source file: mills2.txt Date: Thu, 9 Nov 1995 19:44:05 -0800 Subject: Re: guitars, slides & harmonics - simple help wanted From: Gary Morrison <71670.2576@compuserve.com> The situation you described when you play both sides of a string while moving the partition (slide) between the two strings, is what is called a "monochord", obviously meaning "one string", but clearly implying split into two vibrating sections. As for the relationship of the pitches in this situation, here's the scoop: As we know from how frets work, ONE of the two vibrating sections of the string ALONE while moving the slide can illustrate simple harmonic relationships. If you don't play both sides of the string, you have a definite nut and bridge, the sound you're considering vibrating between the bridge and the slide (that would define the bridge). As you perhaps know, the frequency of a vibrating string is inversely proportional to its length. So if you place the slide half-way up the string, you will have a string half the length, and thus twice the frequency of the pitch of the open string (i.e., no slide on the string at all). If you were to place the slide 1/3 of the entire string's length from the bridge, the vibrating length of the string would be 1/3, so the frequency would be 3 times that of the string if you remove the slide from the string completely. Similarly for 1/4 of the open string length, and so forth. But what if you DO sound the other side as well. Let's place the slide in those same places and see what happens to the other part of the string. Well, if you place the slide in the middle, you have 1/2 on one side, and 1/2 on the other, so the pitches are the same, as you correctly pointed out. If you place the slide 1/3 of the way to one side as before, one part of the string will be at 3-times the open-string frequency. What about the other side? Well, it has 2/3 of the open-string length, so it will sound at 3/2 of the open-string frequency. So, if you place the slide 1/3 of the open string length from one end, you will have pitches at 3 times and 3/2 times the frequency of the string if the slide were removed completely. Clearly, 3 is two times 3/2, so the two frequencies are octaves apart. Similarly, if you place the slide at 1/4 of the open string length from one end, you will have string lengths of 1/4 and 3/4. Those will then sound at frequencies of 4 and 4/3 times the frequency of the open string. Clearly 4 is 3 times 4/3, so the two frequencies will be a factor of three apart. But you asked what are the relationships between all of these frequencies? Well, let's look at those three cases: In terms of lengths: In terms of frequency: In terms of Pitch: 1/1 1/1 P1 |-----------------------| |-----------------------| |-----------------------| 1/2 1/2 2/1 2/1 P8 P8 |-----------0-----------| |-----------0-----------| |-----------0-----------| 1/3 2/3 3/1 3/2 P12 P5 |-------0---------------| |-------0---------------| |-------0---------------| 1/4 3/4 4/1 4/3 dbl 8va P4 |-----0-----------------| |-----0-----------------| |-----0-----------------| (Note: P1 = perfect uni. P8 = perf. octave P12 = perf. 12th P5 = perf. fifth (etc.) ) The frequencies of the shorter parts of the strings are harmonics (1 times, 2 times, 3 times, and 4 times) of the frequency of the open string. The others are somewhat more complexly related to the each other and to the open string. If I recall correctly, they are called "superparticulars". But here's what's neat about the frequencies sounded by the larger parts of the string when you make the play harmonics on the smaller parts: The are as far above the common open-string root pitch as the current harmonic is above the previous harmonic. So first time around, on the shorter half (left) of the string, we went from 1/1 times whatever the frequency of the open string may be, to 2/1 times that frequency, a jump of an octave, which is as far above the open-string root pitch as the harmonic side jumped up. The next time around, the pitch of the harmonic side jumped from an octave above the root pitch to a twelfth above the root pitch, an increase of a perfect fifth, and by golly, the longer end is a perfect fifth above that common root pitch. Similarly, the third time, the pitch of the harmonic side went from a twelfth above the root pitch to a double-octave above that root pitch, a pitch change of a fourth. And by golly, the pitch of the longer side is a perfect forth above the root pitch. This relationship occurs only if you stick with the "sweet spots" - the shorter side sticking with 1/2, 1/3, 1/4, 1/5, and so forth of the open string length. As for the other topic - other fingering patterns to pick out on your jumbush's fretless fingerboard - you might want to try subharmonic pitch relationships. You can get them by placing imaginary frets at exactly equal distances apart all the way up the neck. That rather than the usual ever-decreasing distance as you move your way up the neck toward the bridge. These subharmonic frequencies are 1/2, 1/3, 1/4, 1/5, 1/6, and so forth of a very high fret position. But there's a very important constraint here: You must split the open string length into a whole-number of equal-sized pieces! As I vaguely recall from Ivor Darreg's jumbush, the open string length is about 13 inches long. You can therefore split that up into 13 uniform 1-inch long imaginary fret positions. But if your open string length is 13 and a half inches long, DON'T use a 1-inch sized imaginary fret spacing!, leaving a 1/2-inch piece at the end (especially not at the bridge end!). Lengthen this uniform distance between your imaginary frets slightly so that you've segmented that 13.5-inch distance into an even 13 pieces, or shorten it slightly into an even 14 pieces. Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 10 Nov 1995 18:52 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id IAA06750; Fri, 10 Nov 1995 08:52:32 -0800 Date: Fri, 10 Nov 1995 08:52:32 -0800 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu