Yahoo Tuning Groups Ultimate Backup tuning Guitar Frettings (A Pythagorean View)

In a recent post I spoke of the observation that one must consider
some strategically placed 9/8s in the fretting of a guitar, to be
able to play in-tune 4ths - where the guitar is tuned to open 5ths,
and in-tune 5ths - where it's tuned to open 4ths. This audible
reality on the guitar that I recently fretted (with some sweet
neutral intervals), led me to consider this from a sort of
Pythagorean angle. Below are the ratios for a possible fretting
pattern, with Scala data included, which, at least theoretically
attempts to address the issue (an ancient concept applied to our
contemporary needs). The appeal of this fretting is that one would be
able to play in-tune fourths and fifths from nearly all starting
points with a perpendicular fretting scheme, and would be equally
effective with either an open 5ths or open 4ths tuning. This would
seem to have some value in the quest for the just guitar.

Some noteworthy qualities:
Scale is strictly proper
Scale is a Constant Structure
Most common intervals : 4/3, 498.0450 cents & inv.,
amount: 9
Number of recognizable fifths : 11, average 701.933 cents

Guitar Fretting (Pythagorean) - Ligon 12-2000
0: 1/1 0.000 unison, perfect prime
1: 59/56 90.346
2: 9/8 203.910 major whole tone
3: 32/27 294.135 Pythagorean minor third
4: 81/64 407.820 Pythagorean major third
5: 4/3 498.045 perfect fourth
6: 84/59 611.609
7: 3/2 701.955 perfect fifth
8: 128/81 792.180 Pythagorean minor sixth
9: 27/16 905.865 Pythagorean major sixth
10: 16/9 996.090 Pythagorean minor seventh
11: 112/59 1109.654
12: 2/1 1200.000 octave
|
Number of notes : 12
Smallest interval : 256/243, 90.2250 cents
Average interval (divided octave) : 100.000 cents
Average / Smallest interval : 1.108340
Largest interval of one step : 2187/2048, 113.6850 cents
Largest / Average interval : 1.136850
Largest / Smallest interval : 1.260017
Least squares average interval : 100.2196 cents
Median interval of one step : 59/56, 90.3458 cents
Most common interval of one step : 256/243, 90.2250 cents,
amount: 4
Interval standard deviation : 11.5049 cents
Interval skew :-0.0004 cents
Scale is strictly proper
Scale has monotonic third-sizes over circle of fifths
Scale is a Constant Structure
Number of different intervals : 40 = 3.63636 / class
Smallest interval difference : 14337/14336, 0.1208 cents
Most common intervals : 4/3, 498.0450 cents & inv.,
amount: 9
Number of recognizable fifths : 11, average 701.933 cents
Most common triad is 0.0 498.045 701.955 cents, amount: 7
Rothenberg stability : 1.000000 = 1
Lumma stability : 0.785755
Prime limit : 59
Odd number limit : 6561 (O: 2187 U: 6561)
Fundamental : 1/2140992,-21.030 octaves,
0.000 Hz.
Guide tone : 4281984, 22.030 octaves,
1120276484.61 Hz.
Exponens Consonantiae : 9.167693E+12, 43.0597 octaves
Euler's gradus suavitatis : 158
Wille's k value : 1354846
Vogel's harmonic complexity : 80.58333
Wilson's harmonic complexity : 156
Rectangular lattice diameter : 12
Triangular lattice diameter : 8
Prime exponents' range and average:
2: -6 .. 7 0.66667
3: -4 .. 4 0.08333
7: -1 .. 1 0.08333
59: -1 .. 1 -0.08333
Average exponent except of 2 : 0.08333
Average absolute exponent except of 2: 27 / 12 = 2.25000
Inversional symmetry on intervals :
3-4 9-10
Average distance from equal tempered : 6.3653 cents, 0.063652 steps
Standard deviation from equal tempered : 2.1330 cents, 0.021329 steps
Maximum distance from equal tempered : 11.6092 cents, 0.116092 steps
Geometric average of pitches 0..n : 600.893 cents
Arithmetic average of pitches 0..n : 40315465/27832896, 641.4508
cents
Harmonic average of pitches 0..n : 560.335 cents
Geometric average of pitches 1..n-1: 601.055 cents
Arithmetic average of pitches 1..n-1: 33892489/23550912, 630.2192
cents
Harmonic average of pitches 1..n-1: 571.892 cents
Geometric average of pitches 1..n : 650.967 cents
Arithmetic average of pitches 1..n : 38174473/25691904, 685.5533
cents
Harmonic average of pitches 1..n : 616.382 cents

Any opinions? I still am in contemplation of what neutral intervals
to include.

