Abstract:
The method provides luthiers of fretted instrument with a novel approach for installing frets with increased accuracy. The method is an improvement in calculation of fret placement over the “Rule of 18” because it relies on the length of the vibrating string. This method is more pronounced at the end of the fret board closest to the bridge due to the angle formed by the string when depressed with respect to the axis of the fret board. With respect to the twelve-step octave, the scale length is multiplied by the constant of the twelfth root of 0.5 to calculate the length of the string from fret contact to saddle contact for the next tonal step.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   This application is a continuation-in-part of U.S. patent application Ser. No. 11/164,812, filed on Dec. 6, 2005, titled “PYTHAGOREAN FRET PLACEMENT,” herein incorporated by reference in its entirety. 

   FIELD OF THE INVENTION 
   This invention relates in general to musical instrument construction, specifically with respect to fret placement and fret boards for stringed instruments. More particularly, the invention deals with a Pythagorean approach to fret placement for stringed instruments. 
   BACKGROUND OF THE INVENTION 
   In the construction of the neck of stringed instruments, fret placement is important in order to achieve proper intonation. Much has been done to improve intonation through a variety of methods. One such method is the “Rule of 18.” Under this rule, starting with the first fret from the nut, each fret is placed at 17/18 of the previous fret&#39;s distance to the bridge. However, practice has shown that this rule is flawed. 
   With the forgoing problems and concerns in mind, it is the general object of the present invention to provide a novel approach to fret placement, which overcomes the above-described drawbacks while improving intonation of a stringed instrument in the assembling process. 
   SUMMARY OF THE INVENTION 
   It is an object of the present invention to provide a stringed musical instrument having frets placed according to a method that recognizes a right triangle is formed, outlined by the axis of a fingerboard, a string, and the height of the string above the tangential point of string contact with the fret and perpendicular to a tangential point of string contact at the saddle. 
   It is another object of the present invention to provide a stringed musical instrument having frets placed according to a method that calculates the position of a fret on a fret board by measuring the required distance along the axis of the string, where the full string length will span from the point of contact on the saddle to the point of contact on the fret. 
   It is another object of the present invention to provide a stringed musical instrument having frets placed according to a method that involves multiplying the scale length by the twelfth root of 0.5 and multiplying each successive length by the twelfth root of 0.5 in order to provide the necessary string distances at which to place frets on a fret board for a twelve step octave. 
   These and other objectives of the present invention, and their preferred embodiments, shall become clear by consideration of the specification, claims and drawings taken as a whole. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a simplified schematic illustration of a guitar showing the fret placement of the present invention. 
       FIG. 2  is a side schematic view of an open string on the musical instrument of  FIG. 1 . 
       FIG. 3  is a side schematic view of a fretted string on the musical instrument of  FIG. 1 . 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
   The present invention is used for the manufacturing of fretted stringed instruments. More specifically, the present invention is a method for improving placement of frets on fretted musical instruments. 
   As shown in  FIG. 1 , the method of fret placement of the present invention is applied to a stringed musical instrument  2 . The stringed musical instrument  2  is of a conventional design having a tuning head  4 , neck  6 , and body  8 . The neck  6  is attached to the body  8 . At the distal end of the neck  6  opposing the body  8 , the tuning head  4  is attached. Strings  10  are attached to the tuning head  4  and stretched over a saddle  18  for connection to anchor pins  12 . The tuning head  4  is fitted with tuning keys  5 , which adjust the tension of the strings  10 . Adjusting the tension of the strings  10  affects the pitch of the instrument  2 . 
   Still referring to  FIG. 1 , a nut  20  is attached at the joint wherein the tuning head  4  meets the neck  6 . The neck  6  includes a fingerboard  16 . The nut  20  guides the strings  10  onto the fingerboard  16  to provide consistent lateral string placement. Frets  23  are placed along the major axis of the fingerboard  16 , according to the method of the present invention. 
   The tone of the stringed musical instrument  2  is produced by vibration of the strings  10  and modulated by the hollow body  8 . When the string  10  is depressed to the fingerboard  16 , two specific things happen. First, the vibrating length of the string  10  becomes shorter, which produces a higher pitch. Second, the string  10  forms a right triangle with an axis parallel to the fingerboard  16  and the altitude of the string  10 . Based on these concepts, the present invention addresses the concept of string vibration and the layout of the fingerboard  16  as a three-dimensional exercise designed to achieve improved intonation. The present method is neither a compensation nor a tempering of the strings. In fact, no one compensation can be successful in attaining perfect intonation since the mechanism involved is not linear. The method of the present invention will be described in more detail below. 
   It is therefore an important aspect of the present invention that the present method accounts for string vibration and fingerboard layout in fret placement. Previously, frets were placed based on a linear exercise in math based on the Rule of 18. In other words, fret placement was heretofore based on a fixed point along the axis of the fingerboard. By employing the dimensions of the right triangle formed when a string  10  is depressed, intonation of a fretted stringed musical instrument  2  can be improved by the present invention. As such, fret placement is calculated along the axis of the string  10  from a tangential point of string contact on the saddle  18  to a tangential point of contact on a fret  23 . 
   The traditional method places a fret  23  closer to a nut  20  of the instrument  2 . However, the present method places a fret  23  closer to the saddle  18 . As the fret  23  to be calculated approaches the saddle  18 , the angle created by the axis of the string  10  and the axis of the fingerboard  16  increases. Thus, the location of each fret  23  placed by the present method may differ greatly from the location of a fret placed by traditional methods. The reason for this is due to the difference not being linear but rather based on the string height above the fingerboard  16 . This string height is not accounted for by traditional methods. 
     FIG. 2  illustrates side schematic view of an open, or unfretted, string  10  on the musical instrument  2 , according to one embodiment of the present invention. The fingerboard  16  is also shown. The string  10  extends over the saddle  18  and the nut  20 . The saddle  18  and nut  20  are located on opposing sides of the fingerboard  16 . Although the present invention may be used to place any number of frets, only two frets are shown for illustrative purposes in  FIG. 2 , higher fret  22  and lower fret  24 . The higher fret  22  is located on the fingerboard  16  between the saddle  18  and the nut  20  but closer to the saddle  18 . The lower fret  24  is also located on the fingerboard  16  between the saddle  18  and the nut  20  but is closer to the nut  20 . 
   The preliminary step for calculating fret placement according to the present invention involves calculating the right triangle formed from the open string length. The length of a first side  11  of the right triangle is calculated by determining the height difference between a point  13  and a point  15 . Point  13  is a tangential point of contact between the string  10  and the saddle  18 . Point  15  is a tangential point of contact between the string  10  and the nut  20 . A point  17  represents one end point of side  11 . The hypotenuse  19  of the triangle is the open length of the string  10 , also known as the scale length. With side  11  and hypotenuse  19  known, the final side  21  can be calculated, which is also the effective scale length. 
     FIG. 3  illustrates a side schematic view of a fretted string  10  on the musical instrument  2 , according to one embodiment of the present invention. A finger force depressing the string  10  is represented by two arrows  14 . The fingerboard  16  is also shown in  FIG. 3 . The fretted string  10  extends over the saddle  18  and the nut  20 . 
   Fret placement can now be calculated based on the right triangle formed from the points described below. The first step is to determine a string length corresponding to a note on an open string. The string length is the length of open string  10  of  FIG. 2 . Second, the target string length for each fret based on a known ratio of the open note string length for a selected scale must be determined. This length is represented in  FIG. 3  as the length of fretted string  10  from point  28  to point  30 , or line  34 . 
   For the purpose of this step, the fret placement on an instrument  2  that employs a twelve-step octave will be used as an example. Starting with a scale length and multiplying that scale length by a constant that is less than one, the distance between two points of a shorter string, one step higher in pitch, can be determined. This is the equivalent of the placement of the first fret. If this new string length is multiplied by the constant again, the placement of the second fret can be calculated. This process can be continued to the twelfth fret, where the string length will be exactly one half the scale length. Based on this, the constant is determined to be the twelfth root of 0.5, which is a number less than one in excess of thirty decimal places. For the purposes of this description, the constant will be rounded off to 0.94387431268. 
   In contrast, the Rule of 18 uses a constant that is divided into the scale length. This results in the distance from the nut to the first fret and subsequently from one fret to the next. However, this method does not achieve proper intonation. 
   The present invention improves intonation on a fretted instrument by considering the length of the vibrating string. By multiplying the scale length by 0.94387431268, the length of string necessary to achieve the next higher step in tone for a twelve-tone-equal tempered scale can be determined. 
   Returning to the present method of fret placement and  FIG. 3 , the third step involves calculating a vertical distance between point  30  and point  32 . Point  30  is a tangential point of contact between the fretted string  10  and the saddle  18 . Point  32  is based on a horizontal axis  26  that spans from point  28  to the saddle  18 . The horizontal axis  26  is parallel to the fingerboard  16 . Point  28  is a tangential point of contact between the fretted string  10  and the higher fret  22 . The vertical distance is best calculated as the shortest distance between point  30  and horizontal axis  26 . Point  32  represents the point on the axis  26  where this shortest distance would be calculated. Thus, this distance is represented as line  36  on  FIG. 1 . 
   With line  34  and line  36  determined, the final step is determining the fret placement length on the fingerboard  16 . This length is represented as the distance between point  28  and point  32 , or the length of axis  26 . The fret placement length is calculated by finding the square root of the difference of the target string length squared and the vertical distance between point  30  and point  32  squared. In other words, the length of axis  26  ( z ) is the square root of the difference of line  34 ( x ) squared and line  36 ( y ) squared. The equation is represented as:
 
