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NAFlutomat - Native American Flute Design Tool

This page provides a tool for designing Native American flutes. The basic strategy is to provide inputs such as the key of the flute, the bore diameter, wall thickness, finger hole size, and many other parmeters you will be using, and this tool suggests design parameters such as the length of the sound chamber and the location of the finger holes.

Note that this tool attempts to predict the best bore length, finger hole locations, and other output parameters using a set of rather complex equations. While these equations provide can provide a “reasonable” first approximation for these parameters, they should be considered a starting point. It is typically possible to construct a well-tuned flute by using the results of this tool and then careful tuning of the flute (see the Basic-Six, Basic-Five, and Bob Grealish tuning methods). However, the “rather complex equations” used in this tool do not account for pertubations in the size and shape of the flute, the special considerations and limitations on finger hole position or size, or (in particular) the sound that you are trying to achieve.

In short: You may find this tool a useful starting point.

Also, please consider other (less “equation-based”) approaches to flute design on the Finger Hole Placement page.

Getting Started

If you are new to NAFlutomat, I suggest beginning with the Guided Version of NAFlutomat. It was developed in August 2014 as a kinder, gentler approach to this sometimes-daunting design tool for flute-crafting. The guided version is composed of a series of eight steps, each on their own page, that helps you set the most important inputs to this NAFlutomat tool. Each step provides descriptions and explanations of the inputs on that page.

Then, if you are not up for diving into the Quick Walkthrough or Some Details sections, here are some suggestions for getting started:

  • Set the three parameters shown in green below. They control the key of the flute you are crafting, your Inside bore diameter, and the wall thickness at the finger holes. This should get you started for a flute with no direction holes.
  • After setting the three parameters shown in green, click the button and look over the results show in the yellow boxes. You can begin to learn this tool by changing various values in the white boxes and clicking to see how they effect the results in the yellow boxes.
  • Remember that mid-range flutes are in octave 4 (C4, which is fairly low for a Native American flute, up to B4, which is rather high for a Native American flute).
  • Don't enter values in the yellow boxes. They will get overwritten when you click the button.
  • If you need to convert drill bit sizes to inches or millimeters, check out the Drill Bit Conversion Table (opens in a new window).

Standard 6-hole, Mode 1/4 NAF

by Edward Kort, with user-interface improvements by Clint Goss.

Version 1.38b — August 31, 2014

Units of Measurement

Units of Measurement

       Note that changing this selection does not convert the input values
       on this page between Imperial and Metric.


  Temperature Relative Humidity Speed of Sound
Playing conditions
Tuning conditions


Sound Chamber Parameters
Inside bore diameter:
Drill bit conversions (opens in a new window)
TSH Parameters
Flue depth
Cutting edge thickness 
TSH width
TSH length (N/S)
Wall Thickness at TSH
TSH factor
Block/fetish factor
Calculated k2
Calculated k1

Design Iterative Calculations
TSH fraction
TSH width
Of bore diameter
Of bore width
Lesser of diameter or width
Bore Parameters
For specified inside bore diameter:
Ltheor/diam 2 ratio Lactual/diam 2 ratio Lactual/diam ratio

For Lt/d2 ratio of:
For La/d2 ratio of:
For La/d ratio of:

  Closest standard router bit diameter Your router bit diameter
Bore width (router bit diameter)
Bore height (total bore depth)
Effective inside bore diameter
Drill bit conversions (opens in a new window)
Nodal interference and
minimum hole size
Highest intended note
Maximum k2
Minimum playing hole diameter
The settings and calculations above
affect the coloration of the
Hole Diameter fields below.
This calculator is provided for convenience.
Settings here are local to this box and
do not affect other portions of NAFlutomat.
=>  Hz

Pitch Standard

Pitch Standard
A4 =  Hertz (Hz.)  


Scale Temperament
Equal temperament
Harmonic-1 temperament
Harmonic-2 temperament
Pythagorean temperament
Changing the scale temperament
adjusts the Playing Frequency
values below.


