Yahoo Tuning Groups Ultimate Backup tuning Shake Yer Sruti

My 2-year helped me remove the frets from an old acoustic guitar
yesterday. His enthusiasm for the project continued with a chisel on
a chunk of tree root for some time afterward. Playing the guitar, I
was thinking about the instrument building and fretting I did some
years ago, with Paul Erlich´s paper on 22-EDO lying on the table by
coincidence, leafed over to the part where he compares 22-EDO with
the Indian sruti system.

What follows sprang into my head full-blown; it took longer to hunt
down the measuring tape than it did to verify the first intervals.

My assumptions are:

-The origins of tuning are inseperable from physical and practical
considerations.

-Accuracy and consistency can be achieved with math, tools, and
materials which were available to the ancients.

-Fundamental ideas about the origins of the sruti system, as well-
known in different theories, are all correct, even when they
contradict each other. So much music for so long in such a huge and
wildly diverse area, with so much fundamental unity and integrity...
there is simply no way the deepest roots could be in conflict.

-The sruti are equal... and unequal.

-Anytime you use a ruler when fretting, you´re using equal-divisions-
of-length.

-THE LENGTH OF A STRING CHANGES EVERY TIME YOU FRET IT.

Here is a chart resulting from a simple and palpable way of finding
and fretting specific intervals, using simple math
and 22 "sruti", with intervals found at 4, 3, or 2 "sruti".

The first column gives a kind of "sruti number", the second the
physical size of the "sruti" in millimeters, the third, ideal fret
positions for intervals given. The measurements in the "Fret" column
apply to an open string length of 646 mm (measured from my acoustic
guitar), with fret, nut, and bridge height set to the default string
length calculations in Scala.

0 0 Nut Nut-Fret mm Frets

1 14.68 14.68 mm
2 29.36 29.36
3 44.04 44.04
4 58.72 58.72 mm 10/9 is 63.817 mm

1 12.0123
2 24.024
3 36.0369
4 48.0495 106.769 mm 6/5 is 106.54 mm

1 9.828
2 19.656 126.42 mm 5/4 is 127.9 mm

1 8.935
4 35.74 162.16 mm 4/3 is 159.9 mm

1 7.4136
3 22.241 184.4 mm 45/32 is 184.87 mm

1 6.3
4 25.2 209.6 mm 3/2 is 213.36 mm

And so on...

My initial quick calculations with paper, pencil and ruler were
within 2 mm of these calculator- and Scala-assisted calculations.
There are rounding errors in the chart above, I´ll do more accurate
calculations when I have the time (perhaps kind and precision-loving
souls will help out here). My first rough "analog" measurements
wound up about 1 mm higher than the octave/halfway point on my
guitar, and very clear just intervals throughout (I tried to finger
sarod-style on the now-fretless guitar, the physical envelope of
error and possibility in playing was about 3-4 mm).

(By raising the virtual nut and lowering the bridge and frets in
Scala, i.e. bringing everything closer to the instruments of ancient
descent I have observed, the results seem to be even closer, have to
verify this.)

This is how it works, please try it yourself with different lengths
and possible branching-off points as described below!

First of all, the octave/reference point is the octave as tangibly
present on a stringed instrument: the halfway point (give or take
tiny variations due to the relationships between nut, bridge and
string height, which have to be dealt with on the specific
instrument itself). Therefore our "srutis" will be measured, in this
example, against a length of 323 mm. So we calculate our physical
tuning/fretting units from the octave, but they will show up
rationally against the open string.

We divide our octave by 22. (323/22 in this case) Now we have 22
equal divisions of string length. (EDL). Because the traditional
idea of 4,3 or 2 sruti comprising an interval simply rings true to
me as the kind of idea that would last through the centuries even if
its origins were lost, I measured four of these EDL units. See the
result above- interesting but not conclusive about anything much.

Now- here is the most important part- our string is no longer 646
(323) mm long, it has been shortened by 58.72 mm. (Stopped at the
first fret). So, still measuring against our octave, we have a
string length of (323 minus 58.72), or 264.2727... Once again
divide by 22, giving us our next "sruti" size. Taking four of these
units and adding them to the first fret, we get the result shown
above (once again please verify on your own!).

And now simply repeat the process. Each time, the EDL is from the
new length of the string, in other words, from the last fret.

The first measurent and grouping is, obviously, very crucial.
Hopefully others will help exploring what happens when we branch off
from 2 or three units in the first group, for example. However, any
intervals found with this method are curiously consistent in
character, you can´t just find any damn interval you´d like.

The rule about 4, 3, or two units certainly worked for me here (I
just went on instinct and it turned out as above, on the first try).
There are probably discoverable rules about alternating group sizes
that would guarantee a 3/2, etc.

Now, for the second step I am going to take a liberty, justified so:

a: The fifth as reckoned above is a hair flat (it landed dead on the
center of the former equal-tempered fifth fret-slot in my initial
physical measurements, as far as I could tell from the millimeter
ruler).

b:I know there are tiny rounding errors in the chart above, but they
do accumulate because we´re measuring from fret to fret, not from a
fixed point.

c: This system may not be exact (see below).

d: The default string settings in Scala may be off from the settings
with which this system works best.

e: If this happens to be reinventing some kind of historical system,
it would certainly be "eared" in process, just as intonation is
adjusted to this day by ear, even with lasers, electronics, and
computers assisting us.

f: The beatless fifth has been nailed since time out of mind, by ear.

The small liberty I will take now will be to measure the next
interval from the theoretically perfect fret position of the 3/2
interval, less than 4mm above the spot we landed on previously (in
this case).