Thanks,

Jacky Ligon

P.S. I wanted to compare this scale against the Scala scale archive,
and keep getting a "computation error". Was hoping to see if there
was any past use of 59/56 or 84/59.

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗ligonj@northstate.net

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Jacky -- your fretting gives essentially the usual 12-tone
Pythagorean scale
> on the D string. The following approximations apply:
>
> 11: 112/59 1109.654
> is approximately
> 243/128 1109.775
>
>
> 6: 84/59 611.609
> is approximately
> 729/512 611.730
>
> 1: 59/56 90.346
> is approximately
> 256/243 90.225

Paul,

Yes, and I think the subtle differences must come from the fact that
rather than derive the scale from a chain of reduced 3/2s, I merely:

Invert the 59/56 to get the 112/59.
Add a 9/8 to 81/64 to get the 84/59.
Subtract a 9/8 from 32/27 to get the 59/56.

Obviously, I could have inverted the 84/59, to give 118/84, which
would give another 9/8 between 118/84 and 128/81.

My primary concern with looking at our perpendicular fretting
collaboration in this Pythagorean way, is to consider ways to
optimize the in-tune "fourthness" and "fifthness" across the range of
the guitar, because I feel that when we speak of tempering fifths on
the guitar we will certainly introduce beating into chords. With this
I'm thinking in terms of the audible quality of justness of harmonic
timbres that has been under discussion of late, and my ears tell me
that many times a lack of in-tune fourths and fifths will be
pronouncedly ugly in many chords.

I'm wondering if you would agree that this could represent a good
starting point, to which other intervals could be added to give more
variety?

Where and what might you suggest be added, if anything?

Or do you feel that attempting to optimize the in-tuness of the
fourths and fifths in a perpendicular fretting is a futile compromise
for the thirds or other intervals? As we know, fourths, fifths and
octaves will be much more obviously out of tune than thirds will be
if tempered, so I guess it all depends on the sound one is trying to
achieve on a microtonally fretted guitar.

Thanks,

Jacky Ligon

P.S. Could you please explain what you mean by "the usual 12-tone
Pythagorean scale on the D string". Don't get your meaning about the
D string, since I'm considering these to be perpendicular fret
placements lying under all strings to optimize the tuning of 4ths and
5ths.

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Jacky wrote,

>My primary concern with looking at our perpendicular fretting
>collaboration in this Pythagorean way, is to consider ways to
>optimize the in-tune "fourthness" and "fifthness" across the range of
>the guitar, because I feel that when we speak of tempering fifths on
>the guitar we will certainly introduce beating into chords. With this
>I'm thinking in terms of the audible quality of justness of harmonic
>timbres that has been under discussion of late, and my ears tell me
>that many times a lack of in-tune fourths and fifths will be
>pronouncedly ugly in many chords.

Agreed, and in the context of an open tuning that will involve a lot of
fifths and/or fourths, you will automatically get those same fifths and
fourths on any fret. Other than that, I see no need to insist on more that
rough precision in the placement of frets a fourth or fifth apart, since
you'll never reach that far anyway.

>I'm wondering if you would agree that this could represent a good
>starting point, to which other intervals could be added to give more
>variety?

There are so many ways to go. My recent "Shrutar" proposal, which Dave
Keenan reduced to 22-out-of-46-tET, is especially geared toward a specific
set of desiderata (disappearing diaschismas, raga scales, etc.). Name yours!

>As we know, fourths, fifths and
>octaves will be much more obviously out of tune than thirds will be
>if tempered,

I think that is a bias that is a result of an aesthetic shift in the West in
the early-to-mid 18th century. In the Renaissance and Baroque, the usual
meantone tunings had fifths that were as out-of-tune, and often more so,
than the thirds -- and writers on tuning and aesthetics preferred that this
be so. Later, some of the very same writers (as well as new ones) expressed
a changing aesthetic, in which it was thought that fifths should be tuned
more purely that thirds -- probably as a result of the ascendancy of
well-temperaments in the great music of the time.

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗ligonj@northstate.net

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Jacky wrote,
>
> >My primary concern with looking at our perpendicular fretting
> >collaboration in this Pythagorean way, is to consider ways to
> >optimize the in-tune "fourthness" and "fifthness" across the range
of
> >the guitar, because I feel that when we speak of tempering fifths
on
> >the guitar we will certainly introduce beating into chords. With
this
> >I'm thinking in terms of the audible quality of justness of
harmonic
> >timbres that has been under discussion of late, and my ears tell
me
> >that many times a lack of in-tune fourths and fifths will be
> >pronouncedly ugly in many chords.
>
> Agreed, and in the context of an open tuning that will involve a
lot of
> fifths and/or fourths, you will automatically get those same fifths
and
> fourths on any fret. Other than that, I see no need to insist on
more that
> rough precision in the placement of frets a fourth or fifth apart,
since
> you'll never reach that far anyway.

