Proposal: a high-order septimal schisma

I was a member of this list a while ago; I just decided to come back
since I'm working on a 7-limit just approximation of ET with a large
number of tones (probably 118).

Since I calculated 7/4 as being very close to a chain of 39 fifths
upwards, I found an interval:

4052555153018976267/4035225266123964416

This is an alternative septimal schisma, an alternative to
33554432/33480783, which is based on a chain of fourteen fifths
downward.

This may have been proposed already, but I didn't see it on the
Scala list, so I posted it anyway.

~Danny~

Uh, I might wanna withdraw this suggestion. My interval is
7.419127281 cents, versus the more rational septimal schisma, which
is 3.804081415. I miscalculated somewhere...

~Danny~

--- In tuning@y..., "DaWier" <dawier@h...> wrote:
> I was a member of this list a while ago; I just decided to come
back
> since I'm working on a 7-limit just approximation of ET with a
large
> number of tones (probably 118).
>
> Since I calculated 7/4 as being very close to a chain of 39 fifths
> upwards, I found an interval:
>
> 4052555153018976267/4035225266123964416
[snip]

Hello, there, Danny, and welcome back to the Tuning List.

> Since I calculated 7/4 as being very close to a chain of 39 fifths
> upwards, I found an interval:

> 4052555153018976267/4035225266123964416

> This is an alternative septimal schisma, an alternative to
> 33554432/33480783, which is based on a chain of fourteen fifths
> downward.

Thank you for calling attention to a ratio related to certain other
intervals that I discussed in a post last year, and which deserves
recognition in its own right.

Interestingly, although I posted last year about a Pythagorean
"tricomma" tuning that uses this approximation of 7:4 as 39 pure
fifths up, or a regular major sixth at 27:16 plus three Pythagorean
commas at 541331:524288 (~23.46 cents) or a tricomma, ~70.38 cents, I
didn't consider the question of a name for the ratio you describe.

It's equal to the usual 3-7 schisma of ~3.80 cents plus a comma of
Mercator at ~3.62 cents, the difference between 53 pure fifths and 31
pure octaves.

What I did propose was the name monzisma for the difference between
the 25:24 (~70.67 cents) and the tricomma of ~70.38 cents, about 0.29
cents. Please see

</tuning/topicId_22165.html#22165>

The monzisma can also be defined as the difference between the
septimal kleisma of 225:224 and the ratio which you give for the
difference between 39 fifths up (less the requisite number of octaves)
and a pure 7:4, for example.

From another viewpoint, your interval is equal to a septimal kleisma
less a monzisma -- but not I'm sure that this equation helps in
finding an ideal name.

While the "tricomma" representations of 7-based intervals are rather
less accurate than the more familiar ones differing by a 3-7 schisma,
my motivation for the tricomma tuning was to combine these approximate
ratios of 7 with quite close approximations of 17:14 and 21:17,
represented as a regular major third (81:64) less tricomma, and a
regular minor third (32:27) plus tricomma.

I tend to explain the tricomma tuning, with two 12-note chains of pure
fifths a tricomma apart, as a kind of just or rational emulation of
36-tET, with the latter tuning renowned for its close approximations
of 7-based intervals (9:7, 7:6, 7:4, etc.) and 14:17:21. The tricomma
tuning has narrower regular semitones and more active regular thirds,
but with a very notable compromise in the accuracy of 7-based ratios.
For an article giving some of the background, please see

</tuning/topicId_22164.html#22164>

Another way to define the ratio you describe would be as the
difference between a tricomma at ~70.38 cents and a pure 28:27 at
~62.96 cents.

Have you any idea for a good name?

Most appreciatively,

Margo Schulter
mschulter@value.net

Hello, there, Danny and everyone.

While I consider the tricomma tuning and approximation of 7-based
intervals very useful, I'd agree that if our goal is to get an
closer approximation of 7-based intervals using ratios of 2 and 3
only, we'd want to take a different approach.

In fact, Joe Monz, the incomparable Monz, called my attention last
year to a very attractive solution -- what I call "nanisma thirds."