This gives us, with new "sruti" from our now-perfect 3/2:

1 4.9836
4 19.9345 233.294 mm 128/81 is 235 mm

hohohohoho....

My experiences with fretting and intonating guitars tell me that
these differences are due to the "action".

Notice that four of these variable measurement units at this point
no longer represent an interval of ~10/9, but an interval of
~256/243.

What is actually going on here? We´re squaring the circle, drawing
curves with straight lines... doing algorithms with basic arithmetic
and chunks of hardwood.

This may have nothing to do with Indian music at all, the use of the
number 22 being a sheer coincidence. At any rate, I´m calling these
EDvL´s (see below, :-) ) "Zhivan units", after my son.

Whether or not all of this has anything to do with Indian music, the
number 22, I suspect, is no coincidence at all. For no reason I can
think of, as I was first calculating these intervals, I wrote down
the ratio 22/7- figuring some kind of imaginary EDL diatonic scale
maybe. Calculating this, there was Pi smiling at me, accurate to the
second decimal and very close from there on. Checking on the
Internet today, I find, to no surprise, that 22/7 is an ancient and
quite accurate approximation of Pi.

I suspect that these variable-EDL, or EDvL (equal division of
variable length) units are not length/22, but length/7*Pi in size.
As I find time, I will try to verify this. And I suspect that the 7
represents some kind of limit- prime, odd, something, and other
numbers can be used to find either a more complex, or completely
different, tuning.

This is extremely fun but now I have to prepare for some workshops
and concerts, not to mention have a beer or three, take care.

-Cameron Bobr

🔗misterbobro <misterbobro@yahoo.com>

Here are some more results using the Zhivan unit Zh :-), using the
same string length and settings as yesterday but trying different
groupings of these units. I´m calling these flexible (diminishing)
EDL units Zh now, although the more I poke at this the more it
smells like some kind of practical ur-system for fretting sruti.

Zh = x° length/22
where 1° length is 1/2 of the open string,
2° length is l° length minus 4Zh, 3Zh, or 2Zh
and so on

1° length is 323, 1°Zh is 14.681mm....

3 * 1°Zh:

29.36 mm

( 32.25 mm would be 256/243 with these settings, I'll shorten the
string then a try monochord type action next)

2° length is now 293.6363, in other words, our divisible length is
now from the first fret to the octave.

from there, three of the 2°Zh (Zh now being 293.6363/22, or 13.347mm)

3 * 2°Zh =
added to the fret above, to locate the second fret, giving us...

69.4 mm 70.9 mm is 9/8

Hmmmm....

Let´s shorten the open string to 636 mm, giving us 1° length = 318.
Divide by 22 to give us 1°Zh, giving us 14.454545...

3 of these 1°Zh gives us:

43.36 mm

but 256/243 would be found at 31.7 mm from the nut. How about two
of the 1°Zh?

Gives us 28.9 mm, not bad but hang on a second...

Let´s make a virtual monochord type of action here, with the same
length but a flat action and the "fret" a movable bridge almost
touching the string... (1mm high nut and bridge, .9mm high fret, in
Scala).

22/21 is now at 28.89, and 2 * 1°Zh is 28.9

Whoa Nellie.

Now from there, 318 minus 28.9 gives us 289.1
Divide by 22 to get a 2°Zh of 13.141 (I´m rounding from 13.1409...
these tiny errors are good, I´ve decided, because they test the
resilience of the system).

Let´s take three of those, why not... 39.423mm. Added to the first
fret at 28.9 gives us 68.323 mm.

What interval will we find there? 9/8 is at 70.66, could be worse
but...I have a sudden hunch. This morning I saw the little Pi
animation at Wikipedia and I think I can visualize what is really
happening with these "Zh" units.

I´m going to try the nearest ratio that has a 7-limit denominator
and 11-numerator, let´s see...55/49, which would lie at 69.7 mm on
our monochord. But I don´t think this is the key to the system,
because somehow I just know what the next interval is going to be:
it falls far too naturally under the fingers and ears from the 22/21
minor second, it simply has to be...

Going back to our 2nd fret at 68.32mm, or (2 * 1°Zh) + (3 * 2°Zh), 3°
length is 249.68mm.
Divide by 22 gives us 3°Zh of 11.349mm.

Multiply by two (a hunch)... and add to the second fret...90.621 mm.

7/6 is at 90.85 mm. It simply had to be, and there it is.

Whether or not this is a possible explanation for the origin of the
22 sruti idea, it´s a functional fretting system, whoo-hoo!

My hunch about going up 2 Zh comes from the feeling that the
traditional idea of using 4, 3, or 2 "sruti" applies to the Zh
because you need to make groups of so-many "sruti" to land on
certain key intervals, 3/2 probably. We'll see this week.

More later, back to music making!

-Cameron Bobro

PS. Yes I know 22/21 is a no-brainer here, and my calculations are
more
labored than they need to be, but I think it's important to go
through
the steps illustrating the relation between the tuning and the
physical
act of fretting.

🔗yahya_melb <yahya@melbpc.org.au>

--- In tuning@yahoogroups.com, "misterbobro" wrote:
>
> Here are some more results using the Zhivan unit Zh :-), using the
same string length and settings as yesterday but trying different
groupings of these units. I´m calling these flexible (diminishing)
EDL units Zh now, although the more I poke at this the more it
smells like some kind of practical ur-system for fretting sruti.
...

