Stumbled upon this eggheadery you see below, thought it might make for an interesting discussion. Information Paper, May 23, 2005 SUJECT: The Optimal Number of Frets for an Electric Guitar with a 24.75 Scale Problem Statement: We are building a 24.75 scale guitar, and wanted to choose the optimal number of frets to install on the guitar. The guitar will feature a korina body and neck, a flamed maple top, pao ferro fingerboard, and one humbucking pickup in the bridge position. 1. First, we started with the physical construction of three of our favorite guitars. We examined three examples and took detailed notes on the construction and the tone of each. a. A 1963 Fender Stratocaster from a private collection. Guitar was completely stock and still had all hang tags and the sales slip in the case. Features a 25.5 scale, with 21 frets. Finish: Sonic Blue. b. A 1988 Fender HM Strat. Guitar was a new-old stock (NOS) find by a local music store. We believe this is the best sounding and playing 24-fret guitar ever made by a major manufacturer. This HM has one stock Dimarzio humbucker in the bridge position. Features a 25.5 scale with 24 frets. Finish: Aqua. c. A 1991 Gibson Les Paul Classic. This is a genuine, stock pre-historic Classic, and is an extremely rare example because of the 1991 serial number and finish. This combination did not appear in Gibson literature until 1992. Features a 24.75 scale with 22 frets. Finish: Amber. d. All guitars feature rosewood fingerboards, and tone comparisons were made with a Bogner Überschall head with one matching 4x12 cabinet, a 1980s Marshall JCM-800 and matching 80s straight-front cabinet, and a 1997 Matchless Chieftain 2x12 combo. We used a George Ls cable, Ernie Ball Pink Slinky strings, and a Dunlop 1.00 mm Tortex pick for all testing. Assessment: Our assessment included playing individual notes on each string and a series of three chords: A in the second position, G in the third position, and an E in the first position. We played each guitar through each of the individual amps, using both clean and distorted tones on the Bogner and Marshall amps, and clean and natural distortion on the Matchless. We recorded each of the tones into our ProTools recording setup on an Apple G5 PowerMac and a MOTU Firewire Interface. We used a new Shure SM-57 even though we had a locker full of vintage mics available. We believed the Shure would provide a more realistic and familiar tone for our test. Conclusion: The vintage Strat had particularly strong tone in the upper register, and between the 17th and 21st frets, the clarity of the strings was astounding. The HM Strat sounded strained in the extreme upper register, and its 23rd and 24th frets were particularly unmusical. The Les Paul sounded spongier in the upper register, but the power chords were particularly clear and powerful. 2 Based on our sound test, we concluded that we needed to investigate a guitar with a Gibson 24.75 scale with and extended fretboard. 2. Second, we determined the exact placement of the frets for the 24.75 scale using our above examples. For illustration, we started at the 12th fret and worked toward the bridge. The purpose of this analysis was to determine the physical length of the string, regardless of scale, at various frets. Distances are measured from the nut to the top of the fret. Fret Vintage Strat HM Strat Les Paul Reference, Gibson Scale 12 12.75 12.75 12.375 12.375 13 13.466 13.466 13.07 13.07 14 14.141 14.141 13.725 13.725 15 14.779 14.779 14.344 14.344 16 15.38 15.38 14.928 14.928 17 15.948 15.948 15.479 15.479 18 16.484 16.484 16 16 19 16.99 16.99 16.491 16.491 20 17.468 17.468 16.954 16.954 21 17.919 17.919 17.392 17.392 22 18.344 17.805 17.805 23 18.746 18.195 24 19.125 18.563 Table 1. Distance of frets from the nut Based on our sound test, we came to an initial conclusion that a Les Paul could benefit from an additional fret that roughly corresponds, string length-wise, to the position of the 21st fret on the vintage Stratocaster, as well as the 22nd fret on the HM Strat. Initial analysis points to a 23rd fret, but not a 24th fret. The 24th fret on a Les Paul would roughly correspond to the 23rd and 24th fret on the HM Strat, and we already discounted them as unmusical. 3. Finally, we used custom software to complete an analysis on the frequency response of each of the guitars in an effort to determine the optimal pickup placement, as well as the frequency response of each of our test guitars, as well as our future guitar. The frequency for each guitar string is shown below: Low E 82.41 Hz A 110.0 D 146.8 G 196.0 B 246.9 High E 329.6 3 Description Figures 1-5 on the following pages show the frequency response of our guitars. On top is a diagram showing a vibrating guitar string, a fingerboard with inlays, a nut on the left, a bridge on the right, and pickups. Two rulers above the string display the distance from the bridge in inches and the fret number. In the center is the resulting frequency response plot. The horizontal axis is a 20 Hz to 20,000 Hz log frequency scale and the vertical axis is a +10dB to -40dB amplitude scale. The horizontal red bar shows the pitch range of the fretted notes along the neck from the open string to highest fret. The gold vertical line marks the frequency of the currently fretted note. The actual position of the pickups, in inches from the bridge, is displayed below the pickup. The gold vertical line displays frequency of the current note on the response plot. For the test, all guitars were tested with the A string. The table at the bottom has a set of controls for each pickup; the pickup position (in inches from the bridge), the pickup magnetic aperture width (in inches), the pickup level (in dB) both as a value and as a slider control, the polarity of the pickup, and a button to remove that pickup. Things to look for The low-end response disappears as the pickup gets closer to the bridge. For a single pickup the response is the product of two "comb filters responses". A comb filter response contains a series of notches a constant number of Hz apart and looks like a comb. One of the combs is due to the position of the pickup, the distance the pickup is from the bridge while the other is due to the width of the pickup. Both will scale with the pitch of the open string. Symmetry A few players have pointed out that the frequency response curves for a single pickup ought to be identical when the pickup is placed at symmetrical locations about the center of the string. For example, the curve for a pickup 4.0 inches from the bridge ought to be the same as the curve for a pickup 4.0 inches from the nut. Yet the curves are very different. The frequency response plots are built by shortening the vibrating string length towards the bridge, effectively fretting notes up the neck. This corresponds to the formant filtering model (start with a flat response and filter it) that we are accustomed to hearing in acoustic instruments. But fretting the string immediately loses the symmetry of the pickup locations. The curves of symmetrically placed pickups are the same, however, at the specific frequencies of the open string and its harmonics. The most extreme case of this can be seen by comparing the response of a pickup placed right on the bridge to a pickup placed right on the nut. And sure enough, the frequency response of a pickup placed on the nut has nulls at all harmonics of the open string. 4 4. Frequency Response Plots. These represent the frequency response of the high E string fretted at the last fret closest to the bridge. The bridge pickup was used on all guitars. Figure 1. Frequency plot for vintage Stratocaster Note: Note the frequency at the 21st fret (under the right side of the red bar). Figure 2. Frequency plot for HM Stratocaster Note: Note the difference in harmonics at about the 5 khz range between the vintage Strat and the HM Strat. This is probably the reason for the strained response in our audio tests. 5 Figure 3. Frequency plot for Les Paul Note: Note the harmonics of the Les Paul in the 5-20 kHz range. The response is very similar to the HM Strat. Figure 4. Les Paul Scale with 23 Frets Note: The strong harmonics remain in the 5-10 khz range. Note the harmonics are extremely similar to the 22-fret Les Paul. 6 Figure 5. Les Paul Scale with 24 Frets Note: While the harmonics remain, they are more compressed than with the 23 fret guitar, or the 22-fret Les Paul. Comparison of Plots When reviewing these plots, it is important to remember that the differences in humbucking pickups and single coil pickups will manifest themselves in the sound of the individual guitars. Despite a lot of folklore, we believe the pickups will not generally affect the frequency response of the guitars strings. We reviewed each of the plots to determine which construction features would most accurately reflect the results of our subjective sound tests. Figure 6 on the next page graphically shows what we found. Based on the frequency response of the guitars and the physical length of the strings, we believe a guitar with a 24.75 scale and a 23 fret neck will most resemble the response of the vintage Stratocaster. Refer to the red line on the plots. See how closely matched the frequency response of the vintage Strat (upper plot) aligns with the plot for the 23-fret guitar (lower plot). This validates our analysis in Table 1. 7 Figure 6. Vintage Strat (upper) and 23-fret guitar (lower) plots Note: As stated above, refer to the red line on the above plots. See how closely matched the frequency response of the vintage Strat (upper plot) aligns with the plot for the 23-fret guitar (lower plot). 8 Conclusion Based on our subjective sound tests, as well as our objective scientific tests, we believe the optimal number of frets for a guitar with a neck scale length of 24.75 is 23. Additional Research To validate the research, our next step is to build our subject guitar with a 23-fret neck and compare it, both subjectively and objectively, to our test guitars. DISCLAIMER Disclaimer of Endorsement: Reference herein to any specific commercial products, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the author. The views and opinions of author expressed herein do not necessarily state or reflect those of the sponsor, and shall not be used for advertising or product endorsement purposes. 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