171edo: Difference between revisions
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== Theory == | == Theory == | ||
171edo is a remarkable edo which serves as a [[7-limit]] [[microtemperament]], approximating the [[9-odd-limit]] [[tonality diamond]] within about 2/5 of a cent. The excellence of its 7-limit approximations is good enough to make it the eleventh [[zeta integral edo]] but not enough to make it a [[zeta gap edo|zeta gap]]. It is also almost consistent in the 17-odd-limit, missing [[15/11]] and [[22/15]]. | 171edo is a remarkable edo which serves as a [[7-limit]] [[microtemperament]], approximating the [[9-odd-limit]] [[tonality diamond]] within about 2/5 of a cent. The excellence of its 7-limit approximations is good enough to make it the eleventh [[zeta integral edo]] but not enough to make it a [[zeta gap edo|zeta gap]]. It is also almost consistent in the 17-odd-limit, only missing [[15/11]] and [[22/15]]. | ||
Remarkable 5-limit commas 171et [[tempering out|tempers out]] are 32805/32768 ([[schisma]]), {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -14 -19 19 }} ([[enneadeca]]), and {{monzo| -29 -11 20 }} ([[gammic comma]]), and remarkable 7-limit commas 171et tempers out are 2401/2400 ([[breedsma]]), 4375/4374 ([[ragisma]]), 65625/65536 ([[horwell comma]]), 250047/250000 ([[landscape comma]]), 420175/419904 ([[wizma]]), and 703125/702464 ([[meter]]). | Remarkable 5-limit commas 171et [[tempering out|tempers out]] are 32805/32768 ([[schisma]]), {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -14 -19 19 }} ([[enneadeca]]), and {{monzo| -29 -11 20 }} ([[gammic comma]]), and remarkable 7-limit commas 171et tempers out are 2401/2400 ([[breedsma]]), 4375/4374 ([[ragisma]]), 65625/65536 ([[horwell comma]]), 250047/250000 ([[landscape comma]]), 420175/419904 ([[wizma]]), and 703125/702464 ([[meter]]). Therefore, 171et [[support]]s a number of notable 7-limit rank-2 temperaments: [[pontiac]], [[sesquiquartififths]], [[term]], [[ennealimmal]], [[tertiaseptal]], [[supermajor (temperament)|supermajor]], [[enneadecal]], [[neptune]], [[mitonic]], and [[mutt]]. It notably provides the [[optimal patent val]] for the rank-3 [[horwell]] temperament, and is also an excellent tuning for the 5-limit [[Helmholtz (temperament)|helmholtz]] temperament, tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }}. | ||
171edo is much less accurate in the 11-limit, but still quite useful as it is a good tuning (emphasizing accuracy in the 7-limit) for the important rank-3 temperament [[jove]], which tempers out [[243/242]] (rastma) and [[441/440]], not to mention [[540/539]] and 2401/2400. Jove can be extended by adding [[364/363]] for the 13-limit and [[595/594]] for the 17-limit, which 171edo also supports. Alternatively, the 171e val can be used, which tempers out [[385/384]]. In the 2.3.13 subgroup, it provides the optimal patent val for [[Glacier comma|glacier]], and is generally a | 171edo is notably [[Consistency#Consistency to distance d|consistent to distance ''6'']] in the [[7-limit|7-prime]] [[15-odd-limit]], and to distance ''8'' in the [[9-odd-limit]]. No other edo is so consistent until [[3125edo]]. Because of its accuracy in the 7-limit, the 171et mapping is an excellent and relatively simple way to classify 7-limit commas by size. For example, one step represents [[225/224]], two steps [[126/125]], three steps [[81/80]], and four steps [[64/63]]. | ||
