Curt Lang wrote:
I have been reading your very lucid essays on harmony and cycles and feel that I can follow part of what you say but I suffer from not being able to hear the notes and harmonies you describe and enumerate. Have you ever considered writing an essay on your observations and theories in the form of an audio file that would contain your voice -- narrating the ideas -- and an audible version of the notes and chords you describe? I don't know enough about digital audio and midi to suggest how, and I can't imagine how big an audience there might be, but I would find it interesting.
Ray Tomes wrote:
I did consider making a tape some years ago when I was working on the automatic just intonation idea. Unfortunately the programming effort was substantial to do it with computer wave sounds and it needs good equipment to do it with a keyboard. I have an old Kawai but it cannot vary its frequency on request. The people at Yamaha were interested in the idea but their new model with the necessary features was about $7000 at the time. They said they would maybe get me a second hand model for about $1500 but it never eventuated. I did some work on the Amiga that I had at that time but the quality of sound was not good enough to truly show the effects and programming in BASIC made it a bit slow.
If any of the people in the group with good yamaha MIDI systems (that can tune individual notes on the fly) are interested enough in this to do the work then I would be very happy to supply the details of the automatic just intonation idea and assist in the logic.
Actually I have a son who is a sound engineer but he keeps pretty busy doing jobs that he gets paid for (unlike his old man). He is interested in the idea so one day maybe....
I hope to put all the details together on my WWW pages so maybe some clever person will come along and put it in to practice.
Andrew Green wrote:
I must be crazy - because I know what I'm letting myself in for - but why not? No promises on time scales though, I'm already programming against the clock.
I have an AKAI S1000, and a MAUI, both of which allow pitch bend up to half a semitone. Judicious use of that parameter will allow us to change the pitch of each note "on the fly", and save the more laborious procedure of having to change the tuning of the instrument.
I also have software which can be used more or less directly to include your algorithm, and it already bases its decisions on the harmony etc. of the previous 4 bars - which must be about what you need.
The bad news (if it is bad) is that my database of music only includes music for the Renaissance lute.
Let me look at your algorithm, BASIC will do, and I'll see if there's any hope.
Ray Tomes wrote:
Following Andy's kind (or foolish) offer to have a go at producing music based on my AJI (automatic just intonation) invention I have got together my old documents (produced on an Amiga with a dot matrix printer), scanned them, OCRed them and then nearly retyped them to fix the errors. There are also some associated graphics, some which I have redone.
I will put a copy on my WWW pages.
Curt, you will have to share the blame with me if Andy goes mad :-}
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 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.
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 doIn 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 45What 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 haven't yet worked out whether the 11 and 13 ratios are consistent with the cases where I thought Mozart intended these ratios in his "Concerto in C Major".