sigurdV Posted January 17, 2012 Author Report Share Posted January 17, 2012 As a prisoner of visual art i need only give name rank and... Eh it is with utmost reluctance i admit that... By the way did you notice my reflections on circular work in telepathy? What might the relation between our unconscious and the godspot be? Quote Link to comment Share on other sites More sharing options...
Don Blazys Posted January 17, 2012 Report Share Posted January 17, 2012 This is indeed a most fascinating topic. As a musician in the glorious 60's, I did extensive research on the mathematics of music. Back then, there were no"personal computers", "I phones" or even "calculators". We had to go an actual building called a "library", math was done using our very own brains andmusic was played by people who had talent. Sadly, none of that is true today. Today, all we have are "instant mathematicians"and "instant musicians" who would be utterly lost and helpless without their computers.Today, with the aid of a computer, anyone can take some "on line" courses and become a "musician". As a result, the world is now saturated with "auto tuned" phonies such as "Justin Beaver", "Lady Goo Goo" and "Snoop Doggy Poop". That said, the math involved in real music is a huge subject becausedifferent cultures have different forms of music and make use of different scales.The number of possible good sounding scales is actually quite unlimited. Don. Quote Link to comment Share on other sites More sharing options...
Turtle Posted January 17, 2012 Report Share Posted January 17, 2012 As a prisoner of visual art i need only give name rank and... Eh it is with utmost reluctance i admit that... By the way did you notice my reflections on circular work in telepathy? What might the relation between our unconscious and the godspot be? at liberty of the visual arts i need only give art. ahh it is with the utmost enthusiasm i hide that. i have noticed your ring by the way. say "telephony" what might...oh hell...i g0t nothin'. :lol: ok. i have f12 powers for you to chordify -or not- as the case may be. for crying out loud keep your arms inside the vehicle. Quote Link to comment Share on other sites More sharing options...
sigurdV Posted January 17, 2012 Author Report Share Posted January 17, 2012 at liberty of the visual arts i need only give art. ahh it is with the utmost enthusiasm i hide that. i have noticed your ring by the way. say "telephony" what might...oh hell...i g0t nothin'. :lol: ok. i have f12 powers for you to chordify -or not- as the case may be. for crying out loud keep your arms inside the vehicle. Im taking a ... look at it (just before going to work) oh my! ...doesnt it resemble a piano ...say falling into pieces... Did you notice there has been a musician/mathematician dropping in to say hello? Hi don Quote Link to comment Share on other sites More sharing options...
Turtle Posted January 17, 2012 Report Share Posted January 17, 2012 Im taking a ... look at it (just before going to work) oh my! ...doesnt it resemble a piano ...say falling into pieces... Did you notice there has been a musician/mathematician dropping in to say hello? Hi don yeah...i can see that if i squint. can you play it still? notice the power series n's jump around like the n's of the sequences you started with. f(7)=6 but f(12)=2. i did see don, yes. on the curmudgeonly side this week it seems. :rant: i dare say that i didn't make the piono fall apart in an instant nor did i use a computer. next i'll smash up a well-tempered clavier, waht? :hammer: Quote Link to comment Share on other sites More sharing options...
sigurdV Posted January 17, 2012 Author Report Share Posted January 17, 2012 I think its best to proceed at a slow tempo, or rather concentrating on what has been and is going on. 1 f12=2?2 temperisation... the pythagoreic comma ... as i said ... better taking it easy and check our foundation is solid! btw your knowledge of harmony theory is not non existent!!3 Wouldnt it be nice if Don showed us some of his thoughts on the foundations and the beginning steps of Harmony Theory=HT1 4 I guess that HT1 is all that is done in order to introduce the Circles of fourths and fifths, when they are arrived at the study of tonality begins for real.The concept of "key" is introduced. At the moment i will identify it with the note used to produce the scale.5 So when does a visiting Teacher of Harmony Theory shoes up giving us a lucid and thorough presentation of HT1? Whats done so far is no streamlined end product, though most pieces are here I think :) PS What I said inspires me to add: The circles of fourths and fifths really are the same circle! the difference is direction Fourths: BEADGCFA#D#G#C#F#Fifths : F#C#G#D#A#FCGDAEB Likewise A circle of thirds (the chord) is sixth backwards Whats left is the circle of seconds (the scale) whis is seventh backwards... Now then, why THREE circles? wondered sigurdV beginning this study. PS we old swedes have difficulties with "b", we use "h" instead so if you see a "h" when there should be a "b" you know what mistake has been done. 6 My intention was using 5 numbers in order to point towards pentatonics ...But Why Bother About Formating... Now I remember what I wanted to say: I opened the thread "Is there an unconscious" inside the Phsycology forum and the first entry is approximately finished, but to continue I really would feel safer if I had something to respond to... like the beggar who asked for food and when refused, asked for a kettle of water so he could make soup with a nail. Quote Link to comment Share on other sites More sharing options...