Fingering Playing Frequency
Tuning Frequency
Tuning Note Hole
Hole diameter Wall thickness
at hole
Calculated dist.
from end of flute
Calculated dist.
between holes
Total bore length:   
Finger diagram open open closed open open open 6
Finger diagram closed open closed open open open 5
Finger diagram closed closed open closed open open 4
Finger diagram closed closed closed open open open 3
Finger diagram closed closed closed closed open open 2
Finger diagram closed closed closed closed closed open 1
Finger diagram closed closed closed closed closed closed see Finger Hole Size for hole diameter colors (opens in a new window)

Direction Holes

Check this box if your wish to have direction holes     Number of direction holes: 

With no direction hole(s) or
with direction hole(s) closed …
Diameter of
direction hole(s)
Wall thickness at
direction hole(s)
Calculated distance of
direction hole(s)
from end of flute
Tuning Frequency
Tuning Note Playing Frequency
Playing Note

Quick Walkthrough

This section provides the documentation for using the above NAFlutomat calculator:

  1. NAFlutomat supports tuning a flute under one set of environmental conditions for play at another set. Enter temperature and relative humidity for each of these environments. Enter inside diameter and wall thickness of your flute.
  2. Enter flue, block, and TSH parameters.
  3. Select a Pitch Standard. The default is A4 = 440Hz.
  4. Choose a Scale Temperament. This should be Equal temperament if you are designing a concert-tuned flute.
  5. Enter your desired playing frequencies in hertz (i.e. cycles per second) or use the KEY of FLUTE selector to set frequencies automatically by selecting the fundamental note. If you enter your own playing frequencies, the scale temperament selected above will be ignored.
  6. Enter your desired finger hole sizes. If you are going to modify the wall thickness at any of the holes, enter that thickness; otherwise, enter the same wall thickness at each hole, representing the wall thickness.
  7. Optionally, enter direction hole information, choosing either the frequency or note for the flute before drilling the direction hole(s). A good choice is 2 or 3 demi-tones below the flute fundamental.
  8. Press the button.
  9. Examine the results.
    The “Calculated dist. from end of flute” column provides the calculated distance from the center of the finger hole to the foot end of the flute.
    The “Calculated dist. between holes” column indicates how much stretch is required between adjacent playing fingers.
  10. Adjust the finger hole sizes: make them smaller to move them up the flute, larger to move down toward foot end.
  11. Press again, try various finger hole size schemes until you're happy with the layout.
  12. You now have two options to save all of the values for reuse:
    1. If you want to save values for your personal use (rather than sharing with other people), you can use the Favorites (Internet Explorer) or Bookmarks (Netscape) feature of your browser. First, press the button. The page will be reloaded with a very long address. Using the Favorites/Bookmark feature, save this page with a descriptive name. You can now start NAFlutomat with these values by selecting this favorite/bookmark.
    2. If you want to save the values to send to a friend, press the button. A dialog pops up, with a very long string. Select the entire string (in Internet Explorer, it will already be selected; in Netscape, you will need to double click it). Then copy the string to the clipboard (in Windows,  Ctrl+C ; other operating systems will have similar commands). Open a text editor. The Notepad application works fine in Windows; other operating systems have similar plain-text editors. Paste the string into the text editor ( Ctrl+V  in Windows). Save this text file; annotate it if you like. Add several URLs in the same file. Whenever you want to use NAFlutomat with one of these saved sets, select the entire string in the text file, copy it to the clipboard, and paste it into the “Address” area of your favorite web browser. Hit  Enter  to load this URL and voila, NAFlutomat will be invoked with the saved values filled in. If you receive one of these files from a friend, be aware that the first part of the URL (before naflutomat.htm) may be different on your computer. Look at one of your saved URLs and change the first part of your friend's URL to match; it should now work fine. Also, do not paste these URLs into email messages; the line splitting “feature” of most email clients will make the URL unusable. Instead, send the text file as an attachment.

Some details:

  1. Boxes in which you may enter values are white. Boxes containing calculated values are yellow. If you enter values in the yellow boxes, these values will be ignored and overwritten with the calculated values.
  2. Hole numbers start with 1 at the foot of the flute (farthest from the mouthpiece).
  3. You can measure the bore length to anything you want: splitting edge, back of TSH, or top of backset. But do it consistently. The difference is subsumed in the k2, calculated using the block and TSH factors you determined operationally. Please see The “TSH parameters” determine k2 below.
  4. When tuning the flute, use the “Tuning” columns. Be aware that, unless the tuning and playing conditions are the same, the frequencies in the “Tuning Frequency” column will not represent standard concert-pitch notes, as do the “Playing Frequency” values. The “Tuning Note” column shows the note that is closest to concert pitch and the number of cents the tuning note differs from this concert pitch; this provides support for standard chromatic tuners which don't display frequencies.
    The convention for designating octaves, as of version 1.37.1b, uses the IPN notation style. In this system, C4 is middle C on a piano, and F#4 is the lowest note on a mid-range Native American flute.
    If your playing and tuning conditions are the same, you may:
    1. enter the same values for both sets of environment conditions,
    2. set just the playing conditions, or
    3. enter “NA” (or any other string) into one or more of the tuning condition values.
    If you do B or C, the program will copy the playing conditions to the tuning conditions and perform the calculations as if you had done A. Be aware that if you leave a blank in any of the playing condition fields (or any field other than the tuning conditions), the program will interpret the value as “0”.
  5. The “TSH parameters” determine k2, the virtual extension of the bore above the TSH. The greater k2, the shorter the flute for a given fundamental. The TSH factor reflects the TSH geometry; the block factor reflects the shape of the block/fetish over the TSH. To determine appropriate numbers for these factors, after assembling the flute (about an inch longer than the calculations indicate), making/choosing the block you will use, and voicing the flute, but before drilling any holes, do the following:
    1. measure the length of the bore,
    2. cover the flue with a piece of tape or card stock rather than the block, and
    3. measure the frequency of tone the flute produces.
    Enter that frequency for the “Tuning Frequency, no holes” row; enter the bore length in the “Total bore length” row (I know it is yellow, but the program does not know this). Click the button. Now put the intended block on the flute and measure the tone frequency. Again, enter this value in the same Tuning Frequency box; click the button. All subsequent flutes, with similar TSH and block geometries, will have quite similar TSH and block factors. If you are attempting to craft flutes with accurately determined length/bore ratios, I recommend performing the above procedure. If not, it doesn't really matter how you vary the two factors to get the lengths to match; they are used in combination to determine k2. The easiest way to do this is:
    1. enter 1.0 in the “TSH factor” box,
    2. enter the frequency with your intended block in the “Tuning Frequency, no holes” box,
    3. enter the bore length in the “Total bore length” row, and
    4. click the button.
    Nonetheless, it is simple to set them as you make each flute. If you are using direction holes, you will need to reset the “Tuning/direction holes” checkbox and the “Tuning Frequency, no holes” value.
  6. The “Bore parameters” help you design a flute that has the “proper” bore diameter for the specified KEY of FLUTE. By proper I mean a flute that plays all the notes in its range with the intended tone quality, and at the same time being responsive in note changes and overblowing. The program supplies three methods for calculating the bore diameter. The first method uses the classic sound chamber aspect ratio: actual bore length divided by the bore diameter; the typical value used for this ratio is 18. Lew Paxton Price developed a different aspect ratio. Based upon recent discussions with Lew, the ratio is implemented in the program as the theoretical bore length divided by the bore diameter squared; the recommended value for this ratio is 24. The third ratio yields intermediate values; I like a value for this ratio of 16.5. To accurately achieve any of these ratios, you will typically need to bore an oval hole. This program calculates the ratio based upon your entered parameters; it also calculates the dimensions of the oval bore which gives your entered ratio. It does this by setting the bore width to the closest circular radius, in increments of 18″ (2 mm if you chose metric units), allowing you to use common core box or drill bit sizes. Optionally, you may enter the diameter of the bit you wish to use (in the “Your router bit” column). Two of the ratios are affected by k2, which changes with the TSH width. If you make the TSH width a fixed fraction of the bore diameter (as I do), you may have found that you need to perform the calculation, enter a new value for the TSH width, and recalculate (maybe multiple times). There is now a set of widgets, in the “Design iterative calculations” table that performs these iterations for you. It uses the bore width that you enter in the “Your router bit” column. Enter the effective diameter in the bore diameter box to use the oval bore values. If you used the iteration widget, also copy the TSH width that it contains to the “TSH parameters” table.
  7. There is a relationship between k2 and the highest note that can be played on a flute. When the wavelength of a note divided by 4 is less than k2, the energy of the note is dissipated outside of the flute (above the TSH) so the note does not play. Lew Paxton Price calls this nodal interference. Select the “Highest intended note” you wish your flute to play; the default is one octave plus 3 demi-tones above the fundamental. After pressing the button, the “Maximum k2” represents the largest value for k2 that will allow the intended note to play. If this value is less than the “Calculated k2”, you will be unable to play this note. Either accept this limitation, or modify your sound mechanism. The easiest modification is to increase the width of the TSH.
  8. There is also a relationship between the highest intended note and the minimum playing hole diameters. Under the assumption that all the holes will have similar diameters, the “Minimum playing hole diameter” represents the smallest finger hole size that will allow the highest intended note to play cleanly. This value is an approximation, and so should only be used as a guide. You probably don't want to make holes 1 and 2 smaller than this value; the others can be made slightly smaller. The average wall thickness at the finger holes is used in this calculation (the thinner the wall, the smaller the minimum playing hole diameter).
  9. The background color of the finger hole diameter fields is set for those finger holes that are within the range implied by the “Highest intended note” and the “Key of Flute selections”. The background colors of the corresponding hole diameter fields are set based on how they compare with the “Minimum Playing Hole Diameter” calculation: pink for less than the calculated Minimum playing hole diameter size, and a deeper red for less than 90% of the calculated Minimum playing hole diameter size. See the Finger Hole Size and Nodal Interference pages for a description (they open in a new window).
  10. The frequencies set by using the KEY of FLUTE selector represent mode 1 tuning. The mode 4 notes Finger diagram closed open open closed open open and Finger diagram open open open closed open open will be somewhat sharp. As is common with mode 1/4 flutes, you might want to split the difference, make the mode 1 notes Finger diagram closed open closed open open open and Finger diagram open open closed open open open slightly flat and make the mode 4 notes Finger diagram closed open open closed open open and Finger diagram open open open closed open open slightly sharp. My preference is to tune the flute as indicated, either blowing these notes softer (and in key) in mode 4, or closing hole 2 (as well as 3).
  11. To make a 5-hole flute in mode 1, just set the hole diameter for hole 4 to 0.
  12. To determine whether this program works for you, measure one of your flutes, getting: bore length, bore diameter, TSH parameters, hole diameters, and frequencies for each of the fingerings. Enter these values into the program and adjust the TSH and block factors until the total bore length matches your flute. Then compare the hole positions on the flute to those calculated by the program. If they match, within the errors of your measurements, the program will work for you. I still recommend that you start, as is normal practice, with a longer flute and smaller holes.
  13. I do not advocate using the very analytical approach to flute layout represented by this program over more traditional/organic methods and tunings. I simply offer the program as an option to those who have not yet developed the experience to craft pleasing flutes of different keys without such aids, who wish to explore alternate hole spacings and tunings, like to play with numbers, or just hate to spend a lot of time making firewood because of layout errors.
  14. If the wall thickness at all of the finger holes and direction holes is the same, just enter the thickness in the row for finger hole 6. Then click the button to copy this value for all the other holes.

