The Tao of Walking Ridge

An empirical approach to creating a simple intuitive tool and GUI for determining the hole placement for five hole pentatonic Native American style end blown flutes.  Part 1 covers the basic approach and conclusions for calculating placement of the playing holes.

The following represents the author’s opinions and observations based upon flutes that he owns.  Apply to your flute projects at your own risk.

As a relatively inexperienced flute maker, I was impressed with the luck I had applying Dusty’s (of Tsunami flutes) knuckle method for playing hole placement. I was further inspired by recent conversations with Darrin Mallard who has created some beautiful flutes and who also many of Dusty’s techniques. I wanted to create a handy tuning tool, similar to the flutomat but with a Flash based GUI. However, I was overwhelmed by the equations of Lew Paxton Price.  Since Dusty’s knuckle method worked so well, I became convinced that many factors could probably be ignored for a simple estimation. To quote George Box “All models are wrong, some are useful.”

Armed with an OpenOffice spreadsheet and measured data from flutes I’ve made and acquired, I began by transforming hole location from the typically recorded dimension from end of flute to a “hole ratio”.  The hole ratio is basically just the fraction of barrel length from the sound hole the center line of the playing hole.  Defined this way, as the hole ratio drops, so does the wavelength of the note that is played (frequency rises as wavelength drops).  Expressing this wavelength as a ratio of the flute’s root note, we can look for a relationship between the two that is independent of the key that the flute is in.  It should be noted that an effective barrel length must be determined for flutes with tuning holes.

Regardless of the key, opening the first hole of a Mode 1 flute always plays the minor 3rd of the scale, or a note that has a wavelength of 27/32 of the lowest or fundamental note. Other notes in the scale can be defined in a similar fashion, regardless of what key the flute is.  The hole ratio parameter can then be correlated to empirical data for flutes of all keys.  The hole ratio should be less than the wavelength ratio, for if we placed the hole exactly at the wavelength ratio we would have to cut the flute off entirely to achieve the note.
There are slight differences between tuning for instruments optimized for one key (harmonic temperament) and the more common equal temperament that allows equal compromises in every key.  For more on the Pythagorean method of defining scales, reference this Wikipedia article.  The wavelength ratios for the minor pentatonic scale (Mode 1) are:

holes closed          1/1          1.000      root

1st hole              27/32        0.844      minor 3rd

2nd hole              3/4          0.750      fourth

3rd hole              2/3          0.667      fifth

4th hole             9/16         0.563      flat 7th

5th hole             1/2           0.500      octave

For empirical measurements, I primarily used four flutes that I have purchased and two that I have made.  A couple of my purchased flutes were made by Dusty of Tsunami flutes, one was made by Darrin Mallard, and one was made by Lee LaCroix.  In addition, I made some observations regarding large flutes based upon a low D flute that have unfortunately forgotten the maker of.  For the primary study, keys ranged from F# to C.  All were Mode 1 five hole pentatonic flutes (or a Mode 1/4 where Mode 4 was ignored).

Frequencies for each note were measured, so if a hole played flat or sharp the hole placement and size could be compared to its performance.  In addition, some cross fingering frequencies were measured to determine the sensitivity to hole placement.  By comparing only A open with only B open, the impact on frequency of “moving” hole A could be determined.  Similar techniques were used for other holes.  However, I was soon to discover that this requires the holes to be similar in diameter to work!

One of my favorite somewhat scientific approaches is what I call “looking at plots”.  This is where I discovered my first interesting “epiphany”.  Sensitivity for hole location dropped for holes D and E closest to the TSH!  That is what I thought at first, however it turned out that the flutes where only D and only E closed were similar in pitch was because the E holes was much smaller than the D hole.

Disappointed at first, this mistake led me to some significant understanding about hole D and E which is unique to them, specially in a Mode 1 flute where opening the E hole raises the pitch 1 step (vs 1.5 steps for a Mode 5).  For a comfortable playing spacing, the D and E holes are a bit further apart for a Mode 1 than would be ideal in order to leave room for fingers.  Thus, hole E is slightly too close to the TSH, and hole D too far away.  The result is hole E is small and hole D is sometimes larger than the typical 0.3 inches.

