【文章标题】：Can guitar frets perform multiplication? 吉他琴品能用来做乘法运算吗？
【文章正文】： Can Guitar Frets Perform Multiplication? 吉他琴品能用来做乘法运算吗？ September 4, 2026 2026年9月4日 Roscoe, N.Y. 纽约州罗斯科
I’m sure that some pictures are worth a thousand words, but others trigger a whole lot of puzzlement. 有些图片确实胜过千言万语，但另一些却会引发诸多困惑。
Such was the case with the cover of a book I recently bought entitled Calculating with Tones: The Logarithmic Logic of Music: 我最近购买的《音调计算：音乐的对数逻辑》一书封面就是如此：
This book was published by the Oughtred Society, an organization named in honor of William Oughtred, the Anglican clergyman and mathematician who is credited with inventing the first logarithmic slide rule around 1622. 该书由奥翠德学会出版，这个组织是为纪念英国圣公会牧师兼数学家威廉·奥翠德而命名的，他被认为是约1622年发明首个对数计算尺的人。
The Oughtred Society was founded in 1991, “dedicated to the preservation and history of slide rules and other calculating instruments.” 奥翠德学会成立于1991年，“致力于计算尺和其他计算工具的保存与历史研究”。
Their website contains a wealth of useful information on the subject. 他们的网站包含大量相关主题的有用信息。
Although the Oughtred Society website still has a page dedicated to this book, they don’t seem to be selling it at this time, and a second edition is not available directly through the society. 虽然奥翠德学会网站仍有该书专页，但目前似乎没有销售，也无法直接通过学会获取第二版。
This cover illustration intrigued me because logarithms are involved both in our perception of musical pitch and in the construction of a slide rule. 这个封面插图引起了我的兴趣，因为对数既存在于我们对音高的感知中，也存在于计算尺的结构中。
As I discuss in my web-book-in-progress The Lost Art of Logarithms, logarithms and the slide rule were originally invented to ease the tedious processes of multiplication and other mathematical tasks, but they continue to be vital for many reasons, including the understanding of our perceptions of the natural world. 正如我在正在编写的网络书籍《失传的对数艺术》中讨论的，对数和计算尺最初是为了简化繁琐的乘法等数学运算而发明的，但由于诸多原因（包括帮助我们理解对自然界的感知）它们至今仍至关重要。
But this cover illustration seems to imply a direct correspondence between the spacing of tick marks on a slide rule and the irregular spacing of frets on a guitar, and even suggests that guitar frets are spaced in such a way that they could be used to perform multiplications just like with a slide rule. 但这个封面插图似乎暗示计算尺刻度间距与吉他琴品的不规则间距存在直接对应关系，甚至表明吉他琴品的间距设计使其能像计算尺一样进行乘法运算。
Why else would the book be entitled Calculating with Tones? 否则这本书为何要命名为《音调计算》呢？
The illustration is reproduced on page 34 of the book with a caption that includes a description of both halves of the graphic: 该书第34页复制了这个插图，并配有说明文字描述图形的两个部分：
Guitar: Frets are positioned on the neck of the guitar to help locate the individual notes. The distance from one to the next becomes shorter as the frets get closer to the lower bridge. The reduction is logarithmic.
吉他：琴品位于吉他琴颈上用于定位音符。随着琴品靠近琴桥，间距逐渐缩短。这种缩减遵循对数规律。
Slide Rule: The scale on the slide rule corresponds to a logarithmic function. When using the slide rule, one carries out multiplication by adding distances. The progressive decrease in distance between marks on the slide rule corresponds to the same thing on the neck of the guitar.
