Undim family: Difference between revisions

Undim features a quarter-octave period, which acts as the [[1215/1024|ptolemaic augmented second (1215/1024)]]. That and five [[4/3|perfect fourths]] (i.e. a minor second, ~[[256/243]]) give the interval class of 5. Undim equates the [[Pythagorean comma]] with a stack of four [[schisma]]s. This makes it a member of the [[schismic–Pythagorean equivalence continuum]], with {{nowrap| ''n'' {{=}} 4 }}.

 

Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious extension to prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo|1\4]]. This interpretation is arguably much more harmonically plausible, owing to its simplicity and thereby greater tolerance to mistuning.

* [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6754{{c}}

: [[error map]]: {{val| 0.0000 +0.7204 +0.3092 }}

* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6054{{c}}

: error map: {{val| 0.0000 +0.6504 +0.6591 }}

[[Badness]] (Smith): 0.241703

Septimal undim tempers out the [[dimcomp comma]], mapping ~25/21 to the 1/4-octave period. It can be described as {{nowrap| 12 & 140 }}, and is the unique temperament that equates a syntonic~septimal comma with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma (interval region)|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[292edo]] makes for an excellent tuning.  

* [[CTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7948{{c}}

: [[error map]]: {{val| 0.0000 +0.8398 -0.2879 +0.4308 }}

* [[POTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7362{{c}}

: error map: {{val| 0.0000 +0.7812 +0.0051 +1.0754 }}

{{Optimal ET sequence|legend=1| 140, 152, 292 }}

[[Badness]] (Smith): 0.062754

* CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7433{{c}}

* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.6886{{c}}

{{Optimal ET sequence|legend=0| 140, 152, 292, 444d, 596d }}

Badness (Smith): 0.034837

* CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7792{{c}}

* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7363{{c}}

{{Optimal ET sequence|legend=0| 140, 152f, 292 }}

Badness (Smith): 0.028172

[[Comma list]]: 4375/4374, 2199023255552/2179240250625

* [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5556{{c}}

: [[error map]]: {{val| 0.0000 +0.6006 +0.9081 +0.1761 }}

* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5764{{c}}

: error map: {{val| 0.0000 +0.6214 +0.8043 +0.7369 }}

[[Badness]] (Smith): 0.268206

* CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5582{{c}}

* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5826{{c}}

Badness (Smith): 0.070215

* CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5562{{c}}

* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5741{{c}}

Badness (Smith): 0.058390

* [[CTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.4765{{c}}

: [[error map]]: {{val| 0.0000 +0.5215 +1.3036 +0.9865 }}

* [[POTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5090{{c}}

: error map: {{val| 0.0000 +0.5540 +1.1415 +1.2457 }}

Badness (Smith): 0.213094

* CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4692{{c}}

* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5090{{c}}

Badness (Smith): 0.048007

* CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4168{{c}}

* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4773{{c}}

Badness (Smith): 0.041365

[[Category:Undim family| ]] <!-- main article -->

[[Category:Rank 2]]