Turtle Posted January 17, 2012 Report Share Posted January 17, 2012 dude!! you screwed up my post by editing the post i was quoting!!! i say again, make new posts, don't edit back onto old ones. it is not only a misdirection to readers, it can muck up other's posts. so, i had my reply copied so let's go with that. I think its best to proceed at a slow tempo, or rather concentrating on what has been and is going on. 1 f12=2? the last graph i gave is akin to your f(12) as it has 12 elements in the sequence; notes? you see @ top of graph 1 2 3 4 5 6 7 8 9 A B C. in using base notation >10 mathematicians/programmers begin using capital letters: A=10, B=11. C=12, etc.. anyway, my powers graph has only 2 steps and neither is the starting sequence. quite odd. not fallen apart; never together. recall this was not the case with my f(7) powers graph...the one you were imprisoned by/for/with. 2 temperisation... the pythagoreic comma ... as i said ... better taking it easy and check our foundation is solid! btw your knowledge of harmony theory is not non existent!! i'm letting mr. jones' pages seep into my spongiform gray matter. :note: i'm slow, but i do poor work. :lol: 3 Wouldnt it be nice if Don showed us some of his thoughts on the foundations and the beginning steps of Harmony Theory=HT1 ahh perchance to dream... 4 I guess that HT1 is all that is done in order to introduce the Circles of fourths and fifths, when they are arrived at the study of tonality begins for real.The concept of "key" is introduced. At the moment i will identify it with the note used to produce the scale.5 So when does a visiting Teacher of Harmony Theory shoes up giving us a lucid and thorough presentation of HT1? Whats done so far is no streamlined end product, though most pieces are here I think :) i'm still trying to grok the diagram. 9/8 waht??? isn't there some graphs...i know there are, that i should find that show all the harmonic vibrations in a plucked string? i'm on it like a turtle up a latter. Quote Link to comment Share on other sites More sharing options...
sigurdV Posted January 17, 2012 Author Report Share Posted January 17, 2012 okIll trybut habits die hard!(but now I see a very good reason! thatll work)see 6 in my previous entry, pleeeeeeeeeeeease enter something so I have a reason to proceed :)Im curious as to where Don came from ...The music forum? Nah he is publishing in here. Quote Link to comment Share on other sites More sharing options...
sigurdV Posted January 17, 2012 Author Report Share Posted January 17, 2012 HT 1 (=Harmony theory for Matematicians,Part 1): Lesson 1:Searching for the function Q: True to my own advice im returning close to the origin (1=c)and claims there is a basical musical operation (taking thirds again and again, stop when next result is the beginning) defining a function f on x = a natural number, here follows the first twelve values: x = 1 2 3 4 5 6 7 8 9 10 11 12fx = 1 1 2 2 4 4 3 3 6 6 10 10 The function f is somewhat peculiar but is algorythmically defined. Exercise 1 Prove that fn = fn+1 if n is odd! Exercise 2 Define f as a non algoritmic function. Exercise 3 Define f as a continous function. (=Q) Exercise 4 Prove, or disprove that all values of the exponential function are values of Q. Good Luck! Wishes sigurdV Quote Link to comment Share on other sites More sharing options...
Turtle Posted January 18, 2012 Report Share Posted January 18, 2012 HT 1 (=Harmony theory for Matematicians,Part 1): Lesson 1:Searching for the function Q: True to my own advice im returning close to the origin (1=c)and claims there is a basical musical operation (taking thirds again and again, stop when next result is the beginning) defining a function f on x = a natural number, here follows the first twelve values: x = 1 2 3 4 5 6 7 8 9 10 11 12fx = 1 1 2 2 4 4 3 3 6 6 10 10 The function f is somewhat peculiar but is algorythmically defined. Exercise 1 Prove that fn = fn+1 if n is odd! Exercise 2 Define f as a non algoritmic function. Exercise 3 Define f as a continous function. (=Q) Exercise 4 Prove, or disprove that all values of the exponential function are values of Q. Good Luck! Wishes sigurdV :doh: all functions are algorithms. i don't think you're going to get the response(s) you want because you want a response in agreement with your pet idea(s). (0=1 0r is it 1=∞? non-standard though; oui/no) so, i think you're fn+1 business sounds like russel's principia mathematica. :shrug: i read 6 & it is just another leash. :dogwalk: don came from the non-figurate thread as best i recall. he's as up as you on being down on standards. :shrug: here's the leash i referenced: Vibrating String @ wiki Vibration, standing waves in a string, The fundamental and the first 6 overtones which form a harmonic series. Quote Link to comment Share on other sites More sharing options...
