Equal Temperament

would it be possible to acheive equal temperament with greater ease by starting from the equal tempered tritone and tuning sixth perfect fifths above and sixth perfect fifths below? would this be a sufficient approximation?

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hi Raintree,

--- In tuning@yahoogroups.com, "Raintree Goldbach"
<goldraintree425@h...> wrote:

> would it be possible to acheive equal temperament
> with greater ease by starting from the equal tempered
> tritone and tuning sixth perfect fifths above and
> sixth perfect fifths below? would this be a sufficient
> approximation?

not at all, if i've understood you correctly.
that scale doesn't even resemble equal-temperament.

here it is presented the way you describe it,
with the 12edo tritone in the middle of the chain:

"5th", cents

6 11.73000519
5 509.7750043
4 1007.820003
3 305.8650026
2 803.9100017
1 101.9550009
600
0
-1 498.0449991
-2 996.0899983
-3 294.1349974
-4 792.1799965
-5 90.22499567
-6 588.2699948

to begin with, there are 14 notes, not 12.

and secondly, the notes occur as pairs of a diatonic scale.
here they are in pitch-height order:

"5th", cents

4 1007.820003
-2 996.0899983

2 803.9100017
-4 792.1799965

TT 600
-6 588.2699948

5 509.7750043
-1 498.0449991

3 305.8650026
-3 294.1349974

1 101.9550009
-5 90.22499567

6 11.73000519
ref 0

-monz

you are correct, it should have been 5 perfect fifths above and five below, giving 12 notes as follows

101.955 cents
192.175 cents
305.861 cents
396.078 cents
509.778 cents
599.996 cents
690.212 cents
803.910 cents
894.130 cents
1007.826 cents
1098.038 cents
1200.000 cents

From: "monz" <monz@attglobal.net>
Reply-To: tuning@yahoogroups.com
To: tuning@yahoogroups.com
Subject: [tuning] Re: Equal Temperament
Date: Sun, 28 Dec 2003 17:52:12 -0000

hi Raintree,

--- In tuning@yahoogroups.com, "Raintree Goldbach"
<goldraintree425@h...> wrote:

> would it be possible to acheive equal temperament
> with greater ease by starting from the equal tempered
> tritone and tuning sixth perfect fifths above and
> sixth perfect fifths below? would this be a sufficient
> approximation?

not at all, if i've understood you correctly.
that scale doesn't even resemble equal-temperament.

here it is presented the way you describe it,
with the 12edo tritone in the middle of the chain:

"5th", cents

6 11.73000519
5 509.7750043
4 1007.820003
3 305.8650026
2 803.9100017
1 101.9550009
600
0
-1 498.0449991
-2 996.0899983
-3 294.1349974
-4 792.1799965
-5 90.22499567
-6 588.2699948

to begin with, there are 14 notes, not 12.

and secondly, the notes occur as pairs of a diatonic scale.
here they are in pitch-height order:

"5th", cents

4 1007.820003
-2 996.0899983

2 803.9100017
-4 792.1799965

TT 600
-6 588.2699948

5 509.7750043
-1 498.0449991

3 305.8650026
-3 294.1349974

1 101.9550009
-5 90.22499567

6 11.73000519
ref 0

-monz

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

> here it is presented the way you describe it,
> with the 12edo tritone in the middle of the chain:

I thought he might be describing two chains of five fifths 600 cents
apart.

> > would it be possible to acheive equal temperament
> > with greater ease by starting from the equal tempered
> > tritone and tuning sixth perfect fifths above and
> > sixth perfect fifths below? would this be a sufficient
> > approximation?

//

>you are correct, it should have been 5 perfect fifths above and
>five below, giving 12 notes as follows
>
>101.955 cents
>192.175 cents
>305.861 cents
>396.078 cents
>509.778 cents
>599.996 cents
>690.212 cents
>803.910 cents
>894.130 cents
>1007.826 cents
>1098.038 cents
>1200.000 cents

Wait aminute, how'd you get this, I don't even see 600 cents
here?

-Carl

>> > would it be possible to acheive equal temperament
>> > with greater ease by starting from the equal tempered
>> > tritone and tuning sixth perfect fifths above and
>> > sixth perfect fifths below? would this be a sufficient
>> > approximation?
//
>>you are correct, it should have been 5 perfect fifths above and
>>five below, giving 12 notes as follows
>>
>>101.955 cents
>>192.175 cents
>>305.861 cents
>>396.078 cents
>>509.778 cents
>>599.996 cents
>>690.212 cents
>>803.910 cents
>>894.130 cents
>>1007.826 cents
>>1098.038 cents
>>1200.000 cents
>
>Wait aminute, how'd you get this, I don't even see 600 cents
>here?

Oh, I'm just blind. This is what I thought you meant from your
first message.

This scale has lots of perfect fifths and 2 very flat meantone
fifths. It has 7 pythagorean major thirds (not my idea of a
good time), 3 near-just ones, and 2 near-12-tET ones. Thus, I'd
call it an extreme well temperament.

As far as a bearing plan, it would be very easy to tune if one
could set the tritone somehow.

-Carl

--- In tuning@yahoogroups.com, "Raintree Goldbach"
<goldraintree425@h...> wrote:

> you are correct, it should have been 5 perfect fifths above and
five below,
> giving 12 notes as follows
>
> 101.955 cents
> 192.175 cents
> 305.861 cents
> 396.078 cents
> 509.778 cents
> 599.996 cents
> 690.212 cents
> 803.910 cents
> 894.130 cents
> 1007.826 cents
> 1098.038 cents
> 1200.000 cents

Fifths of 690 cents, 12 cents narrow of just, are a bit too far off
for what was historically considered "in tune" in Western music, but
you may still be able to make beautiful music with this scale. Play
on!

Happy New Year,
Paul