source file: mills2.txt Date: Thu, 22 May 1997 07:59:37 +0200 Subject: Harmonics & Beats are better with JI From: rtomes@kcbbs.gen.nz (Ray Tomes) In the thread "Re: Equal divisions -- of what?" it was pointed out that there are equal linear steps in the JI scale, specifically... JI scale 24 27 30 32 36 40 45 48 Step 3 3 2 4 4 5 3 so there are several places where equal steps occur and more still when larger steps are used. Indeed the linear series 1 2 3 4 5 6 (7) 8 9 10 can be fully formed in the JI scale [(7) needs a special blues flat]. This is something from my WWW site which investigates the steps between notes which are also of course the beats... Harmonics & Beats are better with Just Intonation. One of the really nice things about the just intonation scale is that the beats between notes are in the scale, whereas for 12T equitempered tuning they miss by quite a lot. I am using "beats" here a bit loosely because sometimes these beats are in the range of frequencies of notes, not the wa-wa-wa sound we associate with beats. First the equitempered scale to show why it doesn't work in this way: note do re mi fa so la ti do Freq. 1.0000 1.1225 1.2599 1.3348 1.4983 1.6818 1.8877 2.0000 Beats 0.1225 0.1374 0.0749 0.1625 0.1835 0.2049 0.1123 Beats*8 0.980 1.099 1.198* 1.300 1.468 1.639 1.797* [* means Beats*16] The reason for *8 or *16 is to move the note up 3 or 4 octaves to see where it falls in the scale. In this case the beats are not generally in tune with the key. When beats are considered for notes that are 2 or more apart in the scale the situation is even worse. The same will apply to 19TET etc. Now the just intonation scale note do re mi fa so la ti do Freq. 1.0000 1.1250 1.2500 1.3333 1.5000 1.6667 1.8750 2.0000 Beats 0.1250 0.1250 0.0833 0.1667 0.1667 0.2083 0.1250 Beats*8 1.000 1.000 1.333* 1.333 1.333 1.667 1.000 Note do do fa fa fa la do In this case all the beats are perfectly in tune with the scale itself. When notes 2 or more apart are considered then the situation is still very good. This is more easily expressed by using the ratios of frequencies as shown below: note do re mi fa so la ti do re mi fa so la ti Ratio 24 27 30 32 36 40 45 48 54 60 64 72 80 90 Beats-1 3 3 2 4 4 5 3 6 6 4 8 8 10 Beats-2 6 5 6 8 9 8 9 12 10 12 16 18 Beats-3 8 9 10 13 12 14 15 16 18 20 26 Beats-4 12 13 15 16 18 20 19 24 26 30 Beats-5 16 18 18 22 24 24 27 32 36 Beats-6 21 21 24 28 28 32 35 42 Beats-7 24 27 30 32 36 40 45 What is shown is the relative beat frequency between notes, for example in the "Beats-3" line the 10 is the beats (or difference) between mi=30 and la=40. The figures are always up the diagonals to the top line. Now what is noticeable is that in most cases the beats are actually perfectly in the scale as follows: note do re mi fa so la ti do re mi fa so la ti Beats-1 do do fa fa fa la do ... Beats-2 do la do fa so fa so ... Beats-3 fa so la do# do mib mi ... Beats-4 do do# mi fa so la so# ... Beats-5 fa so so la# do do re ... Beats-6 tib tib do mib mib fa sob ... Where the ... means that the cycle repeats for the next octave. All the notes are perfectly in the key except the ones labelled: do# mib sob so# la# tib. These all fall in the places where there are black notes and the "b" (flat) ones are correct in my view as to what the flat frequencies should be. The "#" (sharp) ones are unusual ratios but then again there is no real meaning to these notes in the key. They do have the unusual ratios including 11, 13 and 19. Some of these are produced by the cases like fa-ti (careful how you pronounce that one :-) which is a dreadful chord anyway, and others by the cases like re-la which is our old friend D-A when in C, that is 27-40, which wants to be 2-3. I would suggest that it would be possible to make some tempered tunings which are not too far from equitempered tunings, but which have beats that are reasonably well in tune. Has anyone ever investigated this? It might be a fruitful exercise. -- Ray Tomes -- rtomes@kcbbs.gen.nz -- Harmonics Theory -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 22 May 1997 10:20 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07520; Thu, 22 May 1997 10:20:03 +0200 Date: Thu, 22 May 1997 10:20:03 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA07521 Received: (qmail 1223 invoked from network); 22 May 1997 08:19:54 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 22 May 1997 08:19:54 -0000 Message-Id: <199705220418_MC2-1712-DDD1@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu