an expansion of Partch's 43-tone scale to 81 notes

While working on a webpage on how to approximate just intonation on fretless
bass by using 72-edo (dividing every 'fret' into sixths), I decided to add
some intervals to Harry Partch's scale, since I was using the intervals from
his 43 tone-per-octave just scale for examples.

Then I noticed various patterns in how Partch picked his 43 tones, and ended
up coming up with a superset of my own, using the following rules:

1) Maximum denominator is 99.
2) For the prime number 3, it can have an exponent as high as 4 in the
numerator or denominator.
3) For the primes 5, 7 and 11, no exponent above one is allowed in the prime
factorization of the numerator and denominator. Though intervals such as 7/5
and 11/7 are allowed, 49/32 is not.
4) Only one of the above three primes can be included each the numerator and
denominator. 35/32 and 55/48 are illegal, for example.

The result is an 81 tone-per-octave scale with the following intervals
(hopefully I didn't make any errors):

1/1 81/80 33/32 21/20 16/15 12/11 11/10 10/9
9/8 8/7 7/6 32/27 6/5 11/9 5/4
14/11 9/7 21/16
4/3 27/20 11/8 7/5 10/7 16/11 40/27
3/2 32/21 14/9 11/7 8/5 18/11 5/3
27/16 12/7 7/4 16/9 9/5 20/11 11/6 15/8
40/21 64/33 160/81

Recommended name: 'SuperPartch'.

Note that some of the pitches are very close together, so much they map to
the same note in 72-edo. But in quarter-comma sagittal notation (217-edo),
they all find themselves expressed as distinct pitches. I haven't checked
third-comma (152-edo) though.

This could be added to the Scala archive, where I rediscovered a ClownTone
tuning I completely forgot about (it's a 19-limit redo of the 12-tone scale
somewhat like Carlos Harmonic). I'm listening to a Scala-retuned MIDI of
Bach's Inventions. It sounds horrible.

From: "Danny Wier" <dawiertx@...>

> 1/1 81/80 33/32 21/20 16/15 12/11 11/10 10/9
> 9/8 8/7 7/6 32/27 6/5 11/9 5/4
> 14/11 9/7 21/16
> 4/3 27/20 11/8 7/5 10/7 16/11 40/27
> 3/2 32/21 14/9 11/7 8/5 18/11 5/3
> 27/16 12/7 7/4 16/9 9/5 20/11 11/6 15/8
> 40/21 64/33 160/81

Uh, that's EXACTLY the same thing as Partch-43. Let's try this again.

1/1 81/80 64/63 45/44 33/32 28/27 22/21 21/20
16/15 15/14 88/81 12/11 11/10 10/9
9/8 112/99 8/7 7/6 33/28 32/27
6/5 40/33 11/9 27/22 99/80 56/45
5/4 81/64 80/63 14/11 9/7 128/99 21/16
4/3 27/20 15/11 11/8 112/81 88/63 7/5
45/32 64/45 10/7 63/44 81/56 16/11 22/15 40/27
3/2 32/21 99/64 14/9 11/7 63/40 128/81
8/5 45/28 160/99 44/27 18/11 33/20
5/3 27/16 56/33 12/7 7/4 99/56 16/9
9/5 20/11 11/6 81/44 28/15
15/8 40/21 21/11 27/14 64/33 88/45 63/32 160/81
2/1

Now aren't you glad I still can't sleep?

Also, forgot to mention, different notes have different 'colors', according
to their prime factorization:

noire: Pythagorean (1/1, 2/1, 3/2, 4/3, 9/8, 16/9)
rouge: 5-limit (5/4, 8/5, 5/3, 6/5, 9/5, 10/9)
bleue: 7-limit without 5 (7/4, 7/6, 8/7, 9/7, 28/27)
pourpre: 7-limit with 5 (7/5, 10/7, 15/14)
verte: 11-limit without 5 and 7 (11/8, 11/6, 12/11, 33/32)
jaune: 11-limit with 5 but not 7 (11/10, 40/33)
aqua: 11-limit with 7 but not 5 (11/7, 14/11, 33/28)
blanche: 11-limit with 5 and 7 (not used in this scale)

In the future, names like 'orange', 'brune' and 'grise' could be applied to
13-limit pitches. They don't have to be French names either; I just chose
that language. I could've just as well picked German, English or Japanese.

