Xenharmonic Wiki

Ennealimmal: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,012 edits
mNo edit summary
Sintel (talk | contribs)
autopatrolled, emailconfirmed
1,625 edits
-TempClean
 
(23 intermediate revisions by 10 users not shown)
Line 1: Line 1:
'''Ennealimmal temperament''' has a period of 1/9 octave and tempers out [[2401/2400]] and [[4375/4374]]. EDOs that support ennealimmal include {{EDOs| 27, 45, 72, 99, 171, 270, 441, and 612 }} EDOs.
{{Infobox regtemp
| Title = Ennealimmal
| Subgroups = 2.3.5.7
| Comma basis = [[2401/2400]], [[4375/4374]]
| Edo join 1 = 27 | Edo join 2 = 45
| Mapping = 9; 2 3 2
| Generators = 5/3 | Generators tuning = 884.322 | Optimization method = CWE
| MOS scales = [[18L 9s]], [[27L 18s]], [[27L 45s]]
| Pergen = (P8/9, P5/2)
| Odd limit 1 = 9 | Mistuning 1 = 0.204 | Complexity 1 = 45
| Odd limit 2 = 7-limit 81 | Mistuning 2 = 0.408 | Complexity 2 = 99
}}
'''Ennealimmal''' is a [[regular temperament|temperament]] with a period of {{frac|1|9}} octave and tempers out [[2401/2400]] and [[4375/4374]]. Edos that support ennealimmal include {{EDOs| 27, 45, 72, 99, 171, 270, 441, and 612 }}.


See [[Ragismic microtemperaments #Ennealimmal]] for more technical data.
See [[Septiennealimmal clan #Ennealimmal]] for technical data.


== Ennealimmal extensions ==
Ennealimmal scales are built from a ''period'' (which is exactly {{frac|1|9}} of an octave), and a ''generator'' (which is approximately 49 cents and represents several small intervals including 36/35). Depending on the size of the generator and the period in steps, the above listed edos make sense:
Ennealimmal temperament has various extensions to the 11-limit. These are all members of the ennealimmal family, but in addition they are linear temperaments:
 
* Ennealimmal (99e & 270) - tempering out 2401/2400, 4375/4374, 5632/5625
{| class="wikitable"
* Ennealimmia (171 & 270) - tempering out 2401/2400, 4375/4374, 131072/130977
|- style="white-space: nowrap;"
* Ennealimnic (72 & 99e) - tempering out 243/242, 441/440, 4375/4356
! Period (steps) !! Generator (steps) !! Generator (cents)<br />(pure octave) !! Edo
* Ennealiminal (72 &amp; 171e) - tempering out 385/384, 1375/1372, 4375/4374
|-
| 3 || 1 || 44.444 || 27
|-
| 11 || 4 || 48.485 || 99
|-
| 30 || 11 || 48.889 || 270
|-
| 19 || 7 || 49.123 || 171
|-
| 8 || 3 || 50.000 || 72
|-
| 5 || 2 || 53.333 || 45
|}
 
Ennealimmal extends less well to the [[11-limit]]. Extensions include enneabiotic (99e & 270), ennealympic (171 & 270), ennealimnic (72 & 99e), and ennealiminal (72 & 171e).
 
See [[Ennealimmal extensions]] for a discussion on 11-limit extensions.
 
== Interval chain ==
In the following table, odd harmonics 1–9 are labeled in '''bold'''.
 
{| class="wikitable center-1 right-2 right-4 right-6 right-8"
! rowspan="2" | Period
! colspan="2" | Generator 0
! colspan="2" | Generator 1
! colspan="2" | Generator 2
! colspan="2" | Generator 3
|-
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
|-
| 0
| 0.000
| '''1/1'''
|
|
|
|
|
|
|-
| 1
| 133.333
| 27/25
| 84.322
| 21/20
| 35.310
| 49/48, 50/49
| 1186.298
| 125/63
|-
| 2
| 266.667
| 7/6
| 217.655
| 245/216
| 168.643
| 54/49
| 119.631
| 15/14
|-
| 3
| 400.000
| 63/50
| 350.988
| 49/40, 60/49
| 301.976
| 25/21
| 252.965
| 81/70, 125/108
|-
| 4
| 533.333
| 49/36
| 484.322
| 250/189
| 435.310
| 9/7
| 386.298
| '''5/4'''
|-
| 5
| 666.667
| 72/49
| 617.655
| 10/7
| 568.643
| 25/18
| 519.631
| 27/20
|-
| 6
| 800.000
| 100/63
| 750.988
| 54/35
| 701.976
| '''3/2'''
| 652.965
| 35/24
|-
| 7
| 933.333
| 12/7
| 884.322
| 5/3
| 835.310
| 81/50
| 786.298
| 63/40
|-
| 8
| 1066.667
| 50/27
| 1017.655
| 9/5
| 968.643
| '''7/4'''
| 919.631
| 245/144
|-
| 9
| 1200.000
| 2/1
| 1150.988
| 35/18
| 1101.976
| 189/100
| 1052.965
| 147/80
|}
<nowiki>*</nowiki> In 7-limit CWE tuning, octave reduced


== Scales ==
== Scales ==
* [[Ennealimmal27]] - proper [[18L 9s]]. Ninth-octave analog of haplotonic scale
* [[Ennealimmal27]] &ndash; proper [[18L 9s]]. Ninth-octave analog of haplotonic scale
* [[Ennealimmal45]] - improper [[27L 18s]]. Ninth-octave analog of mega-haplotonic scale
* [[Ennealimmal45]] &ndash; improper [[27L 18s]]. Ninth-octave analog of mega-haplotonic scale
** [[Ennealimmal45trans]] - symmetric 5-limit transversal version
** [[Ennealimmal45trans]] &ndash; symmetric 5-limit transversal version
* [[Ennealimmal72]] - proper [[27L 45s]]. Ninth-octave analog of albitonic scale
* [[Ennealimmal72]] &ndash; proper [[27L 45s]]. Ninth-octave analog of albitonic scale
* [[Ennealimmal99]] - proper [[72L 27s]]. Ninth-octave analog of chromatic scale
* [[Ennealimmal99]] &ndash; proper [[72L 27s]]. Ninth-octave analog of chromatic scale
* [[Ennealimmal171]] - [[99L 72s]] scale. The boundary of propriety is [[270edo|270EDO]].
* [[Ennealimmal171]] &ndash; [[99L 72s]] scale. The boundary of propriety is [[270edo]].
 
== Music ==
; [[Gene Ward Smith]]
* [https://archive.org/details/fingers_201403 ''The 45000 fingers of Dr. S''] (2003) – Ennealimmal[54] in TOP tuning


[[Category:Ennealimmal| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Microtemperaments]]
[[Category:Breedsmic temperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Temperament]]
[[Category:Landscape microtemperaments]]
[[Category:Ennealimmal| ]]
Retrieved from "https://en.xen.wiki/w/Ennealimmal"