Two *weird* golden-mean tunings

I came up with these two scales today and both are based on the classic 1.618:1.000 ratio also known as the Golden Mean. Except these two scales are based on the ratio being applied to the major triad and the root-fifth-octave, and the results proved to be pretty cacophonic.

The next two files I'm submitting to the group are retunings of that same Bach piece (Badinerie from Suite No. 2 in B Minor). I calculated the pitches of both tunings to 19-tones, all given in cents from the key of C:

Major Triad
C 0.0000
C# 167.5248
Db 51.7680
D 219.2928
D# 386.8176
Eb 271.0608
E 438.5856
E# 606.1104
F 490.3536
F# 657.8784
Gb 542.1216
G 709.6464
G# 877.1711
Ab 761.4144
A 928.9392
A# 1096.4640
Bb 980.7072
B 1148.2320
Cb 1032.4752
C' 1200.0000

Fifth + Octave
C 0.0000
C# 391.4855
Db -108.2039
D 283.2816
D# 674.7671
Eb 175.0776
E 566.5631
E# 958.0487
F 458.3592
F# 849.8447
Gb 350.1553
G 741.6408
G# 1133.1263
Ab 633.4369
A 1024.9224
A# 1416.4079
Bb 916.7184
B 1308.2039
Cb 808.5145
C' 1200.0000

Stay tuned for the MIDI files.

~DaW~

Hi Danny

The Bach is a fun midi file.

That's a pretty nice scale you've made actually.

I've added an improvisation in diatonic mode of your
golden ratio major triad scale to my improvisations page.

1/1 219.293 386.818 490.354 709.646 877.171 1096.46 2/1

http://members.tripod.com/~robertinventor/tunes/improvisations.htm

The major thirds are the consonances to resolve to and the
fourths are comparative dissonances (in harmonic
timbres)

Robert

I just "discovered" golden meantone temperament. I must be slow, but
I never had paid any significant attention to postings on golden
meantone. Now, upon investigating mathematically what happens when
you temper meantone to yield a perfect 1.618034 ratio (in terms of
pitch size, not frequency), I see that amazing things happen (amazing
to me, anyway).

First this meantone is fairly close to the Woolhouse 7/26-comma
meantone. The fifths are 5.741 cents flat, major thirds, 1.456, and
minor thirds 4.285 cents flat. I take the decimals out so far just to
prevent rounding errors that otherwise obscure the exactness of the
golden mean properties of this tuning system.

What I never before realized is that having tempered in this manner,
not only the major and minor seconds are in the ratio of the golden
mean, but also the diatonic and chromatic half-steps, the minor third
to major second, the perfect fourth to minor third, the minor sixth
to perfect fourth, and the minor ninth to minor sixth.

So here we have an admittedly open temperament (closely approximated
by 81-EDO, however) that has the recursively self-referent properties
of the golden mean accessible to the human ear on the level of
ubiquitous golden proportions among PITCH DISTANCES. We hear pitch
logarithmically and golden meantone BOTH respects that and yields
highly consonant harmonic relationships in addition to the
pervasiveness of the golden proportion melodically.

Further, if we substitute 81-tET, we get the Fibonacci sequence
starting from chromatic semitones, then diatonic semitones, whole
steps, minor thirds, perfect fourths, minor sixths and minor ninths
as 5, 8, 13, 21, 34, and 55 steps respectively. Also 81-tET is in the
Fibonacci sequence seeded by the numbers 7 and 12 (i.e., 7, 12, 19,
31, 50, 81, all of which are well-known meantone approximations
excepting the 7 which can be taken as the steps of a perfect fifth
from within 12-tET.

No such magical effects appear either aurally or analytically when
the golden mean is applied directly to frequency, which is not how we
perceive anyway. Considering the recursive, fractal nature of so many
other elements of musical structure, I find all this highly
intriguing.

--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:
> I came up with these two scales today and both are based on the
classic 1.618:1.000 ratio also known as the Golden Mean. Except
these two scales are based on the ratio being applied to the major
triad and the root-fifth-octave, and the results proved to be pretty
cacophonic.

--- In tuning@y..., BobWendell@t... wrote:

> I find all this highly
> intriguing.

Well then you should study the Wilson Horagram stuff that's being
posted here and the Fibonacci and Tribonacci stuff being discussed
here and on the tuning-math list.