sigurdV Posted January 18, 2012 Author Report Share Posted January 18, 2012 A good visual aid! It helps me ordering my thoughts. All functions algorithms? My nomenclature is deficient then. Id like to find a mathematical formula that can be used for determining the values of f. In a similar way as, say , the values of the exponential function is found by: y=xx And...hmmm... You may be right about my pet ideas but i deny you can get them from what has been said in here. The concept im honestly studying in here is: TONALITY Info on it is appreciated. I claim that its first approximation is to notice that it is one of a triad achieved at by a basic operation on 1234567 provided the numbers are identified with the notes cdefgab. also: f1=1 since you take the given and stop (since there is nothing left to jump over), f0=0 since theres no number to take,nor any number to jump over.If this has any bearing on my pet ideas is too early to say:) so the first vibration gives 1=c, then 1/2=c, 1/3=g,1/4=cit gets difficult... i should produce overtones on a string to check what notes i hear but i cant be certain of what divisions the 12 frets stand for... Tuning down fourth string to c means fret 0 gives c fret twelve gives pressed note c overtone cfret seven gives g gfret five f cfret four e efret three d# gfret two d dfret one c# ? Damn! I cant produce or hear the overtone at fret one:(What I got = cdeg?I believe the qustion mark stands for "a" since that would give me the c-pentatonic scale! IF the missing notes f and b can be gotten as overtones they should be found between fret 0 and fret one, but i hear no overtones there at all...(hm.. no good scientist am I? Iforgot checking som frets :fret ten gives A#A# -actually the overtone is between frets, ON fret 10 is D.) Perhaps not so surprising: B and F is the border of the tonality C Major. (here i use the terms "key" and "tonality"as equivalent... but tonality is a deeper concept than key. Im not sure of what editing should be done of this entry, i let it wait. So ive done an empirical experiment...WOW... perhaps its relevant here to mention that a guitar ordinarily is tuned:eadgbe counted from the thickest string. It would be nice to communicate with a musician in here,so dear visitor why dont you get a guitar,learn how to tune it, and how to play it? If you dont have friends to help you out you can find i dont know how many amiable courses over the net. WHAT! You already did? You wonder how this so called composer teaches the guitar? I stand up to the challenge. Lets see what your fingers can do :) First a word of warning: You will only if necessary be taught something that can be learnt elsewhere. So (in the case the topic allows me to say the following): 1 Tune the d-string down a half step to c# 2 strum the guitar,again and again,try out rythms3 If you did the tuning and retuning correct you listen to a rather pleasing chord... it is called A Major nine :) Now you use the index finger to press down all strings at the same time at the first fret to produce the chord A# Major9. If this gives you problems then you truly are a beginner and should not worry, I assure you, just practise, and the A# will eventually sing out as nice as A did.4 You are now at least in principle able to produce all Major nine chords!fret 2 gives Bfret 3 gives Cfret 4 gives C#fret 5 gives Dfret 6 gives D#fret 7 gives Efret 8 gives Ffret 9 gives F#fret 10 gives Gfret 11 gives G#fret 12 returns you to A! The Major nine chord is (especially in blues and rock)an acceptable substitute for the Major chord :)If a real Major chord is needed just play the three thickest stringsGo ahead and practice, in #23 and in eventual entries not written yet, you will find some exercises intended for you. Quote Link to comment Share on other sites More sharing options...
sigurdV Posted January 19, 2012 Author Report Share Posted January 19, 2012 One raises a question (Hi!), another attempts an answer (Whats new?), gets a question in return (Nothing in particular... etc etc. Hi! Im Dr Fraud, and I think Id better introduce a lesser known fact about sigurd: At a very early age he decided his Chamber Pot was explaining things to him: Ma! Today it says "S"... I am afraid that observation produced various dire (Watch out! End in sight! :censored: ) consequenses, among them his mother feeding him on oil in order to suppress the ouija effect. On second thought (Get Thee Behind Me Dr) perhaps communication aint necessarily so simple as its Null Hypothesis (What do you think Mr Jones?) claims it to be. I remember that working as a teacher worked best when I answered a Freely given question with answers carrying hidden statements provoking another question... If the strategy is properly carried out the roles of questioner and answerer actually gets reversed :) Do you have an opinion in thiz matter? Exercise : Explain what Socrates meant by claiming he knew nothing? Quote Link to comment Share on other sites More sharing options...
sigurdV Posted January 19, 2012 Author Report Share Posted January 19, 2012 Hi there Don!We here at the instituto of nothing will be very very happy to take on your case. Butt im wanted at the uNiVeRcItY as a participator in some mysterious experiment going on in there, (Cant imagine why they insisted me to digest nothing but coal since yesterday? ...Must be some really pressing matter i suppose) Anyway that simply means i cant make more than a supperficial analyzis of your text today. 1 This is indeed a most fascinating topic. As a musician in the glorious 60's, I did extensive research on the mathematics of music. Back then, there were no"personal computers", "I phones" or even "calculators". We had to go an actual building called a "library", math was done using our very own brains andmusic was played by people who had talent. Sadly, none of that is true today. Today, all we have are "instant mathematicians"and "instant musicians" who would be utterly lost and helpless without their computers.Today, with the aid of a computer, anyone can take some "on line" courses and become a "musician". As a result, the world is now saturated with "auto tuned" phonies such as "Justin Beaver", "Lady Goo Goo" and "Snoop Doggy Poop". That said, the math involved in real music is a huge subject becausedifferent cultures have different forms of music and make use of different scales.The number of possible good sounding scales is actually quite unlimited. Don. 1 What do you think u mean by "This"? Dr Smullian himself takes an interest in this question (which normally is used to prolong the treatment, in order to enlarge our wallets). Quote Link to comment Share on other sites More sharing options...
CraigD Posted January 21, 2012 Report Share Posted January 21, 2012 ok. try this from 2 terms through 13 terms. f=number of steps to repeat inclusive. f=11 21 2 f=21 2 31 3 21 2 3 f=21 2 3 41 3 2 41 2 3 4 f=41 2 3 4 51 3 5 2 41 5 4 3 21 4 2 5 31 2 3 4 5...I can’t figure out how you’re generating these sequences, Turtle. I get the sequences including most of yours by counting from 0 by (1 to B) modulo b – that is, with generator fuction[math]a_{n+1} = ((a_n - 1 + i) \text{mod} m) + 1[/math]Here are values for m = 1 to 16: 1 1 2 1 1 1 2 3 1 3 2 1 1 1 1 2 3 4 1 3 1 3 1 4 3 2 1 1 1 1 1 2 3 4 5 1 3 5 2 4 1 4 2 5 3 1 5 4 3 2 1 1 1 1 1 1 2 3 4 5 6 1 3 5 1 3 5 1 4 1 4 1 4 1 5 3 1 5 3 1 6 5 4 3 2 1 1 1 1 1 1 1 2 3 4 5 6 7 1 3 5 7 2 4 6 1 4 7 3 6 2 5 1 5 2 6 3 7 4 1 6 4 2 7 5 3 1 7 6 5 4 3 2 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 1 3 5 7 1 3 5 7 1 4 7 2 5 8 3 6 1 5 1 5 1 5 1 5 1 6 3 8 5 2 7 4 1 7 5 3 1 7 5 3 1 8 7 6 5 4 3 2 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 1 3 5 7 9 2 4 6 8 1 4 7 1 4 7 1 4 7 1 5 9 4 8 3 7 2 6 1 6 2 7 3 8 4 9 5 1 7 4 1 7 4 1 7 4 1 8 6 4 2 9 7 5 3 1 9 8 7 6 5 4 3 2 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 1 3 5 7 9 1 3 5 7 9 1 4 7 10 3 6 9 2 5 8 1 5 9 3 7 1 5 9 3 7 1 6 1 6 1 6 1 6 1 6 1 7 3 9 5 1 7 3 9 5 1 8 5 2 9 6 3 10 7 4 1 9 7 5 3 1 9 7 5 3 1 10 9 8 7 6 5 4 3 2 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 1 3 5 7 9 11 2 4 6 8 10 1 4 7 10 2 5 8 11 3 6 9 1 5 9 2 6 10 3 7 11 4 8 1 6 11 5 10 4 9 3 8 2 7 1 7 2 8 3 9 4 10 5 11 6 1 8 4 11 7 3 10 6 2 9 5 1 9 6 3 11 8 5 2 10 7 4 1 10 8 6 4 2 11 9 7 5 3 1 11 10 9 8 7 6 5 4 3 2 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 1 3 5 7 9 11 1 3 5 7 9 11 1 4 7 10 1 4 7 10 1 4 7 10 1 5 9 1 5 9 1 5 9 1 5 9 1 6 11 4 9 2 7 12 5 10 3 8 1 7 1 7 1 7 1 7 1 7 1 7 1 8 3 10 5 12 7 2 9 4 11 6 1 9 5 1 9 5 1 9 5 1 9 5 1 10 7 4 1 10 7 4 1 10 7 4 1 11 9 7 5 3 1 11 9 7 5 3 1 12 11 10 9 8 7 6 5 4 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 1 3 5 7 9 11 13 2 4 6 8 10 12 1 4 7 10 13 3 6 9 12 2 5 8 11 1 5 9 13 4 8 12 3 7 11 2 6 10 1 6 11 3 8 13 5 10 2 7 12 4 9 1 7 13 6 12 5 11 4 10 3 9 2 8 1 8 2 