With the tricomma intervals, we have approximations differing from
pure ratios of 7 by the usual 3-7 schisma of ~3.80 cents _plus_ the
comma of Mercator at ~3.62 cents.

However, as Monz pointed out to me, by going beyond a cycle of 53 pure
fifths, we can arrive at near-7 intervals impure by the usual 3-7
schisma of ~3.80 cents _less_ the comma of Mercator at ~3.62 cents, or
only about 0.19 cents.

For example, let's consider the familiar 3-7 schisma approximation of
a 12:7 major sixth (~933.13 cents) as a regular Pythagorean major
sixth at 27:16 (~905.87 cents) plus a Pythagorean comma, giving a
ratio of 14348907:8388608 (~929.33 cents).

Now suppose we add another 53 pure fifths to the chain, giving 68 in
all, and producing a ratio of

278128389443693511257285776231761:162259276829213363391578010288128

or ~932.94 cents. This ratio differs from a pure 12:7 by only

649037107316853453566312041152512:648966242035284859600333477874109

or ~0.19 cents, the nanisma.

Curiously, although I had proposed the nanisma as an interval in a
rather theoretical discussion of a possible 106-note tuning, it was
the Monz who suggested 68 fourths up or fifths down as an
approximation for 7:6, giving this small ratio a very practical
application.

Thus, if we seek a "higher-order 3-7 schisma" more accurate than the
usual one equal to what George Secor has most aptly called the
comma of Archytas at 64:63 less the Pythagorean comma, the nanisma
seems a worthy choice. With a long enough chain of fifths, of course,
we might find even closer approximations.

Interestingly, 1024-tET (very close to 1024-note Pythagorean tuning)
provides a close model of the "nanisma thirds."

A 7:6 minor third, for example, is approximated as 228/1024 octave,
equal to 68 near-pure fourths (each 425/1024 octave) up less 28
octaves.

Anyway, I'd like to thank you for raising these questions, and the
Monz for calling my attention to what I describe as "nanisma thirds"
and other intervals varying from ratios such as 7:6, 9:7, and 7:4 by
only about 0.19 cents in a tuning with pure fifths and fourths.

Most appreciatively,

Margo Schulter
mschulter@value.net

On Mon, 26 Aug 2002, M. Schulter wrote:

> Interestingly, although I posted last year about a Pythagorean
> "tricomma" tuning that uses this approximation of 7:4 as 39 pure
> fifths up, or a regular major sixth at 27:16 plus three Pythagorean
> commas at 541331:524288 (~23.46 cents) or a tricomma, ~70.38 cents, I
> didn't consider the question of a name for the ratio you describe.

This should, of course, be 531441:524288.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
> Hello, there, Danny and everyone.
>
> While I consider the tricomma tuning and approximation of 7-based
> intervals very useful, I'd agree that if our goal is to get an
> closer approximation of 7-based intervals using ratios of 2 and 3
> only, we'd want to take a different approach.

It seems to me this problem is identical to finding 7-limit linear temperaments which have an approximate 2 and 3 as generator, and in terms of wedgies that is the same as having a wedgie whose first value is a "1". Here are some possibilities, with the tuning-math names of the corresponding temperaments, and the wedgies:

Dominant Seventh [1,4,-2,-16,6,4]

5 ~ 2^(-4) 3^4 7 ~ 2^6 3^(-2)

Flattone [1,4,-9,-32,17,4]

5 ~ 2^(-4) 3^4 7 ~ 2^17 3^(-9)

Meantone [1,4,10,12,-13,4]

5 ~ 2^(-4) 3^4 7 ~ 2^6 3^(-2)

Hexadecimal [1,-3,5,20,-5,-7]

5 ~ 2^7 3^(-3) 7 ~ 2^(-5) 3^5

Superpythagorean [1,9,-2,-30,6,12]

5 ~ 2^(-12) 3^9 7 ~ 2^6 3^(-2)

Schismic [1,-8,-14,-10,25,-15]

