source file: mills3.txt Date: Wed, 17 Dec 1997 23:58:14 +0100 Subject: 19-tone Fret Lengths From: Gregg Gibson Bobby Lee said: > After reading in this group about 19TET (I'd never heard of it until a week > ago!), I've come to realize that it would be fairly easy to retune my > diatonic steel to it. I'd have to paint myself a 19 tone fretboard - that > sounds like fun, actually. Tuning the strings, pedals and knee levers would > be a snap with my Korg tuner. I'm afraid I'm very ignorant about the pedal steel. However, here are the 19-tone equal fret lengths for a guitar with a scale length of 25 inches. These frets give a stretched octave of 1202.7 cents, which is what I recommend for the best harmony. These figures do not include the necessary compensation (necessary on guitars at any rate) which would _decrease_ these values by something on the order of 0.07 inches (more for the bass strings, less for the treble). I give only the first 19 frets, since the little-used higher frets would perhaps be too close together to play easily, and are, as I understand, less used in any event. However, I'm not a guitar player, so someone else on the list who has actually built a guitar or other stringed instrument will probably be a better source of info. However, I can comment on fret placement and compensation, if you have questions. Fret number from nut Fret distance from nut in inches 1 0.90 2 1.76 3 2.60 4 3.40 5 (minor third) 4.18 6 (major third) 4.92 7 5.65 8 (fourth) 6.34 9 7.01 10 7.66 11 (fifth) 8.28 12 8.88 13 (minor sixth) 9.46 14 (major sixth) 10.02 15 10.55 16 11.07 17 11.57 18 12.05 19 (octave) 12.52 I apologize to those to whom what follows is elementary. Here is how to find fret lengths in any temperament you may desire: 1. Convert from the cents of the interval you're interested in, to the equivalent power of 10, that is to say to the logarithm (this is what a logarithm is, a power of 10; for example the log of 100 is 2, because we must raise 10 to the second power, i. e. 10 squared, to get 100) 2. Raise 10 to this power to get the decimal ratio of the cent interval 3. Divide total string length (or scale length, i.e. the length of string from nut to bridge) by this decimal ratio; this will give you the part of the string that will sound the desired interval, i.e. that will sound the note above the note of the open string. I wish I could give a diagram here... 4. Finally, to find the distance from the nut, subtract the distance from the bridge from the total length of the string from nut to bridge For example, to find the fret sounding the pitch 696.3 cents above the note sounded by an open string of length 25 inches: 1. 696.3/(1200/log(2)) (divide 1200 by the log of 2, then take the result and divide 1200 by that result) = .174672654984 2. 10^.174672654984 (raise 10 to the .174672654984 power) = 1.49510830872 3. 25 inches/1.49510830872 (divide the scale length 25 inches by the result) =16.7211966212 inches from the bridge 4. 25 inches - 16.7211966212 inches = 8.27880337882 inches from nut, which we can round to 8.28 inches, as in the above table These are not difficult calculations, and anyone with a pocket calculator can perform them. This does _not_ take into account compensation however, which typically requires that the scale length be lengthened by 1 to 5 millimeters (according to Fletcher & Rossing: Physics of Musical Instruments, p. 228). The bass strings particularly require to be lengthened, hence the angled bridge concept. Electric basses are said to require even more compensation. If this compensation is not made, fretted notes will be sharper than open ones. Perhaps I should do a post on compensation, and see if someone with more experience than I have (I'm not a guitar player or maker) has some more thoughts on the subject. But there are some makers, I am told, including even some professional guitar-makers, who do not properly compensate, so their instruments do not give terribly accurate results. Some of the compensation techniques are patented, I believe. The log of 2 refers to the definition of the cent, and is not to be changed merely because we have stretched the octave. SMTPOriginator: tuning@eartha.mills.edu From: Gregg Gibson Subject: Szende not Szasz PostedDate: 18-12-97 01:54:49 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 18-12-97 01:52:44-18-12-97 01:52:44,18-12-97 01:52:21-18-12-97 01:52:22 DeliveredDate: 18-12-97 01:52:22 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C1256571.0004D1BA; Thu, 18 Dec 1997 01:54:34 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA18266; Thu, 18 Dec 1997 01:54:49 +0100 Date: Thu, 18 Dec 1997 01:54:49 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA18545 Received: (qmail 812 invoked from network); 17 Dec 1997 16:54:46 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 17 Dec 1997 16:54:46 -0800 Message-Id: <3498D678.3D51@ww-interlink.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu