136-et: excellent 11-limit meantone

hello all,

when i made this webpage applet

http://tonalsoft.com/enc/index2.htm?meantone-error/meantone-error.htm

i found empirically that, with a mapping of the
"5th" to 79 of its degrees, 136-et provides a meantone
with an excellent approximation to 11-limit JI.

i've never seen this scale pop up in anyone's analysis.
here's a table of intervals based on the 2^(79/136) mapping:

interval 136edo .. ~cents
. name ...deg

p8 ..... 136 ... 1200
dim8 ... 127 ... 1120.588235
aug7 ... 132 ... 1164.705882
maj7 ... 123 ... 1085.294118
min7 ... 114 ... 1005.882353
dim7 ... 105 .... 926.4705882
aug6 ... 110 .... 970.5882353
maj6 ... 101 .... 891.1764706
min6 .... 92 .... 811.7647059
dim6 .... 83 .... 732.3529412
aug5 .... 88 .... 776.4705882
p5 ...... 79 .... 697.0588235
dim5 .... 70 .... 617.6470588
aug4 .... 66 .... 582.3529412
p4 ...... 57 .... 502.9411765
dim4 .... 48 .... 423.5294118
aug3 .... 53 .... 467.6470588
maj3 .... 44 .... 388.2352941
min3 .... 35 .... 308.8235294
dim3 .... 26 .... 229.4117647
aug2 .... 31 .... 273.5294118
maj2 .... 22 .... 194.1176471
min2 .... 13 .... 114.7058824
dim2 ..... 4 ..... 35.29411765
aug1 ..... 9 ..... 79.41176471
p1 ....... 0 ...... 0

note that 136edo's best mapping of the 3/2 ratio
is 80 degrees ... but that mapping does not give a
meantone.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> i found empirically that, with a mapping of the
> "5th" to 79 of its degrees, 136-et provides a meantone
> with an excellent approximation to 11-limit JI.
>
>
> i've never seen this scale pop up in anyone's analysis.
> here's a table of intervals based on the 2^(79/136) mapping:

I thought we discussed this before. 136 is a semiconvergent to the
minimax tuning of the huygens, or 31&43, version of meantone. The
semiconvergents go 5, 7, 12, 19, 31, 43, 74, 104, 136, 167 ....
However an excellent generic choice in the 11-limit for meantone is 31-et.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> hello all,
>
>
> when i made this webpage applet
>
> http://tonalsoft.com/enc/index2.htm?meantone-error/meantone-error.
htm
>
>
> i found empirically that, with a mapping of the
> "5th" to 79 of its degrees, 136-et provides a meantone
> with an excellent approximation to 11-limit JI.
>
>
> i've never seen this scale pop up in anyone's analysis.
> here's a table of intervals based on the 2^(79/136) mapping:
>
> interval 136edo .. ~cents
> . name ...deg
>
> p8 ..... 136 ... 1200
> dim8 ... 127 ... 1120.588235
> aug7 ... 132 ... 1164.705882
> maj7 ... 123 ... 1085.294118
> min7 ... 114 ... 1005.882353
> dim7 ... 105 .... 926.4705882
> aug6 ... 110 .... 970.5882353
> maj6 ... 101 .... 891.1764706
> min6 .... 92 .... 811.7647059
> dim6 .... 83 .... 732.3529412
> aug5 .... 88 .... 776.4705882
> p5 ...... 79 .... 697.0588235
> dim5 .... 70 .... 617.6470588
> aug4 .... 66 .... 582.3529412
> p4 ...... 57 .... 502.9411765
> dim4 .... 48 .... 423.5294118
> aug3 .... 53 .... 467.6470588
> maj3 .... 44 .... 388.2352941
> min3 .... 35 .... 308.8235294
> dim3 .... 26 .... 229.4117647
> aug2 .... 31 .... 273.5294118
> maj2 .... 22 .... 194.1176471
> min2 .... 13 .... 114.7058824
> dim2 ..... 4 ..... 35.29411765
> aug1 ..... 9 ..... 79.41176471
> p1 ....... 0 ...... 0
>
>
>
> note that 136edo's best mapping of the 3/2 ratio
> is 80 degrees ... but that mapping does not give a
> meantone.
>
>
>
> -monz

of course, the table i provided only includes mappings
of 5-limit intervals, and my whole point was that 136-et
is excellent in the 11-limit ... so i need to show some
of those intervals too.

here's mapping of 136edo generators ("5ths") to
all the main 11-limit intervals:

.. ratio generator 136edo .. ~cents error

.. 11/8 ... + 18 .... 62 ... - 4.259118835
.. 11/6 ... + 17 ... 119 ... + 0.637058501
.. 11/9 ... + 16 .... 40 ... + 5.533235837
........... + 15 .... 97 ...
........... + 14 .... 18 ...
........... + 13 .... 75 ...
........... + 12 ... 132 ...
........... + 11 .... 53 ...
... 7/4 ... + 10 ... 110 ... + 1.762328825
... 7/6 .... + 9 .... 31 ... + 6.658506161
.. 11/7 .... + 8 .... 88 ... - 6.02144766
............ + 7 ..... 9 ...
... 7/5 .... + 6 .... 66 ... - 0.159251428
............ + 5 ... 123 ...
... 5/4 .... + 4 .... 44 ... + 1.921580253
... 5/3 .... + 3 ... 101 ... + 6.817757589
............ + 2 .... 22 ...
... 3/2 .... + 1 .... 79 ... - 4.896177336
... 1/1 ..... 0 ..... 0 ...
... 4/3 .... - 1 .... 57 ... + 4.896177336
............ - 2 ... 114 ...
... 6/5 .... - 3 .... 35 ... - 6.817757589
... 8/5 .... - 4 .... 92 ... - 1.921580253
............ - 5 .... 13 ...
.. 10/7 .... - 6 .... 70 ... + 0.159251428
............ - 7 ... 127 ...
.. 14/11 ... - 8 .... 48 ... + 6.02144766
.. 12/7 .... - 9 ... 105 ... - 6.658506161
... 8/7 ... - 10 .... 26 ... - 1.762328825
........... - 11 .... 83 ...
........... - 12 ..... 4 ...
........... - 13 .... 61 ...
........... - 14 ... 118 ...
........... - 15 .... 39 ...
.. 18/11 .. - 16 .... 96 ... - 5.533235837
.. 12/11 .. - 17 .... 17 ... - 0.637058501
.. 16/11 .. - 18 .... 74 ... + 4.259118835

-monz

From: "monz" <monz@t>

> i found empirically that, with a mapping of the
> "5th" to 79 of its degrees, 136-et provides a meantone
> with an excellent approximation to 11-limit JI.
>
>
> i've never seen this scale pop up in anyone's analysis.

If I understand it right, you mean that the double-augmented third (i.e. a
distance of +18 fifths) is to approximate 11/8.
If you are interested, I can tell you, though I think you already know this,
that 50-equal has a double-diminished fifth (i.e. only -13 fifths) which
approximates the 11/8 even better (with an error of less than a cent).
Petr