9 3 10 4 11 5 12 6 13 7 1 9 4 12 7 2 10 5 13 8 3 11 6 1 10 6 2 11 7 3 12 8 4 13 9 5 1 11 8 5 2 12 9 6 3 13 10 7 4 1 12 10 8 6 4 2 13 11 9 7 5 3 1 13 12 11 10 9 8 7 6 5 4 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 3 5 7 9 11 13 1 3 5 7 9 11 13 1 4 7 10 13 2 5 8 11 14 3 6 9 12 1 5 9 13 3 7 11 1 5 9 13 3 7 11 1 6 11 2 7 12 3 8 13 4 9 14 5 10 1 7 13 5 11 3 9 1 7 13 5 11 3 9 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 9 3 11 5 13 7 1 9 3 11 5 13 7 1 10 5 14 9 4 13 8 3 12 7 2 11 6 1 11 7 3 13 9 5 1 11 7 3 13 9 5 1 12 9 6 3 14 11 8 5 2 13 10 7 4 1 13 11 9 7 5 3 1 13 11 9 7 5 3 1 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 3 5 7 9 11 13 15 2 4 6 8 10 12 14 1 4 7 10 13 1 4 7 10 13 1 4 7 10 13 1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 1 6 11 1 6 11 1 6 11 1 6 11 1 6 11 1 7 13 4 10 1 7 13 4 10 1 7 13 4 10 1 8 15 7 14 6 13 5 12 4 11 3 10 2 9 1 9 2 10 3 11 4 12 5 13 6 14 7 15 8 1 10 4 13 7 1 10 4 13 7 1 10 4 13 7 1 11 6 1 11 6 1 11 6 1 11 6 1 11 6 1 12 8 4 15 11 7 3 14 10 6 2 13 9 5 1 13 10 7 4 1 13 10 7 4 1 13 10 7 4 1 14 12 10 8 6 4 2 15 13 11 9 7 5 3 1 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 3 5 7 9 11 13 15 1 3 5 7 9 11 13 15 1 4 7 10 13 16 3 6 9 12 15 2 5 8 11 14 1 5 9 13 1 5 9 13 1 5 9 13 1 5 9 13 1 6 11 16 5 10 15 4 9 14 3 8 13 2 7 12 1 7 13 3 9 15 5 11 1 7 13 3 9 15 5 11 1 8 15 6 13 4 11 2 9 16 7 14 5 12 3 10 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 10 3 12 5 14 7 16 9 2 11 4 13 6 15 8 1 11 5 15 9 3 13 7 1 11 5 15 9 3 13 7 1 12 7 2 13 8 3 14 9 4 15 10 5 16 11 6 1 13 9 5 1 13 9 5 1 13 9 5 1 13 9 5 1 14 11 8 5 2 15 12 9 6 3 16 13 10 7 4 1 15 13 11 9 7 5 3 1 15 13 11 9 7 5 3 1 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1But mine matches your for some cases, such as m=11, but not others, such as m=6:Yours (f=4):1 2 3 4 5 61 3 5 2 4 61 5 4 3 2 61 4 2 5 3 61 2 3 4 5 6Mine (m=6):1 2 3 4 5 61 3 5 1 3 51 4 1 4 1 41 5 3 1 5 31 6 5 4 3 21 1 1 1 1 1 What’s you algorithm? Quote Link to comment Share on other sites More sharing options...
Turtle Posted January 21, 2012 Report Share Posted January 21, 2012 I can’t figure out how you’re generating these sequences, Turtle. ...But mine matches your for some cases, such as m=11, but not others, such as m=6:Yours (f=4):1 2 3 4 5 61 3 5 2 4 61 5 4 3 2 61 4 2 5 3 61 2 3 4 5 6Mine (m=6):1 2 3 4 5 61 3 5 1 3 51 4 1 4 1 41 5 3 1 5 31 6 5 4 3 21 1 1 1 1 1 What’s you algorithm? beginning with the sequence:1 2 3 4 5 6list the first number in the beginning sequence to get the first number in the new sequence:1skip the second number in the beginning sequence and list the third number of the beginning sequence as second in the new sequence:1 3skip the fourth number in the beginning sequence and list the fifth number of the beginning sequence as the third number in the new sequence:1 3 5skip the sixth number in the beginning sequence and return back to the first skipped number in the beginning sequence and list the first skipped number in the beginning sequence as the fourth number in the new sequence:1 3 5 2skip the third number in the beginning sequence and list the fourth number from the beginning sequence as the fifth number in the new sequence:1 3 5 2 4skip the fifth number in the beginning sequence and list the sixth number from the beginning sequence as the sixth number in the new sequence:1 3 5 2 4 6 the new sequence becomes the beginning sequence. rinse & repeat until the series returns to the original sequence. count the number of steps/sequences, not including the repeat. doing this i got: f(6)=41 2 3 4 5 61 3 5 2 4 61 5 4 3 2 61 4 2 5 3 61 2 3 4 5 6 i did this by hand/eye so i certainly may have made mistakes. i caught a few, but they became obvious rather quickly as they jossle the later patterns. Quote Link to comment Share on other sites More sharing options...