171edo is much less accurate in the 11-limit, but still quite useful as it is a good tuning (emphasizing accuracy in the 7-limit) for the important rank-3 temperament [[jove]], which tempers out [[243/242]] (rastma) and [[441/440]], not to mention [[540/539]] and 2401/2400. Jove can be extended by adding [[364/363]] for the 13-limit and [[595/594]] for the 17-limit, which 171edo also supports. Alternatively, the 171e val can be used, which tempers out [[385/384]]. In the 2.3.13 subgroup, it provides the optimal patent val for [[Glacier comma|glacier]], and is generally a good [[2.3.5.7.13 subgroup|2.3.5.7.13]] and better 2.3.5.7.13.17 subgroup temperament. | |||
171edo is an excellent tuning for the [[Carlos Gamma]] scale, since the difference between 5 steps of 171edo and 1 step of Carlos Gamma is only -0.010823 cents. | 171edo is an excellent tuning for the [[Carlos Gamma]] scale, since the difference between 5 steps of 171edo and 1 step of Carlos Gamma is only -0.010823 cents. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|171|columns= | {{Harmonics in equal|171|columns=11}} | ||
{{Harmonics in equal|171|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 171edo (continued)}} | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
171 factors into primes as {{nowrap| 3<sup>2</sup> × 19 }}, and 171edo shares the nearly pure [[7/6]] of [[9edo]] and the nearly pure [[6/5]] of [[19edo]], with every 7-limit interval expressible in terms of 2, 6/5, 7/6, and any one of primes 3, 5, or 7. [[342edo]], which doubles 171, provides an excellent correction for the 11th harmonic. | 171 factors into primes as {{nowrap| 3<sup>2</sup> × 19 }}, and 171edo shares the nearly pure [[7/6]] of [[9edo]] and the nearly pure [[6/5]] of [[19edo]], with every 7-limit interval expressible in terms of 2, 6/5, 7/6, and any one of primes 3, 5, or 7. | ||
[[342edo]], which doubles 171, provides an excellent correction for the 11th harmonic and is one of the most accurate [[11-limit]] temperaments, with unmatched relative error up until [[1848edo]]. | |||
[[684edo]], which quadruples it, achieves [[17-odd-limit]] consistency. | |||
== Intervals == | == Intervals == | ||
| Line 26: | Line 33: | ||
== Approximation to JI == | == Approximation to JI == | ||
=== | === 15-odd-limit intervals === | ||
{{ | {{Q-odd-limit intervals|171|15}} | ||
| | |||
| | === Consistent circles === | ||
| | 171edo contains consistent circles of [[7/6]], [[6/5]], and [[9/7]], each with 9, 19, and 171 notes respectively. | ||
| | |||
| | {| class="wikitable center-all left-5" | ||
| | |+ style="font-size: 105%;" | Consistent circles in 171edo | ||
| | |- | ||
| | ! Note<br>count | ||
| | ! [[Interval]] | ||
! [[Closing error|Closing<br>error]] | |||
! [[Circle #Definitions|Consistency]] | |||
! Associated<br>edostep | |||
|- | |||
| 9 | |||
| [[7/6]] | |||
| -26.2% | |||
| Normal | |||
| 2\9 = 38\171 | |||
|- | |||
| 19 | |||
| [[6/5]] | |||
| +40.1% | |||
| Normal | |||
| 5\19 = 45\171 | |||
|- | |||
| 171 | |||
| [[9/7]] | |||
| +8.8% | |||
| Strong | |||
| 62\171 | |||
|} | |||
=== 7-prime-limited odd-limit analysis === | |||
171edo is ''distinctly'' [[consistent]] and monotone up to the 7-prime-limited 45-odd-limit: | |||
{{Databox | |||
|collapse=true | |||
|title=The 7-prime-limited 45-odd-limit, by 171edo mapping (SW3 format) | |||
|text= | |||
<pre> | |||
(* | |||
7-PL 45-OL odds: | |||
1 3 5 7 9 15 21 25 27 35 45 | |||
Mapping Ratio Error | |||
*) | |||
(* 7\171*) 36/35 (* +.352c *) | |||
(* 9\171*) 28/27 (* +.197c *) | |||
(* 10\171*) 25/24 (* -.497c *) | |||
