Recently, I became interested in the topic of ancient harp tunings.
This interest was spurred from wondering what kind of tunings King
David may have used on his harps and played while singing and composing
the Psalms. The smaller harp that he played while tending his flocks as
a boy is said by tradition to have had ten strings.
It does not seem unreasonable to speculate that he may have used
subharmonic tunings. These tunings do have an ancient, primitive feel to
them.
While wondering about this, Kraig Grady sent me a pointer to a
fascinating and useful document by Erv Wilson:
-
http://www.anaphoria.com/tres.PDF
In this document, I found a page with some subharmonic based scales
titled:
Subharmonic Species, & Diaphonic Cycles
(Constructed from 2 Equally-Divided Strings)
Issued 19 Jan 65 by Ervin W. Wilson
The third scale on this page divides the octave into ten unequal
parts and is shown as:
-
16 15 14 13 12=18 17 16 15 14 13 12
This type of notation is intended to specify a scale built from
multiple (in this case two) subharmonic series joined together. This can
be confusing the first time you see it but conversion to ratio or cents
is not difficult.
| Getting to the Meat in that Sub |
Conversion Method 1:
To convert the subharmonic scale specification to the more commonly
used absolute ratio notation, divide each subharmonic into the
appropriate numerator needed to start the scale out at 1/1. When you
reach the point where two subharmonics are shown as equivalent, continue
as before with the new numerator, but also multiply each interval by the
ratio at which the equivalence occured.
Here is the result, represented as absolute ratios from 1/1:
-
16/16,16/15,16/14,16/13,16/12*18/18,16/12*18/17,16/12*18/16,16/12*18/15,16/12*18/14,16/12*18/13,16/12*18/12
This reduces to:
-
1/1,16/15,8/7,16/13,4/3,24/17,3/2,8/5,12/7,24/13,2/1
Conversion Method 2:
Instead of trying to figure out the absolute ratios from 1/1, figure
the relative ratios from the previous scale degree. Doing it this
way makes it easy to specify the scale as relative ratios by simply
cascading the subharmonic numbers into ratios:
-
16 15 14 13 12=18 17 16 15 14 13 12
becomes:
-
16:15,15:14,14:13,13:12,18:17,17:16,16:15,15:14,14:13,13:12
Here is a graphical depiction of the scale, specified as absolute
ratios from 1/1:
Note the identical subharmonic pentachords — from 1/1 to 4/3, and
from 3/2 to 2/1. The graph shows clearly how the subharmonic intervals
get larger as they ascend in pitch within each pentachord. This is what
straight run subharmonic scales look like. Straight run harmonic
scales ascend in pitch with each interval getting smaller.
Note also that these two pentachords are separated by 9/8 divided
into 18/17 * 17/16.
| Trimming the Fat from the Sub Scale |
I tuned up Erv’s scale and was very pleased with its sound which I
will describe as ancient and acoustic and primitive.
But I wanted to simplify it a bit and perhaps have the ten notes that
I would use for the harp span more than an octave. I played the scale
for a while with the idea of finding a set of seven notes I liked to
have a heptatonic (seven note) scale I could map to the white keys of a
standard keyboard. I settled on the sound and grooved on the mood of the
following subset of our original scale:
-
1/1,16/15,16/13,4/3,3/2,8/5,24/13,2/1
And so now we have a pseudo-symmetrical scale! I did not set out to
make a pseudo-symmetrical scale when I picked the subset out by ear; it
just turned out that way.
This was a really nice sounding scale but still wasn’t exactly
the right feel I wanted. I decided to try stretching and compressing the
scale, as I often do with new scales, to see how it sounded... if that
livened up things a bit. After trying various scalings of the scale, I
found that I liked the sound of the scale when stretched to have an
octave at around 1217 cents (2.019736) instead of 1200 cents (2/1).
| Scale Stretching Made Easy |
For those of you who would like to try your own stretched scales,
this is a good place to cover the basics of how to do so. The first
two methods describe the math in detail and are best done in a
spreadsheet. Those with LMSO can cut to the chase and use LMSO’s
trivially easy way to stretch any scale, described below as Method 3.
A common mistake is to try to stretch a scale by multiplying all the
absolute ratios by a scaling factor. Unfortunately, this does not work.
The reason it does not work is that multiplying a set of ratios by some
fixed number merely adds the ratio that that fixed number
represents to each of the ratios. In other words, this method just ends
up shifting the scale rather than stretching it.
Method 1: Stretching Ratio Values
If you are working with ratios and you want to apply a stretch, raise
each ratio to the power of the stretching factor instead. That
stretching factor is calculated using a ratio of cents values.
stretched_interval (in cents)
sf (stretch factor) = -----------------------------
original_interval (in cents)
Therefore, if you want to stretch the octave from 1200 to 1217 cents,
set sf = 1217/1200.
As an example of stretching, let’s try this method to stretch my
little 7 note subset of Erv’s scale so that the octave spans 1217 cents.
Using:
-
sf = 1217/1200 = 1.014166666667.
Gives the following results: (You can try it out in a spreadsheet if
you like.)
Ratio Ratio^sf
----- --------
1/1 1.000000
16/15 1.067642
16/13 1.234395
4/3 1.338778
3/2 1.508641
8/5 1.610689
24/13 1.862259
2/1 2.019736
Here the stretched scale is shown relative to 53-limit JI:
Method 2: Stretching Cents Values
If you are working with cents, stretching is done with the same
stretch factor, but you just multiply that stretch factor by each cents
value. For example:
-
sf = 1217/1200 = 1.014166666667.
Ratio Cents Cents * sf
----- -------- ----------
1/1 0.000 0.000
16/15 111.731 113.314
16/13 359.472 364.565
4/3 498.045 505.101
3/2 701.955 711.899
8/5 813.686 825.214
24/13 1061.427 1076.464
2/1 1200.000 1217.000
Method 3: Stretching by converting to an equal tuning
In Li’l Miss’ Scale Oven (LMSO), all tunings can be easily converted
to an equal temperament that is close enough to the original scale so as to be basically equivalent (at
least relative to the tuning resolution of your instrument’s tuning
tables). By using this step, stretching of any scale — whether it is made of
just ratios, measured in cents, or put together some other way — becomes a trivial
two-step operation and does not require a spreadsheet or any error-prone mathematical
gyrations and calisthenics. Without LMSO, doing the stretching requires
a spreadsheet or the adept use of a calculator, as was just covered under Method 1
and Method 2, above.
Step 1: Show the scale as equal. In LMSO, choose
‘Srutis Absolute’ or ‘Srutis Difference’ from the ‘Scale Pattern’
menu. You will see that the Scale Pattern field — whether it was
in cents or ratios before — will now be shown as a equal tuning
(perhaps with some huge number of divisions).
Step 2: Simply type in a new ‘Repeat Ratio’
to be the stretched value you are targeting. You’re done!
If you want to specify the stretched interval in cents, you can use the
cents2ratio() function in the Repeat Ratio field. For example, you can
either set the Repeat Ratio to a decimal value like '2.02' instead of
'2/1', or you can type in ‘cents2ratio(1217)'.
Here are some diagrams comparing our scale to 34 equal and 53 equal
which you may find briefly entertaining:
Compressing a scale
If you are compressing a scale, where the resulting compressed
interval is smaller than the original interval, the stretch factor will
be less than 1.0 and can be calculated the same way as before using
Methods 1 or 2.
compressed_interval (in cents)
sf (stretch factor) = -----------------------------
original_interval (in cents)
Stretching to fit any interval
The stretch factor (sf) can be constructed to alter any interval you
like — not just the octave! This useful and easy technique will be
discussed in more detail and with some examples in the next article in
this series.
Having tuned up my ancient harp tuning with a stretch, I found it
bright, vibrant, more realistic, more organic and invigorating. A
success!
If you like, the stretched tuning can be heard playing on a harp. Go
to:
-
http://geocities.com/nonoctave/
...and download the song titled “A Breeze Stirs the Shade”.
I hope this article was of use to you. Have fun stretching and
compressing your scales and in the next article we’ll go into more depth
with advanced stretching techniques.
— X. J. Scott
[Go to Part II =>]