Topic: Rank 3 for 1200-EDO: simplemint (1200.0, 704.0, 58.0)

1 scales

Thread (2 messages)

From: Margo Schulter (2010-08-26)
Subject: Rank 3 for 1200-EDO: simplemint (1200.0, 704.0, 58.0)

Hello, all.

Since the topic of rank 2 (linear) temperaments in 1200-EDO has been
raised, how about a rank 3 temperament?

Here's what I call simplemint, with a 2/1 octave (1200 cents) as the
period; a fifth at 704 cents as the second generator; and an interval
of 58 cents between two 12-note chains of fifths as the third
generator. For those who may find it more convenient to download a
Scala file directly rather than to cut and paste from this post, I've
included a URL:

<http://www.bestII.com/~mschulter/simplemint24.scl>

! simplemint24.scl
!
Rank 3 temperament (2-3-7-9-11-13), 704c 5th, 58c spacing (1200-EDO)
  24
!
  58.00000
  128.00000
  186.00000
  208.00000
  266.00000
  288.00000
  346.00000
  416.00000
  474.00000
  496.00000
  554.00000
  624.00000
  682.00000
  704.00000
  762.00000
  832.00000
  890.00000
  912.00000
  970.00000
  992.00000
  1050.00000
  1120.00000
  1178.00000
  2/1

One lesson of this tuning for me is that 1200-EDO doesn't have to mean
"inaccurate." We'll see, using for example George Secor's High
Tolerance Temperament in 29 notes (HTT-29) with its tolerance of about
3.25 cents as a standard, that lots of intervals of 2-3-7-9-11-13 are
within that tolerance.

How do we give an overview, at a glance, of what resources are
available? Scala's SHOW INTERVALS or SHOW /LINE INTERVALS provide one
handy solution, of course: but here are some ideas for one alternative
format for sharing some information about a rank 3 system.

We might begin by listing the three generators of a rank 3 temperament:

   (r3=   g1     g2     g3
        1200.0, 704.0, 58.0 )

The next thing we might want to know is what intervals we'll find, and
how many of them in a tuning of a specific size, here 24. The
following two "family analysis" tables do this. The first takes the
bottom note of the lower 12-note chain as the 1/1 (actually the
288-cent step, step 6 in the Scala file) and lists the intervals we
get first moving through this chain (itself a rank 2 temperament), and
then moving from the bottom to the top of the second chain:

   families:
   [ FAM1-r2:  704  208  912  416 1120  624
     FAM2-r2:  128  832  336 1040  544
     FAM3-r3:   58  762  266  970  474
     FAM4-r3: 1178  682  186  890  394  1098 602 ]

Note that the intervals are grouped into four "families." The first
has the "regular diatonic" intervals; the second the "Zalzalian" or
neutral intervals (singly augmented or diminished in conventional
naming); the third the "septimal" intervals; and the fourth the more
remote intervals, which sometimes have a 5-limit or meantone flavor.
While the first two families are present in a single 12-note chain
(rank 2), the third and fourth involve the 58-cent generator of the
rank 3 system.

Additionally, there are some intervals not encountered above which we
can find by setting our 1/1 as the note at the high end of the first
12-note chain of fifths, and then moving from the bottom to the top of
the second chain. These "supplemental" intervals follow the same sequence 
as the rank 2 intervals of the first and second families in the previous
table, with each of these intervals 10 cents larger than its rank 2
counterpart above, except for the 58-cent generator itself which is
the last interval in this table, and in a system with longer rank 2
chains would have the 48-cent enharmonic diesis formed from 12 fifths up
as its counterpart, thus also fitting this pattern:

   families (supplemental r3, from +11=544.0)
     [ 714  218  922   426  1130  634
       138  842  346  1050   554   58 ]

Some of these are very important: our approximations of 13/9, 13/12,
13/8, 11/9, 11/6, and 11/8 -- and all within 3 cents of just.