> My hunch about going up 2 Zh comes from the feeling that the
traditional idea of using 4, 3, or 2 "sruti" applies to the Zh
because you need to make groups of so-many "sruti" to land on
certain key intervals, 3/2 probably. We'll see this week.
>
> More later, back to music making!
>
> -Cameron Bobro
>
> PS. Yes I know 22/21 is a no-brainer here, and my calculations are
more labored than they need to be, but I think it's important to go
through the steps illustrating the relation between the tuning and
the physical act of fretting.

Hey Cameron,

This is looking good so far!

You might call the divisions EDRL - Equal Divisions of Remaining
Length, to distinguish them from EDL. But Zhivan units is good
too ;-)

About the "2, 3, or 4", and your hunches about which one to choose -
the only rationale you've give so far is that they might need to add
up to groups of 7; I didn't understand quite why you think that.

You made a comment about branching; without a particular impetus to
choose just one of the "2, 3, or 4" as your "Zhivan-count"
(or "sruticount" or "stepcount"), we could obviously end up with a
whole forest (OK, mathematically just a tree, but a pretty dense
one) of different tunings, each arising from just one choice out of
3^7 for a heptatonic.

These 2,787 different tunings would:
(1) have some frets fall close enough to others FAPP (=for all
practical purposes, eg within 3mm) (see note *); and
(2) require some slight adjustment to a few of the frets for say,
the major sixth, when changing from one tuning to another.

So it follows that each step (eg Da, the sixth diatonic step above
the tonic Sa) would occupy a range of closely-grouped positions on
the fretboard. This seems to me to be, at least superficially, a
good reason for the existence of this phenomenon in classical Indian
music.

Without detracting from your emphasis on finding a practical tuning
and fretting system, it would be interesting to spreadsheet the
whole tree of Zhivan tunings arising from the different choices
of "Zhivan-count" at each juncture, then plot the resulting fret
locations and inspect the graph for natural coincidences and
groupings.

Of course, we needn't focus only on 7; we could look for the 5, or
12, or 19, or 22, ... best natural groupings to find scales of
different sizes using Zhivan tuning.

---

Notes:
(*) This is easy to see;, for example, the choices ... a, b ...
and ... b, a ... of "Zhivan-count" would necessarily cover the same
proportions of the Remaining Length, RL. The choice ... a, b ...
leaves the proportion (1-a/22)(1-b/22) of RL; similarly the second
choice leaves the same proportion (1-b/22)(1-a/22) of RL. Also,
constraining successive choices so that they (sometimes or always)
add to 7 would also ensure that there would be a fret at either
R1=(1-3/22)(1-4/22) or R2=(1-3/22)(1-2/22)^2 of the RL from any
other fret. Now R1=1-7/22+12/22^2 and R2=R1+4/22^2(1-3/22), which
work out to R1=.7066115 and R2=R1+.0285499. [Both these are larger
than P=(1-1/pi)~=(1-1/3.141592)=.6816901, although R1 is nearer to P
than to R2. And a 7 of 22 division of RL is a lot easier to find
than a (1/pi)-fold division of RL! Though that gives me an idea ...
see note (&) below.]

(&) I know how to place a fret at 1/pi of the RL, but it takes a
fair few words to explain fully. (Funny, considering how the
picture that popped into my head explains it all much more
clearly!) In lieu of the picture, here's the 1,000 words (you have
been warned):

We can do this using simple tools - a cone (maybe turned in wood, or
bent in sheet metal) used as a nomogram, together with a measuring
string, which must be thin, flexible, strong and not at all
stretchy. The cone has a length scale on one side, which represents
the diameter of the cone at that point. The scale has its origin
(zero point) at the tip of the cone.

Inscribe a set of parallel circles around the cone at fixed heights
and the smallest practical equal distance apart, eg about 1 or 2 mm,
depending on the material. Be sure to take the circles as near the
tip of the cone as practical. The circles should make shallow
grooves in the surface. Together, the circular grooves constitute a
scale representing the circumferences of the cone for different
diameters.

If desired, calibrate the scale using a string as a measure on the
cone itself; thus it would be most practical to set the unit of
measure to the maximum diameter on the scale, at or near the lip
(base) of the cone; label this point 1. Establish fractional points
on the diameter scale by repeated division (folding of the measuring
string) and adding. BTW, the only necessary points on the scale are
the origin and the unit diameter.

The cone has to be of such size that the Semilength of the string
being tuned (the length of the string at the natural octave) is
equal to, or slightly less than, the circumference of the cone at
the point marked 1 on the diameter scale.

The tools are now complete.

To set off a distance 1/pi of the RL (Remaining Length), starting
when RL = S (the Semilength), use the measuring string to transfer
the RL from the tuning string to the best fitting one of the set of
parallel circular grooves on the cone's surface. Using this "groove
of best fit" (and don't we all want to find that?), find its
intersection with the diameter scale. From that point to the apex,
set off the diameter on the edge of the cone with the measuring
string; transfer this diameter to the tuning string, from the last-
tuned fret towards the natural octave point, thereby reducing the RL
by RL/pi.

Should you *want* to tune by subtracting (1/pi)th, rather than
(some/22)th, of the remaining string length, these tools would let
you do the job. They would be easily constructed using no
technology more complex than a springy-tree-branch- or foot-powered
lathe. You don't even need a ruler! By carefully calibrating the
diameter scale, you could also use halves, quarters, etc of (1/pi).
Using the tools alone, you could also calibrate the diameters (and
hence the fret positions) in *powers* of (1/pi).

Regards,
Yahya

🔗threesixesinarow <CACCOLA@NET1PLUS.COM>

🔗misterbobro <misterbobro@yahoo.com>

Fantastic, thanks Yahya! I have to print out your post and read it
at home.