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:
> While working on a webpage on how to approximate just intonation on
fretless
> bass by using 72-edo (dividing every 'fret' into sixths), I decided
to add
> some intervals to Harry Partch's scale, since I was using the
intervals from
> his 43 tone-per-octave just scale for examples.
>
> Then I noticed various patterns in how Partch picked his 43 tones,
and ended
> up coming up with a superset of my own, using the following rules:
>
> 1) Maximum denominator is 99.
> 2) For the prime number 3, it can have an exponent as high as 4 in
the
> numerator or denominator.
> 3) For the primes 5, 7 and 11, no exponent above one is allowed in
the prime
> factorization of the numerator and denominator. Though intervals
such as 7/5
> and 11/7 are allowed, 49/32 is not.
> 4) Only one of the above three primes can be included each the
numerator and
> denominator. 35/32 and 55/48 are illegal, for example.
>
> The result is an 81 tone-per-octave scale with the following
intervals
> (hopefully I didn't make any errors):
>
> 1/1 81/80 64/63 45/44 33/32 28/27 22/21 21/20
> 16/15 15/14 88/81 12/11 11/10 10/9
> 9/8 112/99 8/7 7/6 33/28 32/27
> 6/5 40/33 11/9 27/22 99/80 56/45
> 5/4 81/64 80/63 14/11 9/7 128/99 21/16
> 4/3 27/20 15/11 11/8 112/81 88/63 7/5
> 45/32 64/45 10/7 63/44 81/56 16/11 22/15 40/27
> 3/2 32/21 99/64 14/9 11/7 63/40 128/81
> 8/5 45/28 160/99 44/27 18/11 33/20
> 5/3 27/16 56/33 12/7 7/4 99/56 16/9
> 9/5 20/11 11/6 81/44 28/15
> 15/8 40/21 21/11 27/14 64/33 88/45 63/32 160/81
> 2/1
>
> Recommended name: 'SuperPartch'.
>
> Note that some of the pitches are very close together, so much they
map to
> the same note in 72-edo. But in quarter-comma sagittal notation
(217-edo),
> they all find themselves expressed as distinct pitches.

Dave & I have since changed the Sagittal notation for both 11-limit
JI and the 3/2,1/4-comma planar *exact* (i.e., not based on any EDO
whatsoever; of course, these are still expressed as distinct pitches).

> I haven't checked
> third-comma (152-edo) though.

81/64 and 80/63 are both 52deg152. However, 183-ET is an excellent
division (consistent to the 17 limit) that will represent all of the
above pitches uniquely. You'll also find that consecutive degrees of
183 very nicely determine 11-limit consonances as alterations to
tones in a sequence of best fifths:

7/5 is 1deg lower than Gb
14/11 is 2deg higher than E
5/4 is 3deg lower than E
7/4 is 4deg lower than Bb
11/7 is 5deg lower than G#
11/10 is 6deg lower than D
11/8 is 8deg higher than F

Therefore, symbols for each of these degrees of 183 will easily
identify those ratios. (In our updated Sagittal system, there are a
couple of minor differences between the symbols in the 183-ET set and
those for 11-limit JI, which simplifies the 183 notation.)

And in case you wondered what 7deg is good for:

13/8 is 7deg higher than Ab (although 13/8 would be notated more
easily by lowering A by 10deg).

If you want to see what these Sagittal symbols look like in 183, run
Scala and execute the commands:

equal 183
set nota sa183

Then either go into staff view or play the chromatic clavier to see
the actual symbols. If you want the mixed-symbol version of the
notation (using conventional sharp and flat symbols), then:

set sagi mixed

To return to the pure-symbol version of Sagittal:

set sagi pure

To view your SuperPartch scale in Sagittal, make a .scl file
containing your ratios and open it in Scala. Then execute the
following command to set notation to medium-precision (or level one,
athenian) Sagittal JI:

set nota saji1

You can see then the JI symbols either on the chromatic clavier or in
staff view.