5 ~ 2^15 3^(-8) 7 ~ 2^25 3^(-14)

--- In tuning@y..., "DaWier" <dawier@h...> wrote:
> I was a member of this list a while ago; I just decided to come back
> since I'm working on a 7-limit just approximation of ET with a large
> number of tones (probably 118).
>
> Since I calculated 7/4 as being very close to a chain of 39 fifths
> upwards, I found an interval:
>
> 4052555153018976267/4035225266123964416

I've been pondering the corresponding linear temperament a little recently, because of a piece I've written in 171-et called the "Clinton Variations." In tuning-math terms, the temperament is h118 ^ h171 = 32805/32768 ^ 4375/4374 = [1,-8,39,113,-59,-15].
Would you object to "wier" as a name for this temperament? I ended up referring to it as "schismic" on mp3.com, which bothered me because it isn't precisely correct.

From: "genewardsmith"

>I've been pondering the corresponding linear temperament a little recently,
because of a piece I've written in 171-et called the "Clinton Variations." In
tuning-math terms, the temperament is h118 ^ h171 = 32805/32768 ^ 4375/4374 =
[1,-8,39,113,-59,-15].

>Would you object to "wier" as a name for this temperament? I ended up referring
to it as "schismic" on mp3.com, which bothered me because it isn't precisely
correct.I dunno; I haven't worked out 171-equal yet. I'm fairly new at this
game, and right now I'm looking at using Arabic and Turkish tuning and how they
might work out in Western music and jazz (and possibly an attempt at serial
music later). Right now my big thing is 53-tone. I've already come up with a
5-limit just approximation of 53-tet, similar to Fokker periodicity. I haven't
come up with an interval that could be called a "wierisma" yet.I had a 7-limit
53-tone, but it gave me differences of a breedsma (2401/2400) between chromatic
steps in two places. It was also an irregular system where I just used the
smallest factors for each note. In larger scales, I would have cases where the
next chromatic step up being a LOWER pitch in some cases.My approach is similar
to a periodicity block, where a regular pattern schismatic (32805/32768 = -8
fifths) and septimal schismatic (33554432/33480783 = -14 fifths) adjustments in
a regular cycle of fifths, since I'm aiming for what I originally called a
"justified equal temperament". I'm still at work on this thing, and I'll let
y'all know when I come up with something.Of course, the tuning system probably
had been invented or discovered by someone else 800 years ago....~Danny~

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Hello, there, everyone, and this thread might illustrate how an idea
like the search for an alternative "septimal schisma," or 3-7 schisma
as I call it, can lead different people in slightly different
directions when we each seek to define the problem.

For me, the problem is to find ratios of 2 and 3 which approximate as
closely as possible ratios of 2-3-7-9. These are the factor's on which
I'm focusing (e.g. 7:6, 9:7, 7:4, 8:7, 12:7, 14:9).

Here there are three main solutions which occur to me, with ratios
approximated as a given number of pure 3:2 fifths up (+) or down (-):

(1) Usual 3-7 schisma, 33554432:33480783 (~3.804 cents)
7:4 = ~ -14
7:6 = ~ -15
9:7 = ~ +16

(2) Wierisma (tricomma intervals) --
4052555153018976267:4035225266123964416 (~7.419 cents)
9:7 = ~ -37
7:6 = ~ +38
7:4 = ~ +39

(3) Nanisma --
649037107316853453566312041152512:648966242035284859600333477874109
(~0.189 cents)
7:4 = ~ -67
7:6 = ~ -68
9:7 = ~ +69

The term _wierisma_ seems to me very appropriate for the ratio you've
cited, Danny, by which the "tricomma" approximations of ratios of
2-3-7-9 differ from pure versions, about 7.419 cents.

All three types of 2-3-7-9 approximations using pure fifths and
fourths are modelled rather closely in 1024-tET, where something much
like the nanisma versions gives the most accurate results in this
temperament.