Preserving Your Results

If you wish to print the results of the designs you create using this software, you might consider some techniques to elimitate printing the documentation. Here are some suggestions on how to do this, from various sources and suggestions I have received from folks at the Native Flute Woodworking Yahoo Newsgroup.

Please realize that these suggestions are highly dependent of your particular computer setup:

  • On Windows 7, you can use the Snipping Tool. Start ⇒ Search Programs and Files ⇒ “Snipping Tool” … and the draw a box around the area of the screen you wish to save. You end up with a JPEG image that you can edit, enlarge, save, print, etcetera.
  • You can print to PDF and choose the page numbers you wish to print.
  • On Windows, use the  Print Screen  key (possibly while holding down the  Ctrl  key) to capture your screen into the copy/paste buffer, and then paste the image into a drawing program.
  • Many browsers offer the option to print a selected portion of a web page. Select the portion of the page you wish to print, bring up the print menu from the top pulldown menu options, and then choose “Selection” for what you wish to print.
  • On Macintosh systems, there is a Grab application that performs a similar job to the Windows Snipping Tool.


This table shows the fixed ratios used for the various harmonic temperaments and the bias from “ET” (equal temperament):

Fingering Harmonic-1 Harmonic-2 Pythagorean
Ratio Bias from ET Ratio Bias from ET Ratio Bias from ET
Finger diagram open open closed open open open 2:1 0 cents 2:1 0 cents 2:1 0 cents
Finger diagram closed open closed open open open 9:5 +18 cents 16:9 −4 cents 16:9 −4 cents
Finger diagram closed closed open closed open open 8:5 +16 cents 8:5 +16 cents 128:81 −8 cents
Finger diagram closed closed closed open open open 3:2 +2 cents 3:2 +2 cents 3:2 +2 cents
Finger diagram closed closed closed closed open open 4:3 −2 cents 4:3 −2 cents 4:3 −2 cents
Finger diagram closed closed closed closed closed open 6:5 +16 cents 6:5 +16 cents 32:27 −6 cents
Finger diagram closed closed closed closed closed closed 1:1 0 cents 1:1 0 cents 1:1 0 cents


This is version 1.38b (dated August 31, 2014), with formatting and stylistic updates made by Clint Goss to version 1.37.2 of the NAFlutomat web interface written by Edward Kort (email edward.kort@gmail.com) who has graciously allow me to post it here for general use.

If you would like to use the previous version (the “pure” version 1.37.1 dated 19Sep2008 without formatting and stylistic updates) it is available on the NAFlutomat 1.37.1 page. Other previous versions are available as listed in the Revision History below.

Only Edward Kort (edward.kort@gmail.com) should be contacted with questions concerning the operation of NAFlutomat. Feel free to contact Clint Goss regarding any other issues or suggestions about this page.

Pete Kosel (email ph_kosel@cwo.com; web page http://www.cwo.com/~ph_kosel/) wrote the original Flutomat program, which is the basis of this interface and several of the equations used.

The majority of equations used in NAFlutomat are based on a series of flute theory books (or monographs) written by Lew Paxton Price ([Price 1991] and [Price 1997]). He has written flute design software of his own, but bears no responsibility for any aspect of this NAFlutomat software. Accordingly, Lew Paxton Price should not be contacted regarding NAFlutomat.

The complete series of small flute-making books by Lew Paxton Price is described on the Oregon Flute Store site, but are available from other sources or by contacting Lew at lewprice@softcom.net. The Lew Paxton Price web site (primarily educational) is currently at http://www.lewpaxtonprice.us/ (the new site for the older http://www.softcom.net/users/greebo/price.htm).

Members of the Native Flute Woodworking Yahoo Newsgroup made valuable suggestions leading to this release.

If you find any problems or suggestions regarding this page, please contact me.

Revision History

Version 1.38b — August 31, 2014

  • Expand from two choices of temperament to four.
  • Display the ratios of the temperaments dynamically on the interface (for other than equal temperament).

Version 1.38a — August 30, 2014

  • Set background colors of the finger hole diameter fields for those finger holes that are within the range implied by the “Highest intended note” and the “Key of Flute selections”. The background colors of the corresponding hole diameter fields are set based on how they compare with the “Minimum Playing Hole Diameter” calculation. Pink for less than the calculated “Minimum playing hole diameter” size, and a deeper red for less than 90% of the calculated Minimum playing hole diameter size.
  • The interface is expanded so that the actual form-buttons can be used within the documentation.
  • Fix minor bugs and typos.