CraigD Posted January 21, 2012 Report Share Posted January 21, 2012 beginning with the sequence:1 2 3 4 5 6 ...Thanks. I seem of late to be some combination of stupid and lazy, or ... yeah, efficient, that’s it. Much more efficient having you tell me the algorithm rather than trying to puzzling it out myself. :) Now my trusty computer can crank these sequences out – here are the first 11 groups of them: 1 f(1)=1 1 2 f(2)=1 1 3 2 1 2 3 f(3)=2 1 3 2 4 1 2 3 4 f(4)=2 1 3 5 2 4 1 5 4 3 2 1 4 2 5 3 1 2 3 4 5 f(5)=4 1 3 5 2 4 6 1 5 4 3 2 6 1 4 2 5 3 6 1 2 3 4 5 6 f(6)=4 1 3 5 7 2 4 6 1 5 2 6 3 7 4 1 2 3 4 5 6 7 f(7)=3 1 3 5 7 2 4 6 8 1 5 2 6 3 7 4 8 1 2 3 4 5 6 7 8 f(8)=3 1 3 5 7 9 2 4 6 8 1 5 9 4 8 3 7 2 6 1 9 8 7 6 5 4 3 2 1 8 6 4 2 9 7 5 3 1 6 2 7 3 8 4 9 5 1 2 3 4 5 6 7 8 9 f(9)=6 1 3 5 7 9 2 4 6 8 10 1 5 9 4 8 3 7 2 6 10 1 9 8 7 6 5 4 3 2 10 1 8 6 4 2 9 7 5 3 10 1 6 2 7 3 8 4 9 5 10 1 2 3 4 5 6 7 8 9 10 f(10)=6 1 3 5 7 9 11 2 4 6 8 10 1 5 9 2 6 10 3 7 11 4 8 1 9 6 3 11 8 5 2 10 7 4 1 6 11 5 10 4 9 3 8 2 7 1 11 10 9 8 7 6 5 4 3 2 1 10 8 6 4 2 11 9 7 5 3 1 8 4 11 7 3 10 6 2 9 5 1 4 7 10 2 5 8 11 3 6 9 1 7 2 8 3 9 4 10 5 11 6 1 2 3 4 5 6 7 8 9 10 11 f(11)=10Here are the values of f() for the first 1000, ten to a row for viewing ease: n: f(n) 1: 1 1 2 2 4 4 3 3 6 6 11: 10 10 12 12 4 4 8 8 18 18 21: 6 6 11 11 20 20 18 18 28 28 31: 5 5 10 10 12 12 36 36 12 12 41: 20 20 14 14 12 12 23 23 21 21 51: 8 8 52 52 20 20 18 18 58 58 61: 60 60 6 6 12 12 66 66 22 22 71: 35 35 9 9 20 20 30 30 39 39 81: 54 54 82 82 8 8 28 28 11 11 91: 12 12 10 10 36 36 48 48 30 30 101: 100 100 51 51 12 12 106 106 36 36 111: 36 36 28 28 44 44 12 12 24 24 121: 110 110 20 20 100 100 7 7 14 14 131: 130 130 18 18 36 36 68 68 138 138 141: 46 46 60 60 28 28 42 42 148 148 151: 15 15 24 24 20 20 52 52 52 52 161: 33 33 162 162 20 20 83 83 156 156 171: 18 18 172 172 60 60 58 58 178 178 181: 180 180 60 60 36 36 40 40 18 18 191: 95 95 96 96 12 12 196 196 99 99 201: 66 66 84 84 20 20 66 66 90 90 211: 210 210 70 70 28 28 15 15 18 18 221: 24 24 37 37 60 60 226 226 76 76 231: 30 30 29 29 92 92 78 78 119 119 241: 24 24 162 162 84 84 36 36 82 82 251: 50 50 110 110 8 8 16 16 36 36 261: 84 84 131 131 52 52 22 22 268 268 271: 135 135 12 12 20 20 92 92 30 30 281: 70 70 94 94 36 36 60 60 136 136 291: 48 48 292 292 116 116 90 90 132 132 301: 42 42 100 100 60 60 102 102 102 102 311: 155 155 156 156 12 12 316 316 140 140 321: 106 106 72 72 60 60 36 36 69 69 331: 30 30 36 36 132 132 21 21 28 28 341: 10 10 147 147 44 44 346 346 348 348 351: 36 36 88 88 140 140 24 24 179 179 361: 342 342 110 110 36 36 183 183 60 60 371: 156 156 372 372 100 100 84 84 378 378 381: 14 14 191 191 60 60 42 42 388 388 391: 88 88 130 130 156 156 44 44 18 18 401: 200 200 60 60 108 108 180 180 204 204 411: 68 68 174 174 164 164 138 138 418 418 421: 420 420 138 138 40 40 60 60 60 60 431: 43 43 72 72 28 28 198 198 73 73 441: 42 42 442 442 44 44 148 148 224 224 451: 20 20 30 30 12 12 76 76 72 72 461: 460 460 231 231 20 20 466 466 66 66 471: 52 52 70 70 180 180 156 156 239 239 481: 36 36 66 66 48 48 243 243 162 162 491: 490 490 56 56 60 60 105 105 166 166 501: 166 166 251 251 100 100 156 156 508 508 511: 9 9 18 18 