3-7 schisma
-----------
830/1024 octave = ~7:4 (+~3.830 cents)
231/1024 octave = ~7:6 (+~3.832 cents)
368/1024 octave = ~9:7 (-~3.834 cents)

Wierisma (tricomma tuning)
--------------------------
365/1024 octave = ~9:7 (+~7.350 cents)
234/1024 octave = ~7:6 (+~7.348 cents)
833/1024 octave = ~7:4 (+~7.346 cents)

Nanisma
-------
827/1024 octave = ~7:4 (+~0.315 cents)
228/1024 octave = ~7:6 (+~0.317 cents)
371/1024 octave = ~9:7 (-~0.318 cents)

Since the regular fifth of 1024-tET at 599/1024 octave or ~701.953
cents is only about 0.019 cents narrow of a pure 3:2 (~701.955 cents),
it's not surprising that these 2-3-7-9 approximations are very close
to those obtained with pure fifths and fourths.

Most appreciatively,

Margo Schulter
mschulter@value.net

Could someone please explain what the numbers in brackets are? [1,
-8,39,113,-59,-15] For example. Thanks

Paul

--- In tuning@y..., paul.hjelmstad@u... wrote:

> Could someone please explain what the numbers in brackets are? [1,
> -8,39,113,-59,-15] For example. Thanks

It's a wedge invariant, as discussed on tuning-math. Since there is exactly one for each 7-limit linear temperament, one thing it accomplishes is to give a standard identifier for such temperaments. I often search the tuning-math achives using them. They have various other talents as well.

Thanks. Hope this doesn't sound stupid, but could you tell me the
significance of each number in the "wedge invariant"? (Being really literal
please) Are they the powers of 2,3,5,7 or something?

genewardsmith
<genewardsmith@ju To: tuning@yahoogroups.com
no.com> cc:
Subject: [tuning] Re: Proposal: a high-order septimal schisma
09/03/2002 10:44
PM
Please respond to
tuning

--- In tuning@y..., paul.hjelmstad@u... wrote:

> Could someone please explain what the numbers in brackets are? [1,
> -8,39,113,-59,-15] For example. Thanks

It's a wedge invariant, as discussed on tuning-math. Since there is exactly
one for each 7-limit linear temperament, one thing it accomplishes is to
give a standard identifier for such temperaments. I often search the
tuning-math achives using them. They have various other talents as well.

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--- In tuning@y..., paul.hjelmstad@u... wrote:
>
> Thanks. Hope this doesn't sound stupid, but could you tell me the
> significance of each number in the "wedge invariant"? (Being really literal
> please) Are they the powers of 2,3,5,7 or something?

It's hardly stupid, and in fact it's complicated enough I suggest further discussion should take place on tuning-math, and not here. There *are* commas of various kinds hidden in it, for which the numbers are exponents, and as we just saw, if the first number is "1" then 5 and 7 can be expressed in terms of 2 and 3; however it really comes from multilinear algebra, and is not such a good thing to discuss here. There's quite a lot in the tuning-math archives about it.

Let p = 2^u1 3^u2 5^u3 7^u4 and q = 2^v1 3^v2 5^v3 7^v4, then

p ^ q = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2,
u1*v3-v1*u3,u1*v4-v1*u4]

Let r be the mapping to primes of an equal temperament given
by r = [u1, u2, u3, u4], and s be given by [v1, v2, v3, v4]. This
means r has u1 notes to the octave, u2 notes in the approximation of 3, and so forth; hence [12, 19, 28, 24] would be the usual 12-equal, and [31, 49, 72, 87] the usual 31-et. The wedge now is

r ^ s = [u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u3*v4-v3*u4,
u4*v2-u2*v4,u2*v3-v2*u3]

Whether we've computed in terms of commas or ets, the wedge product of the linear temperament is exactly the same, up to sign.

If the wedgie is [u1,u2,u3,u4,u5,u6] then we have commas given by

2^u6 3^(-u2) 5^u1
2^u5 3^u3 7^(-u1)
2^u4 5^(-u3) 7^u2
3^u4 5^u5 7^u6