Or you can use the ratio vector calculator (under Tools) to enter the
ratios. (You don't even need a .scl file to see the Sagittal JI
notation this way.)

Dave and I hope to have some Sagittal documentation available soon
that explains what a lot of the new symbols mean, but I've given you
enough information to get started and hopefully have a little fun
with it.

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> 81/64 and 80/63 are both 52deg152. However, 183-ET is an excellent
> division (consistent to the 17 limit) that will represent all of the
> above pitches uniquely.

There's also tempering to be considered. I took the four smallest
commas which arise as the difference between a scale interval and an
11-limit consonance for both the Partch Genesis scale and Danny's
81-note scale; these turned out to be independent, making life a
little easier. The results suggested trying 72 for Genesis and 94 for
the 81-note scale (of course 94-81=13, so one might feel there's not a
lot of point in stopping there.) Below I give an analysis for the
resulting planar and linear temperaments arising from taking the
smallest commas; in the case of Genesis, I get a linear temperament
which keeps turning up on my 7 and 11 limit lists with no name, so I'm
proposing to name it "Harry"; this of course means the 1/3-fifth
tripling of septimal schismic which arises from the 81-note becomes
"Danny".

Partch Genesis scale

smallest commas: 540/539, 441/440, 8019/8000, 385/384

spacial: 540/539

planar: {540/539,441/440} TM={243/242,441/440}

linear: {540/530,441/440,8019/8000} TM={243/242,441/440,4000/3993}
wedgie=<<12 34 20 30 26 -2 6 -49 -48 15||
mapping=[<2 4 7 7 9|, <0 -6 -17 -10 -15|]
Harry
ets: 58, 72, 130

et: 72

Wier 81-note scale

smallest commas: 32805/32768, 19712/19683, 4000/3993, 540/539

spacial: 32805/32768

planar: {32805/32786, 19712/19683} TM={385/384, 19712/19683}

linear: {32805/32786, 19712/19683, 4000/3993} TM={385/384, 4000/3993,
19712/19683}
wedgie=<<3 -24 52 -25 -45 74 -50 188 25 -250||
mapping=[<1 2 -1 10 0|, <0 -3 24 -52 25|]
Danny
ets: 65, 94, 159, 253

et: 94

It seems if one was going to expand the diamond one could first set them up
on different tonics such as opposite ends of the eikosany of simpler
extensions upon closely related tonics such as 3/2 and 4/3 or increasing
the limit. Other wise there might not be much use even starting with the
diamond in the first place.
The other day i noticed that if one sets up a diamond with the factors
as a sequence of Superparticulars to each other. The layout of the tones
makes much more sense melodically in that one could play an ascending or
descending sequence quite easily. This makes less sense harmonically (
not that it was used that way that much) and also the range of the total
instrument becomes less. But a real example will show more about how the
scale ends up plying out on the instrument.
http://www.anaphoria.com/slimdiamond.gif

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

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

> Partch Genesis scale
>
> smallest commas: 540/539, 441/440, 8019/8000, 385/384

I've mentioned harry and miracle in connection with Genesis, but if
we take the above four commas three at a time, we get four different
temperaments, all compatible with 72. Leaving off the largest,
385/384, gives us harry. Leaving off the second largest, 8019/8000,
gives us miracle. Leaving off 441/440 gives us hanson, and leaving
off 540/539 gives us unidec. The 46 note unidec DE would seem to be
a pretty plausible 11-limit scale of around 43 in size; 53 notes of
11-limit hanson another good possibility. Of course Miracle[41] and
Harry[58] I've already mentioned.

Of course none of these things are versions of Genesis, but tempered
scales of about the same size in temperaments which would be good
choices for tempering Genesis.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > Partch Genesis scale
> >
> > smallest commas: 540/539, 441/440, 8019/8000, 385/384
>
> I've mentioned harry and miracle in connection with Genesis, but if
> we take the above four commas three at a time, we get four
different
> temperaments, all compatible with 72. Leaving off the largest,
> 385/384, gives us harry. Leaving off the second largest, 8019/8000,
> gives us miracle. Leaving off 441/440 gives us hanson,

Since when does Hanson go beyond 5-limit all the way to 11-limit?

