Partial Slide-Ruleby Robert Morss, RPT Here’s a nifty little device for finding the coincident partials of any interval commonly used in tuning. It’s a simple paper slide rule where the fixed part of the rule (i.e. what might be called the “shelf”) represents the upper note of an interval, and the movable portion (the “slide”) is the lower note of the interval. Simply print out the file prule.txt and carefully cut out the two rectangular parts. Fold the larger piece to make the slide rule shelf, and place the other piece inside it. Here is what it should look like properly assembled and set for an octave.
We can see at a glance the coincident partials at 2:1, 4:2, 6:3, 8:4, and 10:5. If you have ever wondered what all the talk is about different kinds of octaves, this is what is meant, i.e. that beats can occur at each set of coincident partials. As a rule, if we tune a higher set of coincident partials beatless (i.e. 6:3) the octave will have more stretch in it than if we tune the 2:1 partial set beatless. Now we switch to a Major-3rd:
The first coincident partial occurs at 5:4 (there is another at 10:8, but it is inconsequential to the tuning process). Since we only have one relevant set of coincident partials, the Major-3rd is useful as a check interval. When we hear beats in a major 3rd, we can be fairly certain they are occurring at the 5:4 level. But with the octave, beats could coming from any of five different sets of partials (although the 8:4 and 10:5 are not normally a noticable source of beats in the temperament area). What is the hardest interval to tune? Based on the preceding comments about coincident partials, it must be (and indeed is) the unison, since beats can occur at each and every point in the partial series from 1:1 on up. This is particularly noticable if the strings of a unison do not have identical physical traits (i.e. speaking length and diameter). In these instances “clean” unisons are a challenge to achieve, if not impossible. I find this a handy tool for use during tuning classes, or indeed in any advanced discussion of tuning theory. It helps me distinguish the intervals from the partials (among other things). Return to Front Page
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