Version 1.38 — August 27, 2014

  • Integration with the new Guided NAFlutomat interface — a step-by-step approach to setting the primary inputs for this tool. Guided NAFlutomat provides a sequence of pages, each of which sets one or a few inputs. Each page has extensive documentation on the inputs on that page, and also provides default values based on previous inputs.
  • Code update by Clint to add a Pitch Standard parameter (A4 = 440Hz by default). This involved changes to the JavaScript code in this page as well as the common AcousticCalcs.js module. However, AcousticCalcs.js remains backwards compatible with earlier versions of dependent code. An associated new parameter in the saved/restored URL of "ps" was needed. This may cause unforeseen issues with re-loading URLs from earlier versions — still needs testing.
  • Move the selection of Scale Temperament to a separate section and table.
  • Explicitly show inches vs. centimeters, °F vs.°C, and feet/sec vs. meters/sec on the interface to reinforce the units that have been selected. Change the label dynamically when the radio buttons are changed. However, automatic conversion of the input values is not done.
  • New introductory text which sets the scene for how NAFlutomat fits into the scheme of flute-design approaches and tools.
  • Add a note on the lack of conversion of input values when changing between Imperial and Metric.

Version 1.37.2e — January 4, 2013

  • Reorganize the sections on this page for better readability.
  • Highlight the most important and most basic parameters for getting started with this tool in soothing green.
  • Add “Getting Started” documentation at the beginning.
  • Reorganize the Direction Holes section for more comprehensibility.

Version 1.37.2d — April 18, 2011

  • Modification only to the documentation … addition of note about measurement of the bore length (new third item in the “Some Details” section).
  • Set math expressions (such as k2) in a different font style for ease of reading.

Version 1.37.2c — January 31, 2011

  • Repair broken calculation of the direction holes, which were off by an octave. Thanks to Jeremy Howell for reporting this problem.

Version 1.37.2b — December 9, 2010

  • Change the interface to use the IPN notation style of, for example, “F#2” rather than “F,,#”.
  • Upgrade to use standard MIDI note numbers (previously used MIDI numbers off by one octave).
  • This incorporates Ed Kort's bug fix that became official with a version change from 1.37.1 to 1.37.2.

Version 1.37.1a — December 9, 2010

  • A bug found during yesterday's development caused a hang of the code if an ultra-low flute key was selected. This bug was fixed by patches to the findBoreParameters() routine from Ed Kort. No version number change in the base code - it's still 1.37.1 in the base code and 1.37.1a in this web version.
  • This version is available on the NAFlutomat 1.37.1a page.

Version 1.37.1a — December 8, 2010

  • Extract common JavaScript code out into a separate file for use in other calculators.
  • Formatting and style changes (including the relocation of this revision history) to match the style of other Flutopedia pages.

Version 1.37.1 — September 19, 2008

  • Minor bug fix to version 37: Fixed reloading of URL parameters when the L/d ratio was saved.
  • This version is available on the NAFlutomat 1.37.1 page.

Version 1.37

  • Adds a new aspect ratio.
  • Metric calculations have been fixed.
  • Added a note-frequency converter.
  • Changes from version 1.34:
    • Added iterative calculations for bore parameters, TSH factor, and Bird factor.
    • Changed direction hole frequency to reference tuning rather than playing frequency.
    • Added wall thickness replication button.


  1. Pete Kosel wrote the original Flutomat program, which is the basis of this interface and several of the equations used.
  2. Lew Paxton Price wrote a series of monographs which include the majority of equations used and the description on determining TSH and block factors. The aspect ratio (bore length / bore diameter ^ 2) is calculated according to personal discussions with Lew and differs from that presented in his books. I thank Lew for these discussions.

    I thank both Pete Kosel and Lew Paxton Price for their contributions to the art and craft of Native American flute construction.

  3. Don Forshag and other members of the Native Flute Woodworking Yahoo Newsgroup who made valuable suggestions leading to this release.

If you identify errors in the program, I can be contacted at: edward.kort@gmail.com

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To cite this page on Wikipedia: <ref name="Goss_2022_naflutomat"> {{cite web |last=Goss |first=Clint |title=NAFlutomat - Native American Flute Design Tool |url=http://www.Flutopedia.com/naflutomat.htm |date=7 June 2022 |website=Flutopedia |access-date=<YOUR RETRIEVAL DATE> }}</ref>