(* 12\171*) 21/20 (* -.257c *) | |||
(* 16\171*) 16/15 (* +.549c *) | |||
(* 17\171*) 15/14 (* -.145c *) | |||
(* 19\171*) 27/25 (* +.096c *) | |||
(* 22\171*) 35/32 (* -.754c *) | |||
(* 26\171*) 10/9 (* +.052c *) | |||
(* 28\171*) 28/25 (* +.293c *) | |||
(* 29\171*) 9/8 (* -.401c *) | |||
(* 33\171*) 8/7 (* -.405c *) | |||
(* 38\171*) 7/6 (* -.204c *) | |||
(* 42\171*) 32/27 (* +.602c *) | |||
(* 43\171*) 25/21 (* -.092c *) | |||
(* 45\171*) 6/5 (* +.148c *) | |||
(* 54\171*) 56/45 (* +.345c *) | |||
(* 55\171*) 5/4 (* -.349c *) | |||
(* 61\171*) 32/25 (* +.698c *) | |||
(* 62\171*) 9/7 (* +.004c *) | |||
(* 64\171*) 35/27 (* -.152c *) | |||
(* 67\171*) 21/16 (* -.605c *) | |||
(* 71\171*) 4/3 (* +.201c *) | |||
(* 74\171*) 27/20 (* -.253c *) | |||
(* 78\171*) 48/35 (* +.553c *) | |||
(* 81\171*) 25/18 (* -.296c *) | |||
(* 83\171*) 7/5 (* -.056c *) | |||
(* 84\171*) 45/32 (* -.750c *) | |||
(* 87\171*) 64/45 | |||
(* 88\171*) 10/7 | |||
(* 90\171*) 36/25 | |||
(* 93\171*) 35/24 | |||
(* 97\171*) 40/27 | |||
(*100\171*) 3/2 | |||
(*104\171*) 32/21 | |||
(*107\171*) 54/35 | |||
(*109\171*) 14/9 | |||
(*110\171*) 25/16 | |||
(*116\171*) 8/5 | |||
(*117\171*) 45/28 | |||
(*126\171*) 5/3 | |||
(*128\171*) 42/25 | |||
(*129\171*) 27/16 | |||
(*133\171*) 12/7 | |||
(*138\171*) 7/4 | |||
(*142\171*) 16/9 | |||
(*143\171*) 25/14 | |||
(*145\171*) 9/5 | |||
(*149\171*) 64/35 | |||
(*152\171*) 50/27 | |||
(*154\171*) 28/15 | |||
(*155\171*) 15/8 | |||
(*159\171*) 40/21 | |||
(*161\171*) 48/25 | |||
(*162\171*) 27/14 | |||
(*164\171*) 35/18 | |||
(*171\171*) 2/1 | |||
</pre> | |||
}} | }} | ||
The 7-prime-limited 49-odd-limit is where non-distinctness first shows up: namely, ~49/48 = ~50/49 (this is characteristic of all ennealimmal tunings). However, 171edo remains consistent up to much higher 7-prime-limited odd-limits (much higher than even [[99edo]]). | |||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 115: | Line 216: | ||
| 2.88 | | 2.88 | ||
|} | |} | ||
* 171et is lower in relative error than any previous equal temperaments in the 7-limit | * 171et is lower in relative error than any previous equal temperaments in the 7-limit. Not until [[441edo|441]] do we find a better equal temperaments in terms of absolute error, and not until [[3125edo|3125]] do we find one in terms of relative error. | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
| Line 161: | Line 262: | ||
| 182.46 | | 182.46 | ||
| 10/9 | | 10/9 | ||
| [[ | | [[Mitonic]] / mineral (171) / ore (171e) / goldmine (171ef) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 221: | Line 322: | ||
| 498.25 | | 498.25 | ||
| 4/3 | | 4/3 | ||
| [[ | | [[Pontiac]] | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 245: | Line 346: | ||
| 182.46 | | 182.46 | ||
| 10/9 | | 10/9 | ||
| [[ | | [[Domain (temperament)|Domain]] | ||
|- | |- | ||
| 3 | | 3 | ||
| Line 263: | Line 364: | ||
| 315.79<br>(49.12) | | 315.79<br>(49.12) | ||
| 6/5<br>(36/35) | | 6/5<br>(36/35) | ||
| [[Ennealimmal]] ( | | [[Ennealimmal]] / enneabiotic (171ef) / ennealympic (171) / ennealimnic (171) / ennealiminal (171ef) | ||
|- | |- | ||
| 9 | | 9 | ||
| Line 277: | Line 378: | ||
| [[Enneadecal]] | | [[Enneadecal]] | ||
|} | |} | ||
<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct | <nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||
* [[Nestoria7]] | * [[Nestoria7]] | ||
* [[Nestoria12]] | * [[Nestoria12]] | ||
* [[ | * [[Diamond9]]: 26 3 4 5 7 10 7 9 12 5 12 9 7 10 7 5 4 3 26 | ||
== Music == | == Music == | ||