Five limit beats seven limit

Petr mentioned temperaments where the 5-limit can be tuned more accurately than the 7-limit, so I'm listing some where the minimax 5-limit tuning is more than twice as accurate than the minimax 7-limit tuning, together with the ratio. Sometimes, as with ennealimmal, this doesn't matter, but for less accuracy it becomes increasingly important.

ennealimmal <<18 27 18 1 -22 -34|| 6.3985653846330668133

pajara <<2 -4 -4 -11 -12 2|| 5.3663968862822073831

dominant <<1 4 -2 4 -6 -16|| 4.7139371549880495165

orwell <<7 -3 8 -21 -7 27|| 4.2406983397474017989

garibaldi <<1 -8 -14 -15 -25 -10|| 19.514316974206710431

diminished <<4 4 4 -3 -5 -2|| 2.1180542493141422185

amity <<5 13 -17 9 -41 -76|| 2.6710348530539254822

sharptone <<1 4 3 4 2 -4|| 14.710230830813350999

mitonic <<17 35 -21 16 -81 -147|| 12.303567263534785150

sensi <<7 9 13 -2 1 5|| 5.0301870969075073098

negri <<4 -3 2 -14 -8 13|| 2.4440373468832942316

keemun <<6 5 3 -6 -12 -7|| 13.209171417500785617

parakleismic <<13 14 35 -8 19 42|| 3.2188105009061810378

rodan <<3 17 -1 20 -10 -50|| 2.7050793146898482707

gamera <<23 40 1 10 -63 -110|| 4.7434971189083936629

unidec <<12 22 -4 7 -40 -71|| 6.9398729871804149779

flattone <<1 4 -9 4 -17 -32|| 2.1764946049621748182

mavila <<1 -3 -4 -7 -9 -1|| 2.1322931526968713900

opossum <<3 5 9 1 6 7|| 2.1342909087448650331

quartonic <<11 18 5 3 -23 -39|| 6.5004777275411140055

hemithirds <<15 -2 -5 -38 -50 -6|| 30.475157619670291925

hystrix <<3 5 1 1 -7 -12|| 6.9597532069075786019

buzzard <<4 21 -3 24 -16 -66|| 19.941360587887336639

grendel <<23 -1 13 -55 -44 33|| 3.4501108400802555103

kwai <<1 33 27 50 40 -30|| 2.7164647791977736713

muggles <<5 1 -7 -10 -25 -19|| 2.2204705846095479832

schism <<1 -8 -2 -15 -6 18|| 128.59462320117956091

ripple <<5 8 2 1 -11 -18|| 3.0408514272807968623

sycamore <<11 6 15 -16 -7 18|| 2.4837892744402398441

quasisuper <<1 -13 -2 -23 -6 32|| 2.1702931054782706762

bisupermajor <<16 -10 34 -53 9 107|| 59.254174075903265686

escapade <<9 -7 4 -32 -19 29|| 16.554920128171761342

keen <<2 -4 18 -11 23 53|| 2.1057197104703512804

leapday <<1 21 15 31 21 -24|| 3.6283842049988639227

marvo <<6 17 46 13 56 59|| 2.4165608720535164297

On 8/19/2011 9:03 AM, genewardsmith wrote:
> Petr mentioned temperaments where the 5-limit can be tuned more
> accurately than the 7-limit, so I'm listing some where the minimax
> 5-limit tuning is more than twice as accurate than the minimax
> 7-limit tuning, together with the ratio. Sometimes, as with
> ennealimmal, this doesn't matter, but for less accuracy it becomes
> increasingly important.

Interesting list. Many of these are not unexpected, as for instance dominant, sharptone, and flattone (alternative 81/80 temperaments, not as accurate as meantone). Similarly there are those that are more associated with 5-limit temperaments (garibaldi, diminished, mavila, ripple, sycamore) or with higher-limit temperaments (orwell, rodan, unidec, buzzard, kwai).

Probably the most unexpected one on the list is pajara; although the 5-limit srutal is also important, and higher-limit versions do exist, pajara is most often thought of as a 7-limit temperament.

> ennealimmal<<18 27 18 1 -22 -34|| 6.3985653846330668133
>
> pajara<<2 -4 -4 -11 -12 2|| 5.3663968862822073831
>
> dominant<<1 4 -2 4 -6 -16|| 4.7139371549880495165
>
> orwell<<7 -3 8 -21 -7 27|| 4.2406983397474017989
>
> garibaldi<<1 -8 -14 -15 -25 -10|| 19.514316974206710431
>
> diminished<<4 4 4 -3 -5 -2|| 2.1180542493141422185
>
> amity<<5 13 -17 9 -41 -76|| 2.6710348530539254822
>
> sharptone<<1 4 3 4 2 -4|| 14.710230830813350999
>
> mitonic<<17 35 -21 16 -81 -147|| 12.303567263534785150
>
> sensi<<7 9 13 -2 1 5|| 5.0301870969075073098
>
> negri<<4 -3 2 -14 -8 13|| 2.4440373468832942316
>
> keemun<<6 5 3 -6 -12 -7|| 13.209171417500785617
>
> parakleismic<<13 14 35 -8 19 42|| 3.2188105009061810378
>
> rodan<<3 17 -1 20 -10 -50|| 2.7050793146898482707
>
> gamera<<23 40 1 10 -63 -110|| 4.7434971189083936629
>
> unidec<<12 22 -4 7 -40 -71|| 6.9398729871804149779
>
> flattone<<1 4 -9 4 -17 -32|| 2.1764946049621748182
>
> mavila<<1 -3 -4 -7 -9 -1|| 2.1322931526968713900
>
> opossum<<3 5 9 1 6 7|| 2.1342909087448650331
>
> quartonic<<11 18 5 3 -23 -39|| 6.5004777275411140055
>
> hemithirds<<15 -2 -5 -38 -50 -6|| 30.475157619670291925
>
> hystrix<<3 5 1 1 -7 -12|| 6.9597532069075786019
>
> buzzard<<4 21 -3 24 -16 -66|| 19.941360587887336639
>
> grendel<<23 -1 13 -55 -44 33|| 3.4501108400802555103
>
> kwai<<1 33 27 50 40 -30|| 2.7164647791977736713
>
> muggles<<5 1 -7 -10 -25 -19|| 2.2204705846095479832
>
> schism<<1 -8 -2 -15 -6 18|| 128.59462320117956091
>
> ripple<<5 8 2 1 -11 -18|| 3.0408514272807968623
>
> sycamore<<11 6 15 -16 -7 18|| 2.4837892744402398441
>
> quasisuper<<1 -13 -2 -23 -6 32|| 2.1702931054782706762
>
> bisupermajor<<16 -10 34 -53 9 107|| 59.254174075903265686
>
> escapade<<9 -7 4 -32 -19 29|| 16.554920128171761342
>
> keen<<2 -4 18 -11 23 53|| 2.1057197104703512804
>
> leapday<<1 21 15 31 21 -24|| 3.6283842049988639227
>
> marvo<<6 17 46 13 56 59|| 2.4165608720535164297

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> Probably the most unexpected one on the list is pajara; although the
> 5-limit srutal is also important, and higher-limit versions do exist,
> pajara is most often thought of as a 7-limit temperament.

Pajara was certainly expected by me; in fact I had in mind I might use 34edo as an example, where srutal, pajara, hanson and keemun are all living happily together, with srutal and hanson much more accurate than pajara and keemun. Since 612 is a lot less familiar to most people than 34, I would have thought ennealimmal popping up would be less expected. Similar comments for sensi and 65.