Tuning Dictionary update: heptamerides

hello all,

i've just updated the Tuning Dictionary entry
for "heptameride"

http://sonic-arts.org/dict/heptameride.htm

giving a large table of heptameride values for many
important JI intervals and also all 23 intervals which
occur in a 12-tone meantone.

i've discovered that heptamerides offer not only integer
values for 43edo (because 301 = 43 * 7), and thus
nearly-integer values for the closely related 1/5-comma
meantone, but also nearly-integer values for the basic
intervals of the central section of the 7-limit lattice.

thus, heptamerides are an excellent system of interval
measurement for use in comparing 7-limit JI and
1/5-comma meantone, without needing decimal places.

if one is wiling to accept small amounts of error
in the decimal places, the comparisons can be generalized
to most of the other important meantone systems (i.e.,
1/6-, 1/4-, 2/7-, and 7/26-comma, and golden meantone).

another useful feature of heptamerides is that if one
uses tables of logarithms to find the log of the ratio,
simply moving the decimal point 3 places to the right
gives the heptamerides, thus eliminating some calculation.

-monz
"all roads lead to n^0"

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

> i've discovered that heptamerides offer not only integer
> values for 43edo (because 301 = 43 * 7), and thus
> nearly-integer values for the closely related 1/5-comma
> meantone, but also nearly-integer values for the basic
> intervals of the central section of the 7-limit lattice.

So when are you going to put up a web page for shismas and 612-et? :)

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...>
wrote:
> --- In tuning@y..., "monz" <monz@a...> wrote:
>
> > i've discovered that heptamerides offer not only integer
> > values for 43edo (because 301 = 43 * 7), and thus
> > nearly-integer values for the closely related 1/5-comma
> > meantone, but also nearly-integer values for the basic
> > intervals of the central section of the 7-limit lattice.
>
> So when are you going to put up a web page for shismas and
612-et? :)

not before monz first qualifies his current claim that schismas
are well-represented in *614*, i hope . . .

--- In tuning@y..., "monz" <monz@a...> wrote:
> hello all,
>
>
>
> i've just updated the Tuning Dictionary entry
> for "heptameride"
>
> http://sonic-arts.org/dict/heptameride.htm
>
> giving a large table of heptameride values for many
> important JI intervals and also all 23 intervals which
> occur in a 12-tone meantone.
>
> i've discovered that heptamerides offer not only integer
> values for 43edo (because 301 = 43 * 7), and thus
> nearly-integer values for the closely related 1/5-comma
> meantone, but also nearly-integer values for the basic
> intervals of the central section of the 7-limit lattice.

i didn't see 7/6 . . .

> thus, heptamerides are an excellent system of interval
> measurement for use in comparing 7-limit JI and
> 1/5-comma meantone, without needing decimal places.

. . . anyway, good stuff as usual, monz!

> if one is wiling to accept small amounts of error
> in the decimal places, the comparisons can be generalized
> to most of the other important meantone systems (i.e.,
> 1/6-, 1/4-, 2/7-, and 7/26-comma, and golden meantone).

hmm . . . it seems those errors would accumulate rapidly until
they're no longer merely in the decimal places . . .

> another useful feature of heptamerides is that if one
> uses tables of logarithms to find the log of the ratio,
> simply moving the decimal point 3 places to the right
> gives the heptamerides, thus eliminating some calculation.

we can understand why this system was so exciting to sauveur,
since 1/5-comma meantone was pretty much what most
musicians were using in 1701, while computers and calculators
did not exist so calculating logs was a constant necessity for
tuning theorists.

besides 301, other ets that have "advocates" on your et page and
offer excellent approximations to ratios (at least 5-limit ones)
compared with other ets around their size, but for which you have
not yet provided "color lattices", include 1000 and 3072. i noticed
these on the "zooms" i prepared for your page, but which you
never followed up on . . . no doubt you're too busy, sorry to nag . .
.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:

>
> i didn't see 7/6 . . .
>

sorry monz -- found it.

but some of the other intervals -- kleisma, schisma, etc. -- these
aren't really harmonic or melodic intervals one is concerned
about approximating. and, as we've seen in cases like 31-equal,
the best approximation to the just version of such intervals is not
necessarily equal to those intervals' functional representation in
the et. so, to avoid confusion, i'd omit such intervals from your
table, or clarify this point somehow.

hi paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, November 11, 2002 8:01 AM
> Subject: [tuning] Re: Tuning Dictionary update: heptamerides
>
>
> > http://sonic-arts.org/dict/heptameride.htm
> >
> > if one is wiling to accept small amounts of error
> > in the decimal places, the comparisons can be generalized
> > to most of the other important meantone systems (i.e.,
> > 1/6-, 1/4-, 2/7-, and 7/26-comma, and golden meantone).
>
> hmm . . . it seems those errors would accumulate rapidly until
> they're no longer merely in the decimal places . . .

yes, i noticed that myself after sending this post. i'd
say that heptamerides would still be pretty useful for
1/6-comma (on the large end) and *maybe* 1/4-comma (on
the small end), but you're correct that the meantones with
"5ths" smaller than 1/4-comma would accumulate errors rapidly.

> besides 301, other ets that have "advocates" on your et page and
> offer excellent approximations to ratios (at least 5-limit ones)
> compared with other ets around their size, but for which you have
> not yet provided "color lattices", include 1000 and 3072. i noticed
> these on the "zooms" i prepared for your page, but which you
> never followed up on . . . no doubt you're too busy, sorry to nag . .

hmmm ... i could have sworn that i added some info about 1000edo
recently on a webpage update, but apparently i didn't. will do.

can you send links to the graphics you'd like me to use?
IIRC, you were hoping that i'd put red lines on "xoomer.gif"
similar to the first graphic on the "equal temperament" page,
and replace that first graphic with it ... yes?

-monz

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

> IIRC, you were hoping that i'd put red lines on "xoomer.gif"
> similar to the first graphic on the "equal temperament" page,
> and replace that first graphic with it ... yes?

yes . . . though if you prefer, i can do the lines myself. there's
also the upcoming "dual" graph which will show ETs as *lines*, and
commas (or equivalently, families of linear temperament) as the
*points* where the lines (corresponding to the ETs in which the comma
in question vanished) intersect.

let's chat off-list about getting you what you need . . .

hi paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, November 11, 2002 2:58 PM
> Subject: [tuning] Re: Tuning Dictionary update: heptamerides
>
>
> --- In tuning@y..., "monz" <monz@a...> wrote:
>
> > IIRC, you were hoping that i'd put red lines on "xoomer.gif"
> > similar to the first graphic on the "equal temperament" page,
> > and replace that first graphic with it ... yes?
>
> yes . . . though if you prefer, i can do the lines myself. there's
> also the upcoming "dual" graph which will show ETs as *lines*, and
> commas (or equivalently, families of linear temperament) as the
> *points* where the lines (corresponding to the ETs in which the comma
> in question vanished) intersect.

yes, both ... side by side. if you want to do the artwork,
feel free, otherwise i'll eventually get to it.

> let's chat off-list about getting you what you need . . .

okey doke.

-monz

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

> > let's chat off-list about getting you what you need . . .
>
> okey doke.

i e-mailed you yesterday but haven't gotten a response . . . is your
e-mail working? or maybe you just haven't gotten to it yet . . .
anyway, you can see some new graphs on tuning-math . . .