Of course, there are the prime approximations! The following table
lists the relevant primes (2-3-7-11-13), shows the rank 2 or rank 3
chain or combination of generators to best approximate each prime, 
the size of the tempered interval in cents used for each
approximation, the difference from JI in cents, and the _number_ of
locations at which this approximation is available.

   primes:
   <     2        3       7        11       13
         r2       r2      r3       r3       r3
       period     +1    +3/+g3   -1/+g3   -4/+g3
     1200.000  704.000  970.000  554.000  842.000
         J      +2.045   +1.174   +2.682   +1.472
                 n=22     n=9      n=11     n=8    >

The third line of this table, showing chain or combination of
generators for each approximated prime, may call for explanation. For
a rank 2 approximation, within a single chain of fifths, we simply
show the number of fifths up (+) or down (-). For rank 3, we show two
factors: the number of fifths to move up or down on a single chain,
plus the addition or subtraction of the third generator (g3), here at
58 cents (+g3 or -g3).

Thus to get a 7/4 minor seventh, we move up three fifths (+3),
arriving at a 912-cent major sixth; plus up the third generator (+g3)
of 58 cents, arriving at 970 cents. The +g3 sign tells us that we must
be able to add this 58 cent generator, and thus we must start from a
note on the _lower_ 12-note chain, and more specifically one which has
a major sixth at 3 fifths up (+3), of which there are 9.

The last item, the number of locations where an interval is available,
is something that David Keenan rightly pointed to as a very important
factor in assessing "chain-of-fifth" tunings; and it is relevant to
rank 2 and rank 3 temperaments alike, especially for smaller tuning
sets. (And on this list, 24 might still be considered a "small" set.)

Finally, in addition to primes, we just might be looking for things
like neutral or middle seconds at or close to superparticular ratios. 
The following

   superparticular middle seconds:
   {  14:13     13:12     12:11     11:10
        r2        r3        r3        r2
        +7     -5/+g3    +2/-g3      -10
     128.000   138.000   150.000   160.000
      -0.298    -0.573    -0.637    -5.004
       n=10      n=7       n=10      n=4   }

Again, the third line tells us, if the approximation is available in a
single rank 2 chain, how many fifths we must move up or down -- for
example, +7 for 14:13 (the apotome or chromatic semitone). And for
rank 3 approximations, we have first motion along a single chain, and
then the addition or subtraction of the 58-cent generator. Thus to get
a 13:12 approximation, we first move 5 fifths down (arriving at a
limma or diatonic semitone at 80 cents), and then up by 58 cents (+g3)
to arrive at 138 cents.

Similarly, to approximate 12:11, we start by moving up two fifths (+2)
on the chain of fifths, arriving at a 208-cent tone or major second,
and then down by the 58-cent third generator (-g3) to arrive at 150
cents.

As with the prime approximations, it's helpful to know not only how,
and how accurately, an interval is approximated, but at how many
locations, and this information is duly included on the last line of
the table.

Best,

Margo Schulter
mschulter@...

From: genewardsmith (2010-08-28)
Subject: Re: Rank 3 for 1200-EDO: simplemint (1200.0, 704.0, 58.0)

--- In [email protected], Margo Schulter <mschulter@...> wrote:

>    primes:
>    <     2        3       7        11       13
>          r2       r2      r3       r3       r3
>        period     +1    +3/+g3   -1/+g3   -4/+g3
>      1200.000  704.000  970.000  554.000  842.000
>          J      +2.045   +1.174   +2.682   +1.472
>                  n=22     n=9      n=11     n=8    >

Another way to denote this is
[<1  0 * 1  4  6|,
 <0  1 * 3 -1 -4|,
 <0  0 * 3  1  1|]

Here the asterisks mean the 5 is omitted, though alternatively you could map to 0. This leads to a rank three no-fives system tempering out 352/351 and 364/363 (and hence their product, 896/891.) Or you could leave 5 in it and get a rank four system instead, which Graham seems to be calling 34 & 41 & 58 & 87.