Well, I'm thinking of playing in many (and any) positions on the
neck, with chords and melodies.

>
> >I'm wondering if you would agree that this could represent a good
> >starting point, to which other intervals could be added to give
more
> >variety?
>
> There are so many ways to go. My recent "Shrutar" proposal, which
Dave
> Keenan reduced to 22-out-of-46-tET, is especially geared toward a
specific
> set of desiderata (disappearing diaschismas, raga scales, etc.).
Name yours!

Could you deepen my understanding of this proposed fretting, by
showing me what the successive (approximate) 3/2s or 4/3s would be on
each fret position; just considering only two strings tuned either a
3/2 or 4/3 apart. This will bear an interesting relationship to this
Pythagorean proposal, in that it will be revealing to see the cents
values for the fourths or fifths, where one must spread their hand
the distance of an approximate fretted 9/8 in order to intone these
intervals. Please note that my ears are extremely accustomed to
altered fourths and fifths as we have discussed in the past (although
mostly from an electronic timbre point of view), and my recent
experience of playing it on the guitar made me realize how out-of-
tune the altered 5ths sounded when combined in chord voicings.

>
> >As we know, fourths, fifths and
> >octaves will be much more obviously out of tune than thirds will
be
> >if tempered,
>
> I think that is a bias that is a result of an aesthetic shift in
the West in
> the early-to-mid 18th century. In the Renaissance and Baroque, the
usual
> meantone tunings had fifths that were as out-of-tune, and often
more so,
> than the thirds -- and writers on tuning and aesthetics preferred
that this
> be so. Later, some of the very same writers (as well as new ones)
expressed
> a changing aesthetic, in which it was thought that fifths should be
tuned
> more purely that thirds -- probably as a result of the ascendancy of
> well-temperaments in the great music of the time.

What I'm learning from this invaluable exchange, is that one must
find a middle ground between the melodic behavior of an isolated
string, and the harmonic/vertical behavior of playing on simultaneous
strings, because it becomes obvious that what you do to one, totally
impacts on the sound of the other. And the question becomes: What
harmonic/melodic structures does one wish to feature and optimize
with a perpendicular fretting scheme? Certainly personal tuning
tastes and stylistic considerations must come into play here.

Thanks,

Jacky Ligon

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

--- In tuning@egroups.com, ligonj@n... wrote:
>
> Could you deepen my understanding of this proposed fretting, by
> showing me what the successive (approximate) 3/2s or 4/3s would be
on
> each fret position; just considering only two strings tuned either
a
> 3/2 or 4/3 apart.

If you look at the big lattice I posted, all the horizontal
connections represent perfect fifths of 704.35 cents. Does that
answer your question?

> What I'm learning from this invaluable exchange, is that one must
> find a middle ground between the melodic behavior of an isolated
> string, and the harmonic/vertical behavior of playing on
simultaneous
> strings, because it becomes obvious that what you do to one,
totally
> impacts on the sound of the other. And the question becomes: What
> harmonic/melodic structures does one wish to feature and optimize
> with a perpendicular fretting scheme? Certainly personal tuning
> tastes and stylistic considerations must come into play here.

Right. For diatonic, 5-limit music of an even vaguely Western
character, some form of meantone tuning is necessary -- such as a 31-
tone well-temperament. For the pseudo-Indian style I'm after, it
seems from the treatises I've read that it would be acceptable to
have 27/16 on one string while having 5/3 on the other string (and
similarly for 9/5 vs. 16/9), obviating the need for frets a comma
apart. I'll have more to say on the subject a little later.

🔗ligonj@northstate.net

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
>
> If you look at the big lattice I posted, all the horizontal
> connections represent perfect fifths of 704.35 cents. Does that
> answer your question?

Oh ok! Yes, if they've got to be tempered, let them be a little sharp.
I generally prefer this myself as opposed to flattening. Does this
mean though that 4ths would be tempered in the opposite direction
because of inversion?

>
> Right. For diatonic, 5-limit music of an even vaguely Western
> character, some form of meantone tuning is necessary -- such as a
31-
> tone well-temperament. For the pseudo-Indian style I'm after, it
> seems from the treatises I've read that it would be acceptable to
> have 27/16 on one string while having 5/3 on the other string (and
> similarly for 9/5 vs. 16/9), obviating the need for frets a comma
> apart. I'll have more to say on the subject a little later.

Eagerly awaiting,

Jacky Ligon