This morning I decided that equal divisions of pi are what's really
going on and I vaguely figured something like "a wooden wheel and a
string" would be a possibility for the ancients, and there you've
described it.

22/7 is an excellent and ancient pi it turns out- I'll continue with
this and check it occaisionally against actual pi measurements
before moving on to other equal divisions of pi on the computer (so
I don't have to deal with horrendous ratios :-) )

By folding some paper, I figured a way with simple geometry to
divide a given length into 7 equal parts- if I'm not mistaken, it
would be very accurate if done with a straightedge and compass, I'll
doublecheck it and try to describe it. Judging by the stonework of
the ancient world, they could divide and measure at will.

Trying to find methods that avoid rulers as much as possible, as you
pointed out in your 1/pi method.

The first attempt in the first post landed on the 3/2 using 21 Zh in
groups of 4,3, and 2 and I immediately thought, it's 3*7.

But today I looked again and it starts out 4, 4, 4... Still,
somehow I feel there's a good reason for limiting how many units you
can grab and it must be more than coincidence that following
the "rule" got such standard intervals so quickly.

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:

> constraining successive choices so that they (sometimes or always)
> add to 7 would also ensure that there would be a fret at either
> R1=(1-3/22)(1-4/22) or R2=(1-3/22)(1-2/22)^2 of the RL from any
> other fret. Now R1=1-7/22+12/22^2 and R2=R1+4/22^2(1-3/22), which
> work out to R1=.7066115 and R2=R1+.0285499.

This I have to print out and experiment with... what "rule" would
guarantee musically useful fret placements.

If you must group to seven before moving to the next string length,
my crude ratio calculations today looked liked they guaranteed some
kind of tritone. Have to check my math, and check out your R1 and R2-
at first glance that looks like tritones, too (isn't 1/.7066115
almost exactly the square root of two?), just a sec¡K 1/R2 is FAPP º
the 15/11 I got this morning.

But! Let's say at each fret you're allowed 4, 3, or 2 units , then
you start again (this is how I've been doing it), different groups
of seven get you, let's see in the first example¡Kthirds? Because the
units keep shrinking at each new fret.

. --- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:

> Of course, we needn't focus only on 7; we could look for the 5, or
> 12, or 19, or 22, ... best natural groupings to find scales of
> different sizes using Zhivan tuning.

Yes I'm very curious about this and the various limits it would put
on the intervals. Sensible groups of units would be different, and
large in the case of 19 or 22 ED, as far as practical fingering and
fretting.

Well I've got a heavy day ahead. Zhivan has bronchitis so I'm home
with him all week. We'll make a clarinet out of a cardboard tube
tomorrow but I'll just poke some finger-holes his size and not worry
about where they land. :-)

Take care and thanks for the great response,

Cameron Bobro

🔗Mohajeri Shahin <shahinm@kayson-ir.com>

Hi dear Cameron

1-……. We divide our octave by 22. (323/22 in this case) Now we have 22
equal divisions of string length. (EDL)…….

>>>> no , you have not any 22-EDL but you have 44-EDL and 22-EDOL (-: ( equal divisions of octave length )because after dividing string length of 646 mm into 44 divisions you can get 22-EDOL in 323 mm (-;

2- remember that EDL systems provide unequal tetrachords , so in 44-EDL of 646 mm string length you have 4/3 as 11 degree.

3-the fourth degree of 44-edl on a string has a ratio of 11/10 or 165.0042 cent , is it the firs real shruti ?

4- if you want to have an 44-EDL system from 323-(4*(646/44))= 264.2727 and for the fourth degree we have also 165.0042 which at last we have (165.0042*2) cent , is it the second shruti?

5-……. Let´s shorten the open string to 636 mm, giving us 1° length = 318.
Divide by 22 to give us 1°Zh, giving us 14.454545...

3 of these 1°Zh gives us:

43.36 mm

but 256/243 would be found at 31.7 mm from the nut. How about two
of the 1°Zh?........

>>>>>> on string length of 636 mm , the real point of 256/243 is found at 32.296 mm from nut but you say that as 31.7 mm , is it your calculation error or measuring error?

6- adding cent of the fourth divisions of remaining lengths with the same 44-EDL ( I'm not agree with EDRL because N-EDL is the same for different lengths) is like that you add 11/10 to each other:

165.0042

330.0084

495.0126

660.0168

825.021

990.0252

1155.0294

1320.0336

1485.0378

1650.042

1815.0462

1980.0504

2145.0546

2310.0588

2475.063

You can see some information about EDL system in : http://240edo.tripod.com/id19.html

This page is going to be up-dated.

And you can hear some musics based on edl system in: http://240edo.tripod.com/my_music.html

Shaahin Mohaajeri

Tombak Player & Researcher , Microtonal Composer

My web site <http://240edo.tripod.com/>

My page in Harmonytalk <http://www.harmonytalk.com/id/908>

My tombak musics in Rhythmweb <http://www.rhythmweb.com/gdg>

My article in DrumDojo <http://www.drumdojo.com/world/persia/tonbak_acoustics.htm>

My musics in Wikipedia, the free encyclopedia :

- A composition based on a folk melody of Shiraz region, in shur-dastgah by Mohajeri Shahin <http://www.xenharmony.org/mp3/shaahin/shur.mp3>

- An experiment in Iranian homayun and chahargah modes by Mohajeri Shahin <http://www.xenharmony.org/mp3/shaahin/homayun.mp3>