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> Since when does Hanson go beyond 5-limit all the way to 11-limit?

Since the discussion of renaming "catakleismic" to "hanson".

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > Since when does Hanson go beyond 5-limit all the way to 11-limit?
>
> Since the discussion of renaming "catakleismic" to "hanson".

Was this your reasoning:

/specmus/topicId_unknown.html#1727

?

As you can see, I didn't buy it.

Now, if you'd care to turn your sharp mathematical mind to the cactus
I posted and the bouquet I'm about to, you might ponder that each can
be seen as a discrete subgroup of the complex multiplicative group,
with the generator G*e^(i*d), where

()d is any of the usual meantone or schismic or other-diatonic
generators (i.e., a fourth or fifth) expressed in units such that
2*pi (the period) is an octave;

()G is ideally a constant, a "growth" parameter for the plant in
question.

Indeed, in complex notation, points of the lattice can be written as
G^k*e^(i*k*d). These form a group isomorphic to the integers (the
isomorphism is k -> G^k*e^(i*k*d).

Some plants also exhibit multiple periods per octave, so the musical
analogy is even stronger (if you've followed my papers and/or the
tuning-math discussion). Maybe you'd like to talk, on tuning-math,
about transformations of the tone group to complex multiplicative
groups. I have a 10&16 cactus I'd like to label, but I don't know
what temperament it corresponds to . . .

P.S. If the distance of the image of k (in the isomorphism above)
from the origin is given by e^(k/X), how do we express G in terms of
X?

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> I have a 10&16 cactus I'd like to label, but I don't know
> what temperament it corresponds to . . .

Let's try 5-limit . . .

The val for 10 (how 10-equal maps the primes 2, 3, and 5) is

<10 16 23]

and for 16

<16 25 37]

The cross product of these two is the monzo

[-17 2 6>

So we are tempering out the comma 2^-17 * 3^2 * 5^6, or

140625/131072

Not a likely candidate for tempering, but there it is.

7-limit might be a bit more reasonable . . . Gene? (Let's move this
to tuning-math.)

I posted on tuning-math and then searched for previous occurrences of
it.

This temperament has been referred to most recently as Gene's Private
Reserve -- 7-limit, #66.

Earlier, Herman devoted an entire post to this temperament:

/tuning-math/message/8872

Herman -- any ideas for notation? Made any music with it? etc.

I was planning to include only 23 7-limit 'linear' temperaments in my
paper, but this one would only be #26 by the ranking I used. Given
the supporting plants, though, I may have to "budge" . . .

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > I have a 10&16 cactus I'd like to label, but I don't know
> > what temperament it corresponds to . . .
>
> Let's try 5-limit . . .
>
> The val for 10 (how 10-equal maps the primes 2, 3, and 5) is
>
> <10 16 23]
>
> and for 16
>
> <16 25 37]
>
> The cross product of these two is the monzo
>
> [-17 2 6>
>
> So we are tempering out the comma 2^-17 * 3^2 * 5^6, or
>
> 140625/131072
>
> Not a likely candidate for tempering, but there it is.
>
> 7-limit might be a bit more reasonable . . . Gene? (Let's move this
> to tuning-math.)

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:

> Was this your reasoning:
>
> /specmus/topicId_unknown.html#1727
>
> ?
>
> As you can see, I didn't buy it.

You didn't like it then, but what I was talking about was the
discussion on tuning-math a little after the TOP explosion; I
proposed this convention then and I thought you favored it. Don't
you think some sort of rhyme or reason in these matters would be a
good thing? What's your beef?

> Now, if you'd care to turn your sharp mathematical mind to the
cactus
> I posted and the bouquet I'm about to, you might ponder that each
can
> be seen as a discrete subgroup of the complex multiplicative
group,
> with the generator G*e^(i*d), where
>
> ()d is any of the usual meantone or schismic or other-diatonic
> generators (i.e., a fourth or fifth) expressed in units such that
> 2*pi (the period) is an octave;

Right, but having said all that why not now just set z = G*e^(i*d)?

