一位研究者通过对比分析吉他品柱与计算尺的数学结构，探讨吉他品柱是否能像计算尺一样执行乘法运算1。研究发现，虽然初期分析似乎表明两者存在对应关系，但深入验证后发现吉他品柱的间距函数与对数函数仅在1和2两点上重合，无法真正充当乘法工具1。
吉他品柱的位置遵循特定的几何规律1。弦长与振动频率成反比，每一品必须将弦长缩短至前一品的约0.944倍以产生半音间隔，而第12品将弦长减半以产生高八度音1。相比之下，计算尺上的刻度基于十进制对数，每个数的位置等于总长乘以该数的常用对数1。
研究人员将吉他品柱映射到1到2的范围后进行了量化对比1。吉他品柱位置的函数曲线（红线）与对数函数曲线（蓝线）仅在n=1和n=2处相同，其他位置存在显著偏差1。虽然在单品柱范围内可以进行如7/6×3/2=7/4这样的乘法运算，但当扩展到双八度（24品）范围后就无法准确进行1。
此外，研究还介绍了文艺复兴时期的历史方法1。音乐家维琴佐·伽利莱建议使用18:17的比例法则来确定品柱位置，这一比例在文艺复兴时期被广泛使用达两个半世纪1。
An analysis comparing the mathematical structure of guitar frets with logarithmic scales reveals that while the two systems appear similar at first glance, they diverge significantly and cannot reliably perform multiplication.1 Guitar fret positioning follows the formula where the distance from the headstock equals the string length multiplied by (1 - 2^(-fret number/12)), a relationship derived from the need to shorten the string by a factor of 2^(-1/12)—approximately 0.944—at each successive fret to produce semitone intervals.1 The 12th fret reduces string length by half and produces a note one octave higher with double the frequency.1
Slide rules, the traditional analog computing devices, function through decimal logarithmic spacing where each number's position equals the total length multiplied by log₁₀ of that number.1 When guitar frets are mapped to a normalized 1-to-2 range, the fret spacing function (1 - (1/n)^(1/12)) aligns with the logarithmic function (log₂(n)) only at the endpoints n=1 and n=2, diverging significantly elsewhere.1 A multiplication example—7/6 × 3/2 = 7/4—holds true within a single octave, but the calculation breaks down when extended across two octaves (24 frets), demonstrating the frets' inability to function as a reliable multiplication tool.1 Renaissance-era musician Vincenzo Galilei had previously proposed using an 18:17 ratio as a geometric method for determining fret positions, a proportional system that remained widely adopted for two and a half centuries.1
评论
还没有评论，欢迎留下第一条。