🔗misterbobro <misterbobro@yahoo.com>

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@...> wrote:
>
> Hi dear Cameron
>
>
>
> 1-â¦â¦. We divide our octave by 22. (323/22 in this case) Now we
have 22
> equal divisions of string length. (EDL)â¦â¦.
>
> >>>> no , you have not any 22-EDL but you have 44-EDL and 22-EDOL
(-: ( equal divisions of octave length )because after dividing
string length of 646 mm into 44 divisions you can get 22-EDOL in
323 mm (-;
>
> 2- remember that EDL systems provide unequal tetrachords , so in
44-EDL of 646 mm string length you have 4/3 as 11 degree.
>
> 3-the fourth degree of 44-edl on a string has a ratio of 11/10 or
165.0042 cent , is it the firs real shruti ?
>
> 4- if you want to have an 44-EDL system from 323-(4*(646/44))=
264.2727 and for the fourth degree we have also 165.0042 which at
last we have (165.0042*2) cent , is it the second shruti?
>
> 5-â¦â¦. LetÂ´s shorten the open string to 636 mm, giving us 1Â°
length = 318.
> Divide by 22 to give us 1Â°Zh, giving us 14.454545...
>
> 3 of these 1Â°Zh gives us:
>
> 43.36 mm
>
> but 256/243 would be found at 31.7 mm from the nut. How about two
> of the 1Â°Zh?........
>
> >>>>>> on string length of 636 mm , the real point of 256/243 is
found at 32.296 mm from nut but you say that as 31.7 mm , is it your
calculation error or measuring error?
>
> 6- adding cent of the fourth divisions of remaining lengths with
the same 44-EDL ( I'm not agree with EDRL because N-EDL is the same
for different lengths) is like that you add 11/10 to each other:
>
> 0
>
> 165.0042
>
> 330.0084
>
> 495.0126
>
> 660.0168
>
> 825.021
>
> 990.0252
>
> 1155.0294
>
> 1320.0336
>
> 1485.0378
>
> 1650.042
>
> 1815.0462
>
> 1980.0504
>
> 2145.0546
>
> 2310.0588
>
> 2475.063
>
>
>
> You can see some information about EDL system in :
http://240edo.tripod.com/id19.html
>
> This page is going to be up-dated.
>
> And you can hear some musics based on edl system in:
http://240edo.tripod.com/my_music.html

Yes, two of the 1st Zh as in the example because I've been measuring
from the octave (1/2) the string lenghth. The interval of 256/243 I
put in as a guess, the nearest reasonable interval.

The point here isn't really equal divisions of length, though. The
first idea was equal division of a variable (shrinking) length
somehow derived from/related to pi and now, taking it further...

Last night I took a step back and thought about where this is
coming from and going.

It's so simple that it's hard to believe.

As far as I can tell from crude geometry at home, using this
approach (even simplified), you have a way to have 22 equal but not
equal units, 9 of which add to 4/3 and 13 of which add to 3/2. Those
are the units I had a chance to map so far.

I hope my errors weren't too great.

The units are equal, but the extremely simple way they're mapped to
the string causes them to shrink towards the octave.

I'll try quickly, maybe you guys can help me verify. Zhivan is busy
playing my synthesizer...

Draw a circle. The diameter is the string length.

Now, 9 o'clock is the nut.

From 9 o'clock to 6 o'clock, draw a line segment (connect the
points, inside the circle.)

Divide this line into 22 equal parts. These are the units in their
equal state.

From 12 o'clcock, draw a line segment through each of the division
points to the outer circumfrence. You might try the 9th and 13th
units first, counting inward from 9 o'clock toward 6 o'clock (from
the nut toward the octave point)

Now where each of these line segments bisect the diameter
(connecting 9 and 3 o'clock), put a fret and measure the interval
you get against the diameter/string length.

The units are no longer equal when mapped to the string, they shrink
towards the octave.

Please help me verify where they land... :-) You guys are great.

Back tonight, gotta run, take care.

-Cameron

🔗yahya_melb <yahya@melbpc.org.au>

Hi Cameron,

--- In tuning@yahoogroups.com, "misterbobro" wrote:
>
> Fantastic, thanks Yahya! I have to print out your post and read it
at home.
>
> This morning I decided that equal divisions of pi are what's
really going on and I vaguely figured something like "a wooden
wheel and a string" would be a possibility for the ancients, and
there you've described it.
>
> 22/7 is an excellent and ancient pi it turns out- ...

Yep! So ancient, that's what they taught us when I was in primary
school ... ;-) Wasn't until about "B Class" (currently "Year 11")
that they taught us it was actually a transcendental, an infinite
non-recurring decimal with first few places 3.14159... .

> ... I'll continue with this and check it occaisionally against
actual pi measurements before moving on to other equal divisions of
pi on the computer (so I don't have to deal with horrendous
ratios :-) )
>
> By folding some paper, I figured a way with simple geometry to
divide a given length into 7 equal parts- if I'm not mistaken, it
would be very accurate if done with a straightedge and compass,
I'll doublecheck it and try to describe it. Judging by the stonework
of the ancient world, they could divide and measure at will.

SEVEN? That's totally unexpected. Let me know how it turns out -
and if not exactly seven, why you thought it might be.

> Trying to find methods that avoid rulers as much as possible, as
you pointed out in your 1/pi method.
>
> The first attempt in the first post landed on the 3/2 using 21 Zh
in groups of 4,3, and 2 and I immediately thought, it's 3*7.
>
> But today I looked again and it starts out 4, 4, 4... Still,
somehow I feel there's a good reason for limiting how many units
you can grab and it must be more than coincidence that following
the "rule" got such standard intervals so quickly.