204 204 230 230 172 172 521: 260 260 522 522 60 60 40 40 253 253 531: 174 174 60 60 212 212 178 178 210 210 541: 540 540 180 180 36 36 546 546 60 60 551: 252 252 39 39 36 36 556 556 84 84 561: 40 40 562 562 28 28 54 54 284 284 571: 114 114 190 190 220 220 144 144 96 96 581: 246 246 260 260 12 12 586 586 90 90 591: 196 196 148 148 24 24 198 198 299 299 601: 25 25 66 66 220 220 303 303 84 84 611: 276 276 612 612 20 20 154 154 618 618 621: 198 198 33 33 500 500 90 90 72 72 631: 45 45 210 210 28 28 84 84 210 210 641: 64 64 214 214 28 28 323 323 290 290 651: 30 30 652 652 260 260 18 18 658 658 661: 660 660 24 24 36 36 308 308 74 74 671: 60 60 48 48 180 180 676 676 48 48 681: 226 226 22 22 68 68 76 76 156 156 691: 230 230 30 30 276 276 40 40 58 58 701: 700 700 36 36 92 92 300 300 708 708 711: 78 78 55 55 60 60 238 238 359 359 721: 51 51 24 24 140 140 121 121 486 486 731: 56 56 244 244 84 84 330 330 246 246 741: 36 36 371 371 148 148 246 246 318 318 751: 375 375 50 50 60 60 756 756 110 110 761: 380 380 36 36 24 24 348 348 384 384 771: 16 16 772 772 20 20 36 36 180 180 781: 70 70 252 252 52 52 786 786 262 262 791: 84 84 60 60 52 52 796 796 184 184 801: 66 66 90 90 132 132 268 268 404 404 811: 270 270 270 270 324 324 126 126 12 12 821: 820 820 411 411 20 20 826 826 828 828 831: 92 92 168 168 332 332 90 90 419 419 841: 812 812 70 70 156 156 330 330 94 94 851: 396 396 852 852 36 36 428 428 858 858 861: 60 60 431 431 172 172 136 136 390 390 871: 132 132 48 48 300 300 876 876 292 292 881: 55 55 882 882 116 116 443 443 21 21 891: 270 270 414 414 356 356 132 132 140 140 901: 104 104 42 42 180 180 906 906 300 300 911: 91 91 410 410 60 60 390 390 153 153 921: 102 102 420 420 180 180 102 102 464 464 931: 126 126 310 310 40 40 117 117 156 156 941: 940 940 220 220 36 36 946 946 36 36 951: 316 316 68 68 380 380 140 140 204 204 961: 155 155 318 318 96 96 483 483 72 72 971: 194 194 138 138 60 60 488 488 110 110 981: 36 36 491 491 196 196 138 138 154 154 991: 495 495 30 30 396 396 332 332 36 36I find interesting that, periodically, f(n)=n-1, for example, for the following values of n:2 3 5 11 13 19 29 37 53 59 61 67 83 101 107 131 139 149 163 173 179 181 197 211 227 269 293 317 347 349 373 379 389 419 421 443 461 467 491 509 523 541 547 557 563 587 613 619 653 659 661 677 701 709 757 773 787 797 821 827 829 853 859 877 883 907 941 947 1019For these 1st 69 cases of f(n)=n-1, n is always prime, but follows no pattern of skipping primes that I can intuit:1:2 2:3 3:5 4:7 5:11 6:13 7:17 8:19 9:23 10:29 11:31 12:37 13:41 14:43 15:47 16:53 17:59 18:61 19:67 20:71 21:73 22:79 23:83 24:89 25:97 26:101 27:103 28:107 29:109 30:113 31:127 32:131 33:137 34:139 35:149 36:151 37:157 38:163 39:167 40:173 41:179 42:181 43:191 44:193 45:197 46:199 47:211 48:223 49:227 50:229 51:233 52:239 53:241 54:251 55:257 56:263 57:269 58:271 59:277 60:281 61:283 62:293 63:307 64:311 65:313 66:317 67:331 68:337 69:347 70:349 71:353 72:359 73:367 74:373 75:379 76:383 77:389 78:397 79:401 80:409 81:419 82:421 83:431 84:433 85:439 86:443 87:449 88:457 89:461 90:463 91:467 92:479 93:487 94:491 95:499 96:503 97:509 98:521 99:523 100:541 101:547 102:557 103:563 104:569 105:571 