计算尺：尺上的刻度对应对数函数。使用计算尺时，通过相加距离来完成乘法运算。计算尺上标记之间距离的渐进缩减与吉他琴颈上的情况相同。
Despite how compelling that illustration and description are, something bugged me. It seemed fundamentally wrong, and yet I couldn’t pinpoint the exact problem. 尽管插图和描述很有说服力，但某些地方让我困惑。这似乎从根本上就是错的，但我无法准确指出问题所在。
Multiplying with Guitar Frets 用吉他琴品做乘法
I knew what I had to do. First I had to get a guitar such as this one: 我知道该怎么做。首先需要准备这样一把吉他：
Guitar image from Wikipedia by Martin Möller, CC BY-SA 2.0 DE via Wikimedia Commons; modifications by me 吉他图片来自维基百科，Martin Möller拍摄，CC BY-SA 2.0 DE授权；经我修改
If your video display is not wide enough to show this entire guitar, you can use touch or the mouse to drag it horizontally into view. 如果屏幕宽度不足以显示整把吉他，可以用触摸或鼠标横向拖动查看。
If this guitar is truly capable of multiplying numbers, some numeric labels need to be added to it. 如果这把吉他真能用来计算乘法，就需要给它添加数字标签。
Contrary to the terminology on the cover of Calculating with Tones, the guitar string is suspended between the nut at the left of this illustration and the saddle at the right. 与《音调计算》封面术语不同，吉他弦实际是架在图片左侧的弦枕和右侧的琴桥之间。
Those mounts hold the strings in place and govern their effective lengths when they are strummed or plucked when the strings are not pressed against any frets. 这些固定装置决定了空弦弹奏时弦的有效振动长度。
Midway between the nut and saddle is the 12th fret, which effectively divides the string in half to play a pitch an octave higher than the string alone. 弦枕与琴桥中间是第12品，它将弦长一分为二，能弹出比空弦高八度的音高。
It’s a little hard to see, but in the cover illustration, the nut is lined up with the 1 on the all-important C and D scales of the slide rule, and the 12th fret is lined up with the 2. 虽然不太明显，但在封面插图中，弦枕与计算尺关键C/D刻度上的1对齐，第12品则与2对齐。
Let’s transfer these numbers to the guitar: 让我们把这些数字转移到吉他上：
Between these two points are 11 frets, which must obviously be labeled as fractions of 12 between 1 and 2: 这两点之间有11个琴品，显然需要标记为1到2之间的12等分分数：
Many of these fractions can be expressed in reduced forms, and that’s what I’ve done here: 许多分数可以约分，这就是我在此处所做的：
The next step is to saw the guitar in half: 下一步是把吉他锯成两半：
Sorry, but it has to be done. Make a nice clean cut from top to bottom. 抱歉，但必须这么做。从上到下整齐地切开。
Now the two pieces can be put back together but with the freedom to slide one part relative to the other: 现在可以把两部分重新组合，但可以自由滑动其中一部分：
Use touch or the mouse to drag the top half of the guitar to the right to multiply two numbers. (If the bottom half of the guitar can’t fit on your screen, you can continue to horizontally scroll it.) 用触摸或鼠标将上半部分向右拖动来进行乘法运算。（如果下半部分无法完整显示，可以继续横向滚动。）
For example, suppose you want to multiply 7/6 by 3/2. Move the top half of the guitar so that the 1 on the top is aligned with the 7/6 on the bottom: 例如要计算7/6乘以3/2，移动上半部分使顶部的1与底部的7/6对齐：
Now find 3/2 on the top half. Opposite that on the bottom is the product: 7/4, which is indeed 7/6 times 3/2. 然后在顶部找到3/2，其底部对应位置就是乘积：7/4，确实是7/6乘以3/2的结果。
Most of the other number combinations require some interpolation between the frets, and that’s not always easy, particularly because I’ve labeled the frets with fractions rather than decimals. 大多数其他数字组合需要在琴品间进行插值估算，这并不容易，尤其因为我用的是分数而非小数标记。
But a couple other combinations work well, such as 5/4 times 4/3 equaling 5/3, and 4/3 times 3/2 equaling 2. 但也有几个组合效果很好，比如5/4乘以4/3等于5/3，4/3乘以3/2等于2。