In order to understand how much to adjust the hole placement I needed to understand sensitivity to hole diameter.  Fortunately, estimating how much it changes from a typical average is easier to estimate than the calculation needed to determine what the average should be.  Using the hole sensitivity plot above, when the D and E hole was nearly the same size the pitch change was 0.7%.  When the holes were different sizes, the change in pitch was less.  Plotting this and finding the slope gave a sensitivity to pitch of playing hole diameter (d) in inches that surprisingly followed an even clearer trend than the are of the playing hole for the few samples I had.

This trend can only be considered a rough approximation around 0.29 inches for hole E.  In the future, additional data points could be added (perhaps for each playing hole) and many flutes to improve it.  But for now, it will be used to make small adjustments in the playing hole location and estimate the effect on diameter.  If we stay within typical ranges and apply the rule to moderate sized flutes (F# to Bb) then it should be useful to say that the pitch will drop 0.31% for every 0.01 inch increase in hole size.

The hole placement can then be adjusted assuming a typical sensitivity of 0.7, or a .7% change in pitch or wavelength with a 1% change in the hole ratio.  (1% of the barrel length).  So to reduce the D hole size from .32 to .29 on average, the pitch change would be about 1%.  Divided by 0.7, the hole ratio would have to be adjusted 0.014 or 1.4%.  For the D hole to be smaller the hole ratio would need to be reduced by 0.014 from the average.  However, if the D hole is moved it could become smaller than 0.29 on any given flute.  In my opinion, this presents more risk to tone than benefit.  Therefore, in the hole placement tool, I adjusted 1/3 of this theoretical ideal for “closer D and E hole spacing”.  The adjustment was a shift of the hole ratio of .005 (which is about 0.99 * initial hole ratio).

The average E hole was 0.26 inches, so if we wanted to adjust this to 0.29 we would need to increase the hole ratio by 1.4% on average.  These should be considered approximations for “nudging” the hole location from a typical.  To be conservative, the hole placement tool currently nudges the E hole about 0.01 (which is about 1.3 x the typical E hole ratio).  As more data is gathered, the “nudge factor” may be increased.  But the initial hole positions resulted in such a playable instrument, that conservative adjustments from this nominal were implemented in the tool.

Here my personal preference for flutes is an advantage.  I like Native American flutes that play primarily in a one octave range, achieving a couple notes above that with occasional cross fingering and a strong blow.  Thus, I can focus on hole placement for the primary note and ignore the effect on notes and mode shapes in the second octave.  If a particular flute doesn’t sound well cross fingered, I consider that the Zen of the process. The chart above provides a guide to hole placement and an idea of the the acceptable range to experiment in for holes D and E which are the most sensitive to placement.

I tried looking for many possible correlations, including the effect of thickness, TSH dimensions, bird shapes and many others.  Many of these clearly affect the root note of the flute.  But once cut to length, everything becomes a ratio.  After days of searching, I was rather stunned to see the best approximation ignores all data except the desired note.  (George Box was right, all models are wrong, but some are useful.  Expecially simple models.)  Eventually I discovered that a log curve fit very well through all the data points for all the holes once the dimensions were transformed into hole ratios.  This curve then provides a simple way of creating a flute with whatever scale you like.  For a typical target hole of about 0.29 to 0.3 inches, the hole ratio for each equation should be:

Hole Ratio = 0.7752*LN(Wave Ratio) + 0.8202

Hole Ratio = (TSH to CL of playing hole / Total Barrel Length*)

Wave Ratio = (Wavelength of desired note / Root note of flute)

or Wave Ratio = (Root Frequency / Desired Frequency)

*Barrel length for angled end flutes was measured as the midpoint.  If tuning holes are used, and effective barrel length must be calculated.

Thus, a simple estimate for the playing hole locations a five hole style Native American flute can be shown graphically as a log curve.

I am inspired to carve a Mode 5 flute now and try the equation out.  I also have some thoughts about a “Blues Flute” that would add a flat fifth to the scale.

These suggested hole placements are suggested for the typical hand crafted tuning approach of starting with the holes undersized and slowly increasing the size of each hole (starting with hole A) until each hole is in tune.  The guide is not accurate enough to enable drilling directly to a specified diameter.  As more data is gathered, the method will be revised over time.  I hope the simplifed approach helps other flute makers who are starting out and need a reference point for starting their projects or making flutes with altered scales.

       - the Muse

Tags: flutes

This entry was posted on Sunday, April 19th, 2009 at 1:18 pm and is filed under Gear. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.