re : Analysis

> From: "Raintree Goldbach" <goldraintree425@hotmail.com>
> Subject: Analysis
>
> are there any faults, theoretically or acoustically, with the following
> tuning?
>
> thanks
>
> 1/1 15/14 8/7 6/5 5/4 4/3 7/5 3/2 8/5 5/3 7/4 15/8
>

It is perfect if it solves your musical problem.

Now for some analysis along the usual "approximate it with this and
you'll have this many more psuedo-harmonies and fewer wolves" sort
of things. (Note I'm not really a math guy, but thats what I'll
try to do.

First I move to the dreaded land of cents and notice four different
cents values for the semi-tones and a different number of such tones
each.

ratio cents amount
16/15 112 4
15/14 119 3

21/20 84 3
25/24 71 2

Firstly, they look close enough together that I'll make the thing
diatonic and see what comes out of that. This will lead to a large
semitone of around 115 and a small semitone of around 77 (I'm not
going to do the final fudging to get it exact). Now we can notice that
this semi-tone ratio is approximately 3:2, and 7*3+5*2 = 31 suggesting
that 31 et would be a reasonable approximation.

Here is the original scale

0 0
119 119
112 231
84 316
71 386
112 498
84 583
119 702
112 814
71 884
84 969
119 1088
112 1200

and here it is rounded down to two sizes of semi-tone (and
pretty close to how it would look in 31)(and I didn't
do the final bit of math cleanup, so its two cents sharp).

0 0
116 116
116 232
78 310
78 388
116 504
78 582
116 698
116 814
78 892
78 970
116 1086
116 1202

Fine. But lets leave the tuning as is and think of it in
a psuedo-diatonic way, just to see what is happenning
structurally. The scale structure is

LLssLsLLssLL (as a rounded approach to LMstMsLMtsLM)

Assuming you've retuned a keyboard with these notes starting
on C, I bet the region from Bb to D is a bit different to work
in since it is all large intervals. Not that a scale like this
needs to support transposition, but it starts behaving like a
different creature (maybe thats great).

Also the reversal of s and t after the G probably does some
strange things, since the fifths break.

Another thing that this sort of lumpiness leads to is that sums
of intervals in different parts of the scale may not have quite
the properties you would expect.

For example the range of four steps ("major third") ranges
from 351 to 462 (with six 5/4). The range of five steps ("perfect
fourth") goes from 463 to 548 (with five 4/3). The difference
between the wide four step and thin five step is negligible.

This is a feature, a pun. On the other hand, it cuts into the
uniqueness of an interval, however many personalitys it has,
when some personalities are the same as those of other intervals
(yes, the diminished fifth and augmented fourth have this property
in 12et).

Bob

--- In tuning@yahoogroups.com, Bob Valentine <BVAL@I...> wrote:

> First I move to the dreaded land of cents and notice four different
> cents values for the semi-tones and a different number of such tones
> each.
>
> ratio cents amount
> 16/15 112 4
> 15/14 119 3
>
> 21/20 84 3
> 25/24 71 2
>
> Firstly, they look close enough together that I'll make the thing
> diatonic and see what comes out of that. This will lead to a large
> semitone of around 115 and a small semitone of around 77 (I'm not
> going to do the final fudging to get it exact).

Hi Bob,

You may have a misunderstanding of the etymology of the
word "diatonic". The prefix there is not "di-" meaning two, it's "dia-
" meaning through. "Making the thing diatonic" would imply giving it
some quality having to do with the seven-tone-per-octave diatonic
scale, but it appears you were simply making it have two step sizes,
which is something entirely different.

Your humble pedant,
Paul

Now we can notice that
> this semi-tone ratio is approximately 3:2, and 7*3+5*2 = 31
suggesting
> that 31 et would be a reasonable approximation.
>
> Here is the original scale
>
> 0 0
> 119 119
> 112 231
> 84 316
> 71 386
> 112 498
> 84 583
> 119 702
> 112 814
> 71 884
> 84 969
> 119 1088
> 112 1200
>
> and here it is rounded down to two sizes of semi-tone (and
> pretty close to how it would look in 31)(and I didn't
> do the final bit of math cleanup, so its two cents sharp).
>
> 0 0
> 116 116
> 116 232
> 78 310
> 78 388
> 116 504
> 78 582
> 116 698
> 116 814
> 78 892
> 78 970
> 116 1086
> 116 1202
>
> Fine. But lets leave the tuning as is and think of it in
> a psuedo-diatonic way, just to see what is happenning
> structurally. The scale structure is
>
> LLssLsLLssLL (as a rounded approach to LMstMsLMtsLM)
>
> Assuming you've retuned a keyboard with these notes starting
> on C, I bet the region from Bb to D is a bit different to work
> in since it is all large intervals. Not that a scale like this
> needs to support transposition, but it starts behaving like a
> different creature (maybe thats great).
>
> Also the reversal of s and t after the G probably does some
> strange things, since the fifths break.
>
> Another thing that this sort of lumpiness leads to is that sums
> of intervals in different parts of the scale may not have quite
> the properties you would expect.
>
> For example the range of four steps ("major third") ranges
> from 351 to 462 (with six 5/4). The range of five steps ("perfect
> fourth") goes from 463 to 548 (with five 4/3). The difference
> between the wide four step and thin five step is negligible.
>
> This is a feature, a pun. On the other hand, it cuts into the
> uniqueness of an interval, however many personalitys it has,
> when some personalities are the same as those of other intervals
> (yes, the diminished fifth and augmented fourth have this property
> in 12et).
>
> Bob