Re: Stretched tuning experiments

[I wrote:]
>>>Oh, I meant that, despite fewer words. Although, I see that Herman
>>>is saying "1/7-comma meantone with 1/7-comma tempered octaves";
>>>perhaps if the word "tempered" were changed to "stretched", it'd be
>>>clearer...

[Herman Miller:]
>>Done.

[Dave Keenan:]
>No. I think maybe John still doesn't get it. It's definitely "with
>octaves tempered 1/7-comma wide" and _not_ "with a stretch of 1/7
>comma per octave". These are very different things.

You're right, Dave! I'm gonna study the rest of what you wrote and try
to clear up my understanding. Thanks!

JdL

[Dave Keenan wrote:]
>Here are the three scales to the nearest cent
>1. Ordinary 1/7 comma meantone
>2. 1/7 comma meantone with a stretch of 1/7 comma per octave
>3. 1/7 comma meantone with octaves tempered 1/7 comma wide.
>
>Name Octaves Fifths 1/7 with with
> comma stretch tempered octaves
>-----------------------------------------------
>C 0 0 0 0 0
>C# -4 7 92 92 80
>D -1 2 198 198 195
>Eb 2 -3 303 304 309
>E -2 4 396 397 389
>F 1 -1 501 502 504
>F# -3 6 593 595 584
>G 0 1 699 701 699
>G# -4 8 791 793 779
>A -1 3 897 899 894
>Bb 2 -2 1002 1005 1008
>B -2 5 1094 1097 1088
>C 1 0 1200 1203 1203
>
>Look at E. Notice how stretch makes the major thirds (4:5 = 386c)
>slightly worse, but wide-tempered-octaves makes them significantly
>better.

Thanks again, Dave, for the clear and detailed explanation. So... the
tempered octave tuning _could_ be reformulated as an octave stretch,
howbeit the scale to be stretched would be arrived at in a convoluted
manner.

JdL

> Thanks again, Dave, for the clear and detailed explanation. So...
the
> tempered octave tuning _could_ be reformulated as an octave stretch,
> howbeit the scale to be stretched would be arrived at in a
convoluted
> manner.

Er, yes. _Very_ convoluted.

Instead the final scale is arrived at by (for each note) multiplying
the number in the "Octaves" column by the size of the tempered
octave (1200c plus 1/7 comma) and multiplying the number in the
"Fifths" column by the size of the tempered fifth (701.955c minus 1/7
comma) and adding these together.

-- Dave Keenan

[I wrote:]
>>Thanks again, Dave, for the clear and detailed explanation. So... the
>>tempered octave tuning _could_ be reformulated as an octave stretch,
>>howbeit the scale to be stretched would be arrived at in a convoluted
>>manner.

[Dave Keenan:]
>Er, yes. _Very_ convoluted.

Oh c'mon, Dave; it's only _fairly_ convoluted. ;->

>Instead the final scale is arrived at by (for each note) multiplying
>the number in the "Octaves" column by the size of the tempered
>octave (1200c plus 1/7 comma) and multiplying the number in the
>"Fifths" column by the size of the tempered fifth (701.955c minus 1/7
>comma) and adding these together.

Yes, I think I could finally pass a test on it... My interest in
conceptualizing this kind of scale in a reformulated way has to do with
comparing apples to apples against about-to-be stretched tunings.
Certainly the direct route you outline is the one that would normally be
followed.

JdL