I agree, and feel that the "good reason" is probably mathematical.

> --- In tuning@yahoogroups.com, "yahya_melb" wrote:
>
> > constraining successive choices so that they (sometimes or
always) add to 7 would also ensure that there would be a fret at
either R1=(1-3/22)(1-4/22) or R2=(1-3/22)(1-2/22)^2 of the RL from
any other fret. Now R1=1-7/22+12/22^2 and R2=R1+4/22^2(1-3/22),
which work out to R1=.7066115 and R2=R1+.0285499.
>
> This I have to print out and experiment with... what "rule" would
guarantee musically useful fret placements.

On what *someone* may find "musically useful", there are no
guarantees! A slightly less ambitious goal might seek to answer
this: what rules provide _playable_ fret placements with between
(say) 5 and 25 frets per octave. But what is playable?

Playable fret placements
------------------------
allow the player to stop (and preferably also to bend) the string
with the fingertip cleanly on the fingerboard behind each fret,
rather than straddling the next fret behind it. This mandates a
minimum distance between frets for any particular player, which
needs to be larger for those player with big hands, so unless it's
a bespoke instrument for one player, should be chosen to
accommodate the vast majority of players' hands.

Similarly, the guitar fingerboard can't be much longer than the
usual two handspans, or 430mm (@), without making it impossible for
many players to reach all the frets. Nor can the string be much
longer than the usual three handspans, or 645mm (@), without both
moving the bridge too far towards the edge of the soundbox to
create appropriate resonances (#) and making the notes of a major
third far wider than the player's stretch (ability to reach from
one hand position). In short, the maximum string length and fret
separation both depend on the player's handspan and arm length (tho
a guitar tuned in major thirds instead of fourths might accommodate
a proportionately longer string and fingerboard).

These are essentially physical limitations which any observer, even
without special knowledge of instrument making, could note. My
impression is that, even if we stretched the fingerboard by a
quarter, and tuned the strings in major thirds, few guitar players
could make much use of frets placed more than twice as closely as on
a 12-EDO guitar. In other words, the smallest interval we can
practically fret at the high end, nearest the sound hole and
bridge, is about a quartertone, with fret separation around 6 to 7
mm, instead of the current 13mm (@). At the low end, near the nut,
we could use the same fret separation, instead of the current 35 to
36mm (@), giving pitch resolution of about a tenth- or twelfth-
tone.

So it's the top end that limits our playable resolution, which I
reckon at about 24-EDO for most players. Providing extra frets at
the bottom end would certainly give better resolution there, but
one that could not be maintained in higher registers. Eg frets
spaced at 1/53-EDO at the bottom end might be (just) playable, but
for the second octave to be playable, we'd have to space them at
2/53- or 3/53-EDO.

This is my (fairly naive) take on what frettings would be playable
by enough players to be practicable. I'm sure I've missed some
relevant fact well-known to luthiers!

(@) measured on my flat-top guitar. I have fairly large hands; YMMV.

(#) Though the banjo's bridge is much closer to the edge.

> If you must group to seven before moving to the next string
length, my crude ratio calculations today looked liked they
guaranteed some kind of tritone. Have to check my math, and check
out your R1 and R2- at first glance that looks like tritones, too
(isn't 1/.7066115 almost exactly the square root of two?), just a
sec¡K 1/R2 is FAPP º the 15/11 I got this morning.

Yeah, root 2 is about 1.4142 and half that, 0.7071 is also its
reciprocal.

> But! Let's say at each fret you're allowed 4, 3, or 2 units , then
you start again (this is how I've been doing it), different groups
of seven get you, let's see in the first example¡Kthirds? Because
the units keep shrinking at each new fret.

You're right. I intend to follow thru with that spreadsheet I
mentioned.

> . --- In tuning@yahoogroups.com, "yahya_melb" wrote:
>
> > Of course, we needn't focus only on 7; we could look for the 5,
or 12, or 19, or 22, ... best natural groupings to find scales of
different sizes using Zhivan tuning.
>
> Yes I'm very curious about this and the various limits it would
put on the intervals. Sensible groups of units would be different,
and large in the case of 19 or 22 ED, as far as practical fingering
and fretting.

---

Uhuh ... Cain't he'p myself ... a minor monograph follows ;-)

Zhivan scales with s notes per octave
-------------------------------------
If we take 3 Zhivan scale steps adding up, to first approximation,
to 7 steps of 22 EDO, ie, to 2^(7/22), then each of them is, on
average, of size 2^((7/22)/3), as a ratio. This is 7/3 Zh or
2.333' Zh. (The Zhivan is a logarithmic unit.)

[For non-mathematicians: this is just a shorthand way of saying that
the average scale step (and fret) size in Cameron's Zhivan tuning
is about one-third of the interval formed by 7 steps of 22-EDO, or
two and a third steps of 22-EDO.]

Also, since this average lies between the two consecutive integers 2
and 3, the smallest number of Zh in a scale step cannot be greater
than 2, and the largest number cannot be less than 3.

---

Similarly, 2 Zhivan scale steps adding up, to first approximation,
to 7 steps of 22 EDO, ie, to 2^(7/22), are each on average of size
2^((7/22)/2), as a ratio. This is 7/2 Zh or 3.5 Zh.

Also, since this average lies between the two consecutive integers 3
and 4, the smallest number of Zh in a scale step cannot be greater
than 3, and the largest number cannot be less than 4.

