~1701/1600
Mapping generators: ~2, ~3, ~1701/1600


{{Val list|legend=1| 9c, 12, 34, 46, 58, 104c, 113, 125, 159, 171, 742, 913, 1084, 1255, 2681d, 2864, 3035, 3206, 3377, 4290d, 4461bd }}
[[Tuning ranges]]:
 
{{Optimal ET sequence|legend=1| 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<sup>-3</sup>
[[Badness]]: 0.718 × 10<sup>-3</sup>


From here, this temperament has different 11-limit extensions.
==== Eleven limit children ====
 
The next comma of the [[Normal lists #Normal interOptimal ET sequence|normal comma list]] defines which 11-limit family member we are looking at. Some of the options are:
== Rank-2 temperaments ==
Many of the rank-2 temperaments that temper out the Forge comma are controrted, considered below is the only one found so far that isn't.
 
=== Forger ===
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 2401/2400, 14348907/14336000
 
[[Mapping]]: [{{val| 1 -3 -17 -8 }}, {{val| 0 14 59 33 }}]
 
Mapping generators: ~2, ~6400/5103
 
{{Val list|legend=1| 55c, 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
 
[[Badness]]: 0.016736
 
==== 11-limit ====
 
[[Comma list]]: 243/242, 441/440, 3584000/3557763
 
[[Mapping]]: [{{val| 1 -3 -17 -8 -8 }}, {{val| 0 14 59 33 35 }}]
 
Mapping generators: ~2, ~6400/5103
 
{{Val list|legend=1| 55c, 58, 113, 171 }}
 
[[Badness]]: 0.052358
 
==== 13-limit ====
 
[[Comma list]]: 243/242, 364/363, 441/440, 2200/2197
 
[[Mapping]]: [{{val| 1 -3 -17 -8 -8 -13 }}, {{val| 0 14 59 33 35 51 }}]
 
Mapping generators: ~2, ~6400/5103
 
{{Val list|legend=1| 55cf, 58, 113, 171 }}
 
[[Badness]]: 0.026974
 
==== 17-limit ====
 
[[Comma list]]: 243/242, 363/363, 441/440, 595/594, 2200/2197
 
[[Mapping]]: [{{val| 1 -3 -17 -8 -8 -13 9 }}, {{val| 0 14 59 33 35 51 -15 }}]
 
Mapping generators: ~2, ~64/51
 
{{Val list|legend=1| 58, 113, 171 }}


[[Badness]]: 0.023205
* [[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"
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]

Latest revision as of 00:27, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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"
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