106:577 107:587 108:593 109:599 110:601 111:607 112:613 113:617 114:619 115:631 116:641 117:643 118:647 119:653 120:659 121:661 122:673 123:677 124:683 125:691 126:701 127:709 128:719 129:727 130:733 131:739 132:743 133:751 134:757 135:761 136:769 137:773 138:787 139:797 140:809 141:811 142:821 143:823 144:827 145:829 146:839 147:853 148:857 149:859 150:863 151:877 152:881 153:883 154:887 155:907 156:911 157:919 158:929 159:937 160:941 161:947 162:953 163:967 164:971 165:977 166:983 167:991 168:997 169:1009 170:1013 171:1019I’m pretty badly off the topic of the mathematics of sound and music, but can’t resist a discrete math indulgence. ;) Quote Link to comment Share on other sites More sharing options...
Turtle Posted January 21, 2012 Report Share Posted January 21, 2012 Thanks. I seem of late to be some combination of stupid and lazy, or ... yeah, efficient, that’s it. Much more efficient having you tell me the algorithm rather than trying to puzzling it out myself. :) ahh the foe is on the other shoot. :lol: i had to ask too. Now my trusty computer can crank these sequences out – here are the first 11 groups of them: 1 f(1)=1 1 2 f(2)=1 1 3 2 1 2 3 f(3)=2 1 3 2 4 1 2 3 4 f(4)=2 1 3 5 2 4 1 5 4 3 2 1 4 2 5 3 1 2 3 4 5 f(5)=4 1 3 5 2 4 6 1 5 4 3 2 6 1 4 2 5 3 6 1 2 3 4 5 6 f(6)=4 1 3 5 7 2 4 6 1 5 2 6 3 7 4 1 2 3 4 5 6 7 f(7)=3 1 3 5 7 2 4 6 8 1 5 2 6 3 7 4 8 1 2 3 4 5 6 7 8 f(8)=3 1 3 5 7 9 2 4 6 8 1 5 9 4 8 3 7 2 6 1 9 8 7 6 5 4 3 2 1 8 6 4 2 9 7 5 3 1 6 2 7 3 8 4 9 5 1 2 3 4 5 6 7 8 9 ...Here are the values of f() for the first 1000, ten to a row for viewing ease: n: f(n) 1: 1 1 2 2 4 4 3 3 6 6 11: 10 10 12 12 4 4 8 8 18 18 21: 6 6 11 11 20 20 18 18 28 28 31: 5 5 10 10 12 12 36 36 12 12 41: 20 20 14 14 12 12 23 23 21 21 ...I find interesting that, periodically, f(n)=n-1, for example, for the following values of n:2 3 5 11 13 19 29 37 53 59 61 67 83 101 107 131 139 149 163 173 179 181 197 211 227 269 293 317 347 349 373 379 389 419 421 443 461 467 491 509 523 541 547 557 563 587 613 619 653 659 661 677 701 709 757 773 787 797 821 827 829 853 859 877 883 907 941 947 1019 at first i thought you had a repeating error in there, but now i see that i find it interesting that for all f(n) when n is odd, f(n)=f(n+1). For these 1st 69 cases of f(n)=n-1, n is always prime, but follows no pattern of skipping primes that I can intuit:1:2 2:3 3:5 4:7 5:11 6:13 7:17 8:19 9:23 10:29 11:31 12:37 13:41 14:43 15:47 16:53 17:59 18:61 19:67 20:71 21:73 22:79 23:83 24:89 25:97 ...I’m pretty badly off the topic of the mathematics of sound and music, but can’t resist a discrete math indulgence. ;) nothing jumps right out at me either. will go off and stare some more. as to music, sigurd originally set f(7) as an octave and equated the steps and the number of steps, f(7)=3, to the musical qualities "scale", "chord", and "tonality". then, i think, he was asking if other sequences/length-of-scales have matching or otherwise musically interesting steps and/or numbers of steps. i can't quite tell what the conclusion is/was. Quote Link to comment Share on other sites More sharing options...
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