---

So assuming that 2 or 3 Zhivan scale steps add up, to first
approximation, to 7 steps of 22 EDO, ie, to 2^(7/22), each is on
average (%) of size 2^((7/22)/root(2*3)), or about 2^((7/22)/2.5).
Also, the smallest number of Zh in a scale step cannot be greater
than 2, and the largest number cannot be less than 4.

(%) That's the geometric average, not the arithmetic average.

---

How many of these average frets make an octave? It needs to be of
the order of 2.5/(7/22), or 55/7, which is about eight. Allowing
for the contracting RL, second order effects would push this number
up to at least 9, maybe even 12 - I haven't done exact calculations
yet.

So, to summarise, using Cameron's Zhivan tuning technique, with:
- 1 Zh = RL/22;
- (&) scale steps of 2, 3 or 4 Zh; and
- (*) most groups of 2 to 3 successive scale steps totalling, on
average, 7 Zh;

we will get about 9 to 12 scale steps (and thus frets) per octave.
We could label this the Zh(22;2,4;2,3,7) cluster of tunings.

(*) This expression has two points of vagueness - "most" and "on
average". What I'm getting at is that the average of every group,
except possibly for a very few, relatively, of them, falls nearer
to 7 Zh than to either 6 or 8 Zh or any other number of Zh. A couple
of outliers will not change the picture greatly, provided their
deviation from average is "not too great". This loose statement
could be tightened up and formalised if necessary.

(&) If this condition is not specified, a similar, but generally
weaker, pair of limits results. In the absence of this
restriction, from what I wrote earlier it follows that some (^)
scale steps are no bigger than 2 Zh and some (^) scale steps are no
smaller than 4
Zh.

(^) "some" here has its meaning in logic: "at least one".

---

Suppose we want to divide the octave into, say, 22 Zhivan scale
steps, rather than 9 to 12.

What size should the Zhivan unit be?

Let's generalise the above assumptions to:
- 1 Zh = RL/m;
- scale steps of a, a+1, ..., b-1 or b Zh; and
- most groups of c to d successive scale steps total, on average, n
Zh;
for some natural numbers m,a,b,c,d,n resulting in about s scale
steps (and thus frets) per octave. In this case we want s to be
22. Label this the Zh(m;a,b;c,d,n) cluster of tunings.

Its average scale step size is 2^((n/m)/(product(c,c+1,...,d))^(1/(d-
c+1))), and to first-order, s = m(product(c,c+1,...,d))^(1/(d-
c+1)))/n. In particular, for Zh(m;a,b;2,3,7), s = m(product(2,3))^
(1/2))/7 ~= 0.35m = 7m/20; so m ~= 20s/7, which is not much short
of 22s/7 or (pi) times s.

In plain(er) English, this means:
If:
- The Zhyler Zh unit is an Equal Division of Remaining Length into
pi times s parts;
- all scale steps contain between a and b Zh units; and
- most groups of 2 to 3 successive scale steps total, on average, 7
Zh units;

we will get about s scale steps (and thus frets) per octave. Note
that this result depends only on c, d and n. It does not depend on
the values of a and b at all; however, the third condition does
constrain the values of a and b for which we can get a Zhyler scale
at all, as we saw above at note (&).

To get a Zhyler scale of 22 steps, we need a Zhyler unit of about
22pi parts - an EDRL (Equal Division of Remaining Length) into 69
parts. This formula is only approximate, and can be improved, but
it gives a starting point for the tuning. It also applies only to
the Zh(m;a,b;2,3,7) cluster.

Here are some "first-cut" tunings for Zhyler scales in the
Zh(m;2,4;2,3,7) cluster:
- s*pi-EDRL gives s steps
- 16-EDRL gives 5 steps
- 22-EDRL gives 7 steps
- 38-EDRL gives 12 steps
- 53-EDRL gives 17 steps
- 60-EDRL gives 19 steps
- 69-EDRL gives 22 steps
- 97-EDRL gives 31 steps
- 129-EDRL gives 41 steps
- 167-EDRL gives 53 steps

For the general Zh(m;a,b;c,d,n) cluster of tunings, an s-step scale
arises when m ~= m0 s, where m0 = n/(product(c,c+1,...,d))^(1/(d-
c+1))). This simplifies, when c=d, to m0 = n/c, so that m ~=
sn/c. Generally, a and b bracket m0, ie a < m0 < b, where m0 is
given by the equation above. Also, in many cases, if a and b are
near to each other, there will be an s-step scale in the Zh
(sm0;a,b;c,d,n) cluster of tunings. The further a and b are from
each other, and from m0, the less even the scales will appear.

For example, if:
- 1 Zh = RL/m; and
- most groups of 4 successive scale steps total, on average, 38 Zh;
then we have:
- m0 = 38/4 = 9.5; and
- most scale steps average m0=9.5 Zh, and have between 1 and (38-
4+1)=35 Zh; it should be easy to find Zhyler tunings with a,b =
8,10, ie with between 8 and 10 Zh per scale step.

Then some "first-cut" tunings of the Zh(m;a,b;4,4,38) cluster are:
- 48-EDRL gives 5 steps
- 67-EDRL gives 7 steps
- 114-EDRL gives 12 steps
- 162-EDRL gives 17 steps
- 181-EDRL gives 19 steps
- 209-EDRL gives 22 steps
- 295-EDRL gives 31 steps
- 390-EDRL gives 41 steps
- 504-EDRL gives 53 steps

> Well I've got a heavy day ahead. Zhivan has bronchitis so I'm home
with him all week. We'll make a clarinet out of a cardboard tube
tomorrow but I'll just poke some finger-holes his size and not
worry about where they land. :-)

I had a heavy day, thank you, first of all, bounding out of bed just
in time to catch (most of) the 17tone piano concert live, then
trying to answer the thoughts inspired by your email.

