More 270

In case anyone is inspired to consider 270 in the light of a 13-limit
notation system, this might help: the TM basis is 676/675, 1001/1000,
1716/1715, 3025/3024 and 4096/4095. I used this to compute the first
twenty notes of a Fokker block, and used that to compute all the
13-limit intervals with Tenney height less than a million up to twenty
270-et steps, with the following result:

1 {625/624, 352/351, 351/350, 847/845, 385/384, 441/440, 540/539,
364/363, 729/728, 325/324}
2 {196/195, 225/224, 176/175, 896/891, 243/242, 169/168, 640/637}
3 {245/243, 126/125, 121/120, 975/968, 275/273, 144/143, 832/825, 572/567}
4 {99/98, 100/99, 105/104, 512/507, 91/90}
5 {78/77, 81/80, 875/864, 245/242, 507/500}
6 {65/64, 686/675, 64/63, 343/338, 66/65, 625/616}
7 {891/875, 648/637, 847/832, 55/54, 56/55, 637/625}
8 {49/48, 50/49, 729/715, 910/891, 875/858, 143/140, 864/847}
9 {864/845, 169/165, 351/343, 819/800, 968/945, 128/125, 45/44}
10 {352/343, 525/512, 77/75, 693/676, 416/405, 40/39}
11 {1001/972, 605/588, 250/243, 36/35, 147/143}
12 {375/364, 567/550, 624/605, 65/63, 33/32}
13 {121/117, 405/392, 91/88, 512/495, 507/490, 125/121, 336/325}
14 {28/27, 175/169, 648/625, 729/704, 660/637, 539/520, 363/350}
15 {26/25, 27/26, 80/77, 343/330}
16 {176/169, 729/700, 169/162, 126/121, 715/686, 1001/960, 704/675, 25/24}
17 {847/810, 392/375, 256/245, 735/704, 882/845, 117/112, 448/429}
18 {507/484, 245/234, 288/275, 22/21}
19 {150/143, 104/99, 360/343, 605/576, 1024/975, 21/20}
20 {625/594, 637/605, 455/432, 256/243, 825/784, 81/77, 616/585,
539/512, 875/832}

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> In case anyone is inspired to consider 270 in the light of a 13-
limit
> notation system, this might help: the TM basis is 676/675,
1001/1000,
> 1716/1715, 3025/3024 and 4096/4095. I used this to compute the first
> twenty notes of a Fokker block, and used that to compute all the
> 13-limit intervals with Tenney height less than a million up to
twenty
> 270-et steps, with the following result:
>
> 1 {625/624, 352/351, 351/350, 847/845, 385/384, 441/440, 540/539,
364/363, 729/728, 325/324}
> ...
> 8 {49/48, 50/49, 729/715, 910/891, 875/858, 143/140, 864/847}

This list misses a couple of intervals that Dave and I have found to
be valuable for notating 11-limit consonances, and we have chosen the
symbols for these two intervals to notate 1deg and 8deg of 311:

1deg 5103:5120 (3^6*7:2^10*5), the 5:7 kleisma, symbol |(, notates
7/5 and 10/7
8deg 45056:45927 (2^12*11:3^8*7), the 7:11 comma, symbol (|, notates
11/7 and 14/11

The ratios 11/7 and 14/11 may also be notated with:
2deg 891:896, the 7:11 kleisma, symbol )|(
but both symbols are useful to provide alternate spellings.

Should you question the advisability of using a ratio with such large
numbers as 45056:45927 for a common symbol-element (Sagittal flag) in
a notation, let me point out that when it is combined with other
commonly used Sagittal flags, the resulting ratios are much simpler:

|( + (| = (|(, 5103:5120 * 45056:45927 = 44:45
|) + (| = (|), 63:64 * 45056:45927 = 704:729
|\ + (| = (|\, 54:55 * 45056:45927 = 8192:8505, ~26:27 (which uses
the same symbol)

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > In case anyone is inspired to consider 270 in the light of a 13-
> limit
> > notation system, this might help: the TM basis is 676/675,
> 1001/1000,
> > 1716/1715, 3025/3024 and 4096/4095. I used this to compute the
first
> > twenty notes of a Fokker block, and used that to compute all the
> > 13-limit intervals with Tenney height less than a million up to
> twenty
> > 270-et steps, with the following result:
> >
> > 1 {625/624, 352/351, 351/350, 847/845, 385/384, 441/440, 540/539,
> 364/363, 729/728, 325/324}
> > ...
> > 8 {49/48, 50/49, 729/715, 910/891, 875/858, 143/140, 864/847}
>
> This list misses a couple of intervals that Dave and I have found
to
> be valuable for notating 11-limit consonances, and we have chosen
the
> symbols for these two intervals to notate 1deg and 8deg of 311:
>
> 1deg 5103:5120 (3^6*7:2^10*5), the 5:7 kleisma, symbol |(, notates
> 7/5 and 10/7
> 8deg 45056:45927 (2^12*11:3^8*7), the 7:11 comma, symbol (|,
notates
> 11/7 and 14/11

Well, these have more than three digits in the numerator and
denominator, so were explicitly excluded from Gene's list.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> > > In case anyone is inspired to consider 270 in the light of a 13-
limit
> > > notation system, this might help: the TM basis is 676/675,
1001/1000,
> > > 1716/1715, 3025/3024 and 4096/4095. I used this to compute the
first
> > > twenty notes of a Fokker block, and used that to compute all the
> > > 13-limit intervals with Tenney height less than a million up to
twenty
> > > 270-et steps, with the following result:
> > >
> > > 1 {625/624, 352/351, 351/350, 847/845, 385/384, 441/440,
540/539, 364/363, 729/728, 325/324}
> > > ...
> > > 8 {49/48, 50/49, 729/715, 910/891, 875/858, 143/140, 864/847}
> >
> > This list misses a couple of intervals that Dave and I have found
to
> > be valuable for notating 11-limit consonances, and we have chosen
the
> > symbols for these two intervals to notate 1deg and 8deg of 311:
> >
> > 1deg 5103:5120 (3^6*7:2^10*5), the 5:7 kleisma, symbol |(,
notates 7/5 and 10/7
> > 8deg 45056:45927 (2^12*11:3^8*7), the 7:11 comma, symbol (|,
notates
> > 11/7 and 14/11
>
> Well, these have more than three digits in the numerator and
> denominator, so were explicitly excluded from Gene's list.

But I assume that you and Gene would both agree that it is essential
that symbols be provided to notate 7/5, 10/7, 11/7, and 14/11 in JI.
So if these are the principal ratios that will be mapped to 131, 139,
176, and 94 degrees of 270, respectively, then the notational
semantics should take this into account.

For example, taking C as 1/1, if a G-flat of 132deg (arrived at by a
chain of fifths) is lowered by 1deg to arrive at 7/5, then the
interval of 1deg alteration will be 5103:5120.

The only other 15-limit consonances requiring a 1-deg alteration
(from tones in a chain of fifths) are 13/11 and 22/13. Raising the A
of 204deg by 1deg gives 22/13, with the interval of 1deg alteration
as 351:352. While the interval of alteration has smaller integers,
the ratios being notated are less simple.

I submit that the semantics of a notation should be determined by the
simplicity of the ratios for the resulting pitches rather than those
for the intervals of alteration, so I question the imposition of a 3-
digit cutoff for the latter.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "George D. Secor"
> <gdsecor@y...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...> wrote:
> > > > In case anyone is inspired to consider 270 in the light of a
13-
> limit
> > > > notation system, this might help: the TM basis is 676/675,
> 1001/1000,
> > > > 1716/1715, 3025/3024 and 4096/4095. I used this to compute
the
> first
> > > > twenty notes of a Fokker block, and used that to compute all
the
> > > > 13-limit intervals with Tenney height less than a million up
to
> twenty
> > > > 270-et steps, with the following result:
> > > >
> > > > 1 {625/624, 352/351, 351/350, 847/845, 385/384, 441/440,
> 540/539, 364/363, 729/728, 325/324}
> > > > ...
> > > > 8 {49/48, 50/49, 729/715, 910/891, 875/858, 143/140, 864/847}
> > >
> > > This list misses a couple of intervals that Dave and I have
found
> to
> > > be valuable for notating 11-limit consonances, and we have
chosen
> the
> > > symbols for these two intervals to notate 1deg and 8deg of 311:
> > >
> > > 1deg 5103:5120 (3^6*7:2^10*5), the 5:7 kleisma, symbol |(,
> notates 7/5 and 10/7
> > > 8deg 45056:45927 (2^12*11:3^8*7), the 7:11 comma, symbol (|,
> notates
> > > 11/7 and 14/11
> >
> > Well, these have more than three digits in the numerator and
> > denominator, so were explicitly excluded from Gene's list.
>
> But I assume that you and Gene would both agree that it is
essential
> that symbols be provided to notate 7/5, 10/7, 11/7, and 14/11 in
JI.
> So if these are the principal ratios that will be mapped to 131,
139,
> 176, and 94 degrees of 270, respectively, then the notational
> semantics should take this into account.
>
> For example, taking C as 1/1, if a G-flat of 132deg (arrived at by
a
> chain of fifths) is lowered by 1deg to arrive at 7/5, then the
> interval of 1deg alteration will be 5103:5120.
>
> The only other 15-limit consonances requiring a 1-deg alteration
> (from tones in a chain of fifths) are 13/11 and 22/13. Raising the
A
> of 204deg by 1deg gives 22/13, with the interval of 1deg alteration
> as 351:352. While the interval of alteration has smaller integers,
> the ratios being notated are less simple.
>
> I submit that the semantics of a notation should be determined by
the
> simplicity of the ratios for the resulting pitches rather than
those
> for the intervals of alteration, so I question the imposition of a
3-
> digit cutoff for the latter.
>
> --George

I think Gene was just using it for purposes of illustration and
certainly not considering the question of notation in the same way
you and Dave have been.