> ()G is ideally a constant, a "growth" parameter for the plant in
> question.
>
> Indeed, in complex notation, points of the lattice can be written
as
> G^k*e^(i*k*d). These form a group isomorphic to the integers (the
> isomorphism is k -> G^k*e^(i*k*d).

Then the isomorphism is more simply written as k |--> z^k, which is
like a rank one group (equal temperament) mapping, except that z is
a complex number.

> Some plants also exhibit multiple periods per octave, so the
musical
> analogy is even stronger (if you've followed my papers and/or the
> tuning-math discussion). Maybe you'd like to talk, on tuning-math,
> about transformations of the tone group to complex multiplicative
> groups. I have a 10&16 cactus I'd like to label, but I don't know
> what temperament it corresponds to . . .
>
> P.S. If the distance of the image of k (in the isomorphism above)
> from the origin is given by e^(k/X), how do we express G in terms
of
> X?

The image of k is z^k, and the log of that is k*z; the real part of
this is the log of the absolute value of z^k, which is the distance
from the origin of z^k.

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

> The image of k is z^k, and the log of that is k*z; the real part of
> this is the log of the absolute value of z^k, which is the distance
> from the origin of z^k.

Sorry, I was trying to say ln(z^k) = k ln(z). The real part of ln(z)
is ln(G), so the distance to the orgin of z^k is G^k; in other words,
we have "notes" situated on a logarithmic spiral.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> I posted on tuning-math and then searched for previous occurrences of
> it.
>
> This temperament has been referred to most recently as Gene's Private
> Reserve -- 7-limit, #66.
>
> Earlier, Herman devoted an entire post to this temperament:
>
> /tuning-math/message/8872
>
> Herman -- any ideas for notation? Made any music with it? etc.
>
> I was planning to include only 23 7-limit 'linear' temperaments in my
> paper, but this one would only be #26 by the ranking I used. Given
> the supporting plants, though, I may have to "budge" . . .

You are already suggesting a temperament list which will be pretty
thin fare for anyone trying to compose music along my lines; your list
as I recall it refects your own practice, which isn't everyone's. It
seems to me if you are going to stick in something with a TM basis of
50/49 and 525/512 you could also have found some space for some higher
accuracy temperaments, or at least ennealimmal.

This baby has a poptimal generator of 5/26, which serves (quite well)
as an 8/7; which is, incidentally, *also* poptimal. In other words, we
can go this in 26 equal or we can revel in pure 7s by taking two
chains of 8/7s a half-octave apart. All of which means you or Herman
ought to be able to come up with a nifty name for it, and also means
it really does, as you suggested, make more sense as a 7-limit
temperament.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > The image of k is z^k, and the log of that is k*z; the real part
of
> > this is the log of the absolute value of z^k, which is the
distance
> > from the origin of z^k.
>
> Sorry, I was trying to say ln(z^k) = k ln(z). The real part of ln
(z)
> is ln(G), so the distance to the orgin of z^k is G^k; in other
words,
> we have "notes" situated on a logarithmic spiral.

I still don't know how to express G in terms of X.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> I still don't know how to express G in terms of X.

Sure you do; if the distance to the origin is G^k and also e^(k/X),
then G^k = e^(k/X), and taking k-th roots tells us G = e^(1/X). What
the point of knowing that is, I don't know.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > I still don't know how to express G in terms of X.
>
> Sure you do; if the distance to the origin is G^k and also e^(k/X),
> then G^k = e^(k/X), and taking k-th roots tells us G = e^(1/X).
What
> the point of knowing that is, I don't know.

Thanks for your help, Gene.

wallyesterpaulrus wrote:

> I posted on tuning-math and then searched for previous occurrences of > it.
> > This temperament has been referred to most recently as Gene's Private > Reserve -- 7-limit, #66.
> > Earlier, Herman devoted an entire post to this temperament:
> > /tuning-math/message/8872
> > Herman -- any ideas for notation? Made any music with it? etc.

I've been too busy to do much of anything with music lately. Then the last week or so I've had a cold and I've been getting behind with my email. But I was basically looking for something that would work well with sharp octaves, seeing what would show up that you might overlook if you assume 1200-cent octaves.