Hope Zhivan gets over it soon, but not so soon you don't spend more
time with him. ;-)

Very cool if he can learn to make notes on a (presumably soggy)
cardboard tube. What will you use for the reed?

> Take care and thanks for the great response,
> Cameron Bobro

Likewise!
Yahya

🔗misterbobro <misterbobro@yahoo.com>

Yahya, once again I have to print out your post, to study it,
excellent.

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>

(About 22/7 as Pi)
>
> Yep! So ancient, that's what they taught us when I was in primary
> school ... ;-) Wasn't until about "B Class" (currently "Year
11")
> that they taught us it was actually a transcendental, an infinite
> non-recurring decimal with first few places 3.14159... .

We learned "about 3.14" with extra points for memorizing more
digits, and a vague introduction to irrational and transcendental
numbers (Achilles and the tortoise), along with some impressive
statistics on someone's memory of Pi, but I don't remember anything
about it being expressed in ratios other than "in the Bible it's 3
to 1".

----Bobro wrote---------
> > By folding some paper, I figured a way with simple geometry to
> divide a given length into 7 equal parts- if I'm not mistaken, it
> would be very accurate if done with a straightedge and compass,

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
> SEVEN? That's totally unexpected. Let me know how it turns out -
> and if not exactly seven, why you thought it might be.

Didn't divide a length into seven, but a square piece of paper 1
length on each side in such a way that adding a couple of segments
together could be divded evenly into 1/7 bits... (I figure that I
can only do even folds and folding into 3 like a letter
accurately).

It's not important, just wanted to test the speed and simplicity of
dividing by seven geometrically (also something I never learned in
school).

But let's see, the paper is at home, I'll try to do it again
quickly...

A4 letter paper, turn sideways, fold down one corner diagonally,
tear off reaminder (make a square). Here at the studio I have a
transparent ruler, it's going to be fun to see how well I did.

Side of the square is the length we want to divide into 7.

Let's see 210 mm on a side...hmm got lucky, well within 1mm accuracy.

Turn it point up (diamond). Fold into four and partly open, ie. the
left and right "arms" of the diamond now touching in the center.

Now fold this into three, like a letter.

Unfold one third, ie. the "leaf" on top.

All these folds are done towards you so when you open your "letter",
still pointing longways up, you see it's guts so to speak.

Now get another sheet of paper (paper #2)and use it like you'd add
distances on a map, ie. put a tick and turn, put a tick. The length
of the topmost edge of the part folded inside, it's angling down to
the right or left to where it meets the "hinge" fold of the opened
1/3, is a hypoteneuse- add it's length to the leg of the triangle
(or go out the upper-outer corner of the opened "wing", it's the
same).

Cut paper #2 at the second tick and fold that length in four. Unfold-
one unit at each fold should be 1/7 of your original length (the
side of the original square.).

Now let's see with the ruler.

Original length, 21cm
Hypoteneuse 7cm +
base 5cm = 12cm.

Folded in four...3cm, or 1/7 or 21cm.

Dead on as far as this ruler and my eyes can tell.

>
. But what is playable?

I agree with your general assessments about playability dimensions.
The most I've ever tested this in practice sawing and hammering in a
27th fret on a 12-EDO guitar. Even though it was a long-scale 7-
string, more of a baritone instrument, and only 12-EDO, the spacing
at that point was already indicating where practical limits might
lie- not least for the guy fretting it. :-)

> This is my (fairly naive) take on what frettings would be playable
> by enough players to be practicable. I'm sure I've missed some
> relevant fact well-known to luthiers!

Well, what a bitch it is to file and sand densely spaced frets, and
there's a point where the frets get so close together at which you'd
have to ask, what's actually holding these things on?

> You're right. I intend to follow thru with that spreadsheet I
> mentioned.

Looking forward to it very much!
>
> Uhuh ... Cain't he'p myself ... a minor monograph follows ;-)

> 2.333' Zh. (The Zhivan is a logarithmic unit.) ðð

Yes, that's what we're striving for, logarithmic units derived
and/or mapped from simple measurements.

>
> Here are some "first-cut" tunings for Zhyler scales in the
> Zh(m;2,4;2,3,7) cluster:
> - s*pi-EDRL gives s steps
> - 16-EDRL gives 5 steps
> - 22-EDRL gives 7 steps
> - 38-EDRL gives 12 steps
> - 53-EDRL gives 17 steps
> - 60-EDRL gives 19 steps
> - 69-EDRL gives 22 steps
> - 97-EDRL gives 31 steps
> - 129-EDRL gives 41 steps
> - 167-EDRL gives 53 steps

Yes! I guessed at the first four of these as I was cooking today,
from a different and very crude approach, ie. what would be the next
ratios that would "smell" like 22/7? But I smelled out 34-11, not
38-12, I have to study what you wrote.

> Hope Zhivan gets over it soon, but not so soon you don't spend
more time with him. ;-)

Thanks! He seems well but we're afraid of another relapse.
>
> Very cool if he can learn to make notes on a (presumably soggy)
> cardboard tube. What will you use for the reed?

Squeeky kazoo-like thing from a rubber toy. Unfortunately it doesn't
work like the last, larger-diameter, one I used from another toy, so
it's annoying as all get out, he loves it.

til tomorrow, gotta get back to work, good night,

Cameron Bobro