 z =√( x   2   −y   2 )  (1)
 
   Unlike traditional methods, it is an important aspect of the present invention that it accounts for differing frets. Not all frets have tangential points of contact in the same position. As the fret  22  approaches the saddle  18  of the instrument, the angle created by the string  10  and the horizontal axis  26  increases. In turn, the tangential point of contact of the string  10  with the fret  22  offsets slightly. The higher the string height is above the finger board  16 ; the greater the disparity between the traditional method of fret placement and the present method. In addition, as the fret  23  approaches the tail of the instrument  2 , the angle created by line  34  and the horizontal axis  26  increases. As a result, the difference between the two methods of fret placement also increases. 
   By multiplying each successive target string length by the twelfth root of 0.5, the necessary length of line  34  for each successive fret  23  can be calculated. Then, the length of line  36  can be determined based on the tangential point of contact between the fretted string  10  and the next successive fret  23 . Finally, the fret placement for the next successive fret  23  can be calculated according to Equation (1), i.e., from the square root of the difference of new line  34  squared and new line  36  squared. This method can be repeated for each fret  23  to determine the distance of each respective fret  23  from the saddle  18 . 
   The method can be applied to an actual stringed instrument. However, the present method may also be applied to a full-scale drawing of the stringed instrument for the calculations, and then, the frets placed on the actual instrument based on the measurements made on the drawing. 
   While the invention has been described with reference to the preferred embodiments, it will be understood by those skilled in the art that various obvious changes may be made, and equivalents may be substituted for elements thereof, without departing from the essential scope of the present invention. Therefore, it is intended that the invention not be limited to the particular embodiments disclosed, but that the invention includes all equivalent embodiments.