Xenharmonic Wiki

30th-octave temperaments: Difference between revisions

VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,458 edits
zinc is better addressed here and so there's continuous navigation 28 - 29 - 30 - 31 - 32
 
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,458 edits
→Oyster: strong vs weak distinction
 
(13 intermediate revisions by 4 users not shown)
Line 1: Line 1:
{{Fractional-octave navigation|30}}
{{Infobox fractional-octave|30}}
30edo is not a harmonically notable temperament, but some of its multiples are.
30edo is by many metrics 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 [[10th-octave temperaments#Neon|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.


[[Subgroup]]: 2.3.5.7
Due to [[landscape comma]] being a notable comma that divides the octave in 3, many temperaments whose wheel is a multiple of 3 (such as 12, 15, 24, etc.), end up belonging in the [[landscape microtemperaments]]. This page therefore describes theoretical interest specific to [[30edo]], beyond just belonging in that group.


[[Comma list]]: 250047/250000, {{monzo| -45 -24 14 18 }}
[[Zinc]] is a prominent 30th-octave temperament, see [[Landscape microtemperaments#Zinc]] for technical details. From a fractional-octave perspective, MOS scales of zinc are significantly distinct from either landscape rank-3 scales or the 10th-octave mos scales that produce neon or its descendant temperaments like [[deca]]. The 120-tone mos has [[65/64]] within reach. [[Poe]] is another temperament of interest.


{{Mapping|legend=1| 30 2 15 66 | 0 5 6 2 }}
Considered below is oyster.


[[Optimal tuning]] ([[CTE]]): ~53747712/52521875 = 1\30, ~216/175 = 364.389
== Oyster ==
Oyster is a 2.7.11.19 weak restriction of zinc of reasonably low badness and contains all the harmonics within the reach of a standard MIDI 128-note keyboard ― 120 notes is sufficient to cover all the temperament for the subgroup. Furthermore, [[7/4]] is the generator. As harmonics 3 and 5 are not used, oyster is neither a landscape nor neon temperament.


{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}
Overall, this temperament is a simple augmentation of 30edo with harmonics 7, 11, and 19 that is unique and of decent badness.


[[Badness]]: 0.0742
Named so because oysters are the biggest source of the element zinc, the namesake of the temperament it is derived of.


=== 11-limit ===
Subgroup: 2.7.11.19
Subgroup: 2.3.5.7.11


Comma list: 9801/9800, 250047/250000, 97869261875/97844723712
Comma list: 184877/184832, 2.7.11.19 {{monzo|-67 0 12  6}}


Mapping: {{mapping| 30 2 15 66 122 | 0 5 6 2 -2 }}
{{mapping|legend=2| 30 60 128 79 | 0 1 -1 2 }}


Optimal tuning (CTE): ~18865/18432 = 1\30, ~216/175 = 364.389
Optimal tunings:
* WE: ~102760448/100422619 = 40.000{{c}}, ~7/4 = 968.757{{c}}
* CWE: ~102760448/100422619 = 40.000{{c}}, ~7/4 = 968.750{{c}}


{{Optimal ET sequence|legend=1| 270, 1110, 1380, 1650, 1920, 2190 }}
[[Support]]ing [[ET]]s: {{EDOs|30, 120, 150, 270, 390, 420, 570, 660, 690, 960, 1110, 1230, 1500, 1650}}


Badness: 0.0220
Badness (Sintel): 0.499


=== 13-limit ===
{{Navbox fractional-octave}}
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 123201/123200, 250047/250000, 1990656/1990625
 
Mapping: {{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|legend=1| 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de }}
 
Badness: 0.0150
Retrieved from "https://en.xen.wiki/w/30th-octave_temperaments"