205, 217, and 224

These are all now being mentioned, so I thought I'd look at a few of
their properties.

205 = 5 * 41

5-limit basis: [1600000/1594323, 274877906944/274658203125]

7-limit basis: [4375/4374, 5120/5103, 2500000/2470629]

characteristic temperament: amity

meantone fifth: 119/205, 696.585 cents

secor: 4/41 = 117.073 cents

217 = 7 * 31

5-limit basis: [1224440064/1220703125, 4294967296/4271484375]

7-limit basis: [3136/3125, 4375/4374, 2100875/2097152]

characteristic temperament: parakleismic

meantone fifth: 18/31 = 696.774 cents

secor: 3/31 = 116.129 cents

224 = 2^5 * 7

5-limit basis: [32805/32768, 59604644775390625/59296646043258912]

7-limit basis: [4375/4374, 16875/16807, 32805/32768]

characteristic temperament: schismic in the 5-limit, octoid in the 7
and 11 limits

meantone fifth: 65/112 = 696.429 (poptimal)

secor: 11/112 = 117.857 (way sharp)

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> These are all now being mentioned, so I thought I'd look at a few of
> their properties.
>
> 205 = 5 * 41
>
> 5-limit basis: [1600000/1594323, 274877906944/274658203125]
>
> 7-limit basis: [4375/4374, 5120/5103, 2500000/2470629]
>
> <etc. -- snip>

thanks for providing this data!

... is there some good reason why you give ratios for the
bases and don't give the monzos? not a big deal, because
they can easily be calculated ... but that seems to me to
be the more relevant form for giving a lattice basis.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> ... is there some good reason why you give ratios for the
> bases and don't give the monzos? not a big deal, because
> they can easily be calculated ... but that seems to me to
> be the more relevant form for giving a lattice basis.

Just lazy I guess.

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> ... is there some good reason why you give ratios for the
> bases and don't give the monzos? not a big deal, because
> they can easily be calculated ... but that seems to me to
> be the more relevant form for giving a lattice basis.

the ratios, as opposed to the monzos, make it easy to guesstimate the
size of the unison vectors in ji. also, they can give you a clear
indication of the complexity of the unison vectors (just look at the
size of the numbers in the ratio), and an idea as to the minimum
error that tempering them out could impart to the consonant intervals
(look at the difference between the terms in the ratio in conjunction
with their size).