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi Danny
>
> The Bach is a fun midi file.
>
> That's a pretty nice scale you've made actually.
>
> I've added an improvisation in diatonic mode of your
> golden ratio major triad scale to my improvisations page.
>
> 1/1 219.293 386.818 490.354 709.646 877.171 1096.46 2/1
>
> http://members.tripod.com/~robertinventor/tunes/improvisations.htm
>
> The major thirds are the consonances to resolve to and the
> fourths are comparative dissonances (in harmonic
> timbres)
>
> Robert

I don't find these fourths to be dissonant at all (except
the "fourth" from 877.171 to 1419.293). In fact this is virtually
identical to 22-tET, which I've worked with a lot. I find this a poor
diatonic scale, very off-kilter sounding. It's fun to ride the I-IV-
ii-V-I "pump" over and over again, drifting down one 22-tET step at a
time (which happens because 22-tET is _not_ a meantone).

--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:
> I came up with these two scales today and both are based on the
classic 1.618:1.000 ratio also known as the Golden Mean. Except
these two scales are based on the ratio being applied to the major
triad and the root-fifth-octave, and the results proved to be pretty
cacophonic.
>
> The next two files I'm submitting to the group are retunings of
that same Bach piece (Badinerie from Suite No. 2 in B Minor). I
calculated the pitches of both tunings to 19-tones, all given in
cents from the key of C:
>
> Major Triad
> C 0.0000
> C# 167.5248
> Db 51.7680
> D 219.2928
> D# 386.8176
> Eb 271.0608
> E 438.5856
> E# 606.1104
> F 490.3536
> F# 657.8784
> Gb 542.1216
> G 709.6464
> G# 877.1711
> Ab 761.4144
> A 928.9392
> A# 1096.4640
> Bb 980.7072
> B 1148.2320
> Cb 1032.4752
> C' 1200.0000

This is basically what's known around here as the "22-tET
Pythagorean" tuning and notation. It's a lot of fun for moving around
with 6:7:9 and 4:6:7:9 chords, and pretty diatonic melodies.

--- In tuning@y..., BobWendell@t... wrote:

> First this meantone is fairly close to the Woolhouse 7/26-comma
> meantone. The fifths are 5.741 cents flat, major thirds, 1.456, and
> minor thirds 4.285 cents flat. I take the decimals out so far just
to
> prevent rounding errors that otherwise obscure the exactness of the
> golden mean properties of this tuning system.

You should check out the Osmium tuning I discussed on the math list--
the fifth is 3/4 cent flat, the third is 2.85 cents flat, and the 7
is 1/2 cent sharp. These are, obviously, very usable values.

> What I never before realized is that having tempered in this
manner,
> not only the major and minor seconds are in the ratio of the golden
> mean, but also the diatonic and chromatic half-steps, the minor
third
> to major second, the perfect fourth to minor third, the minor sixth
> to perfect fourth, and the minor ninth to minor sixth.

You might want to ponder what happens in the Miracle-Magic square in
Osmium tuning. Let s be the small step, m the medium step, and l the
large step, and let z = 1.3247... be the real root of z^3-z-1=0. Then
s = (43 - 11z - 3z^2)/241, m = (-3 + 40z - 11z^2)/241, and
L = (-11 - 14z + 40z^2)/241, with values in cents of s = 115.337,
m = 152.789, and L = 202.402. It now happens that (in terms of cents)
we have m/s = L/m = z = 1.324... so the three sizes of steps are in a
geometric progression. Moreover, when we split the large step by
L = s + (L-s), it turns out that s/(L-s) = z also, so that L-s,s, and
m become the new small, medium and large steps, and the pattern
continues to scales of size 10,12,19,22,31,41,53,72,94...

The Miracle-Magic Square in Osmium tuning has step pattern smssLssms,
and is approximately

1-s-16/15-m-7/6-s-5/4-s-4/3-L-9/8-s-8/5-s-12/7-m-15/8-s-2

Hi Paul,

> I don't find these fourths to be dissonant at all (except
> the "fourth" from 877.171 to 1419.293). In fact this is virtually
> identical to 22-tET, which I've worked with a lot. I find this a poor
> diatonic scale, very off-kilter sounding. It's fun to ride the I-IV-
> ii-V-I "pump" over and over again, drifting down one 22-tET step at a
> time (which happens because 22-tET is _not_ a meantone).

May depend on the timbre on the soundcard.

I've done a real audio clip of it now.

I was thinking of the fourths such as C - F to C - E rather than the
"wolf".

The 22-tet major third is four cents out while this one is pretty
much exact. I expect that makes a difference to the way it works.

However, didn't want to suggest that it has to work that way for
this scale, just that it felt that way when improvising on this
part. occasion.

I've edited what I said about the scale to say so.

http://members.tripod.com/~robertinventor/tunes/improvisations.htm

Robert