// title: The Collatz Scale // author: henklass // description: // Some time ago a did a thing on a series of numbers, that, as I learned later, was officially called the Collatz problem and it can be found on the Online Encyclopedia of Integer Sequences: http://oeis.org/A070165 . One day I was playing with those numbers on a keyboard and I had the impression, that some notes occur more frequent than others. This implies some kind of tonality. So I rewrote the code to use notenumbers in stead of frequencies. And here is the result. The fourth step, which coincidentally is a C, is dominant and the 10th, F# follows. There is quite some A, B flat, C#, E and G and very little E flat. A flat, B, D, and F are almost absent. So I guess the Collatz scale is: C, C#, E, F#, G, A, B flat, C and you can use E flat as some kind of Blue note or something. // code: //series of numbers: Collatz-scale: http://oeis.org/A070165 //from 1 to 100 //while number>1 //if even: divide by 2, else multiply by 3 and add 1 //remember highest value, remember longest series, both with their startnumber //startnumber->cutoff, number-> frequency, pan mid //startnumber->cutoff, highest value -> frequency, pan right //startnumber->cutoff, longest series -> cutoff, pan left //There seemd to be some tonality in the numbers when used as MIDI-notenumbers //To find that out you can count the occurrences of all the numbers wrapped between 0 en 11 Server.default.boot; ( SynthDef(\beep, { |freq=440, cutoff=1000, pan=0| Out.ar(0, (Pan2.ar(LPF.ar(VarSaw.ar(freq, width: 0.01), cutoff), pan))*0.4 )}).send(s); ) ( var highest=1, longest=1, starthighest=1, startlongest=1; var numberbeep, highestbeep, longestbeep; u=Array.new(12); //array to store the numbers, wrapped to 0..11 {for(0, 12, {u.add(0)})}.value; numberbeep=Synth(\beep, [\freq, 200, \cutoff, 200, \pan, 0]); longestbeep=Synth(\beep, [\freq, 200, \cutoff, 200, \pan, -1]); highestbeep=Synth(\beep, [\freq, 200, \cutoff, 200, \pan, 1]); Routine({ for (1, 100, { arg startnumber; var number, length=1, maximum; number=startnumber; maximum=startnumber; u[startnumber%12]=u[startnumber%12]+1; (number.asString+" ").post; numberbeep.set(\cutoff, 100*startnumber); while ( {number>1},{ if (number.asInteger.even, {number=(number/2)}, {number=3*number+1}); length=length+1; if (number>maximum, {maximum=number}); numberbeep.set(\freq, (32+(number%96)).midicps); //frequency as MIDI-notenumber between 32 and 128, that is 8 octaves longestbeep.set(\freq, (32+(length%96)).midicps); highestbeep.set(\freq, (32+(maximum%96)).midicps); u[number%12]=u[number%12]+1; //count the occurrences of this number wrapped between 0 and 11 0.1.wait; }); if(length>longest, {longest=length; startlongest=startnumber}); if (maximum>highest, {highest=maximum; starthighest=startnumber}); longestbeep.set(\cutoff, 100*longest); highestbeep.set(\cutoff, highest*2); ("length: " + length.asString).post; (" maximum: " + maximum.asString).postln; }); postf("longest series % at % \n", longest, startlongest); postf("highest value % at % \n", highest, starthighest); numberbeep.set(\freq, 33.midicps, \cutoff, 10000); longestbeep.set(\freq, ((32+longest)%96).midicps, \cutoff, 100*startlongest); highestbeep.set(\freq, ((32+highest)%128).midicps, \cutoff, 100*starthighest); 10.wait; numberbeep.free; highestbeep.free; longestbeep.free; }).play; ) ( /*This shows the Collatz-scale: The fourth step, which coincidentally is a C, is dominant and the 10th, F# follows. There is quite some A, B flat, C#, E and G and very little E flat. A flat, B, D, and F are almost absent. So I guess the Collatz scale is: C, C#, E, F#, G, A, B flat, C and you can use E flat as some kind of Blue note or something.*/ u.plot(bounds: Rect(10, 100, 400, 400));//show as graph for(0, 12, {arg i; //show as list if( (u[i] > 0), {(i.asString+" ").post; u[i].postln}) } ); ) numberbeep.set(\freq, (32+(0%96)).midicps); {VarSaw.ar(440, [0.1, 1], 0.05, 1, 0)}.scope