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badness? [was:] The <2,5/3,7> and <2,5/3,7,9> groups

🔗klaus schmirler <kschmir@z.zgs.de>

5/4/2002 2:48:45 PM

hi,
normally i would have read "badness measure", taken it for
what the word says and gone on, but no, not here: the
next-to-nil
badness for 1et baffles me a little. Gene, are you serious?
what
measure is this really? anything to do with music?
my guess: the number of tones itself counts as badness and
is somehow computed with deviations from just; this gives
lower-numbered ets a not quite deserved head start?

klaus

Gene W Smith schrieb:
...
> A search for ets supporting <2,5/3,7> turns up, for a log-flat badness
> measure less than 0.5 and n to 5000, the following:
>
> 1 .2630344055
> 4 .45883938
> 11 .3949962817
> 15 .4272823563
> 57 .1451878461
> 114 .4106521743
> 213 .4874803032
> 270 .3169588282
> 327 .2215293311
> 384 .4759848469
> 3779 .4314215252
>
> The 57-et stands out here. If we go to the index 2 subgroup <2,5/3,7,9>,
> we get for badness less than 0.75 up to n = 2000 the following:
>
> 1 .3625700797
> 5 .5389365888
> 11 .2909521576
> 16 .7085795380
> 22 .7331534985
> 30 .6856035815
> 41 .7433989479
> 83 .7334574068
> 88 .6539733996
> 171 .3202175071
> 182 .7238985678
> 753 .5905099365
> 924 .4265227361
> 1095 .6996134010
...

🔗genewardsmith <genewardsmith@juno.com>

5/5/2002 12:17:21 AM

--- In tuning-math@y..., klaus schmirler <kschmir@z...> wrote:
> hi,
> normally i would have read "badness measure", taken it for
> what the word says and gone on, but no, not here: the
> next-to-nil
> badness for 1et baffles me a little. Gene, are you serious?
> what
> measure is this really? anything to do with music?
> my guess: the number of tones itself counts as badness and
> is somehow computed with deviations from just; this gives
> lower-numbered ets a not quite deserved head start?

It gives it a head start; whether it deserves it or not is a matter of controvery, but you need to go to twice as many notes to the octave to have a chance of beating it, which sounds pretty good, really. I suggest not worrying about the first few entries when considering a log-flat scale, they have some theoretical applications but are not directly usable as temperaments, obviously.