Logaritmic scale

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Permalink Reply by David Stuetzel on March 3, 2011 at 2:55pm

Hallo Sauli,

I should be able to answer that kind of question, it is what i teach music students...
the only problem is that my english is not the best, i am from germany, and all the books and links i know are german, perhaps some are translated to english as well, but it is actually easyer for me to just trie to answer your questiones..
the other problem is, what exactely are your questiones, whom do you want to present them to, what do you know, understand allready? if you could just come over it would be so easy, but it is really hard to expain this type of thing, if i cannt do it face to face... because it is hard to know where to start, how far to go, or how deep...
for example: most people dont know what logaritmic is, most of those who know, still dont really understand. so i know, with my music students, i cannot use the term to make them understand something, onely to frighten most of them, so i have to find a way arround it, and i can rather say in the end: this is what mathematicians call logarithmic.. i can use overtones to make them understand "logaritmic", but not the other way round...

so, if you would like me to answer questiones, you would allwayes have to answer back, as soon as there is another question coming up, or if something is not clear to you...
perhaps scype would be a good thing too...

so. i put one first thing here, does it help? what do you think of it?

the numbers: 1 2 3 4 5 6 7 8 9 10 ....
can look like equal numbers, only getting one mor to the right, and one less to the left, but besides that, they are all the "same", there is no other difference betwen them....
so what about odd numbers? primes? as long as we onely add and substract, there will be nothing special, no difference betwen odd and even, or prime numbers!
once we start deviding and multiplicating everything changes: there is a great difference betwen odd and even, and prim numbers stand out as something verry special, as the "elements" of numbers, out of which we get all the others by multiplication... (this goes to numbertheorie, if followed..).
for multiplication this will be an equel row:

1 2 4 8 16 23 as it is allwayes the same multiplication from one number to the next: allwayes two times as much... the same "verhältniss" (dont know the english word)

1 2 3 4 5 6 7 8 9 is only "eaquel" for adition and substraction, but not for multiplication and division!

1x 2x 3x 4x 5x 6x the frequency of the fundamental, is the overtoneseries (there are questiones hidden here, as most people dont really understand why)

our ear hears the same intervall if there is the same division or multiplication betwenn two frequencies, not if there is the same difference!

so this is eaquel in difference in herz

100hz 200hz 300hz 400hz 500hz, but not in intervalls!

1 2 4 8 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 4 8 is equel distance to our ear equel intervalls, because it is the same divisione or multiplication
1 2 3 4 is equel in difference of frequencies

to spread the overtoneseries to make the distance betwen 1 2 4 8.. similar shows how our ear "sees" it. that is how it gets to smaller intervalls if you go higher up...

1 2 4 8 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
this is the other way round, equal frequenc difference means equal distance, but our earhears 1 2 4 8 as equal, as octaves, because it is the same relation (ah, that is the word: relation = "Verhältniss")

so it comes down to: we hear same relations as same intervalls, actually intervalls are relations! that is something many people know, but most dont really understand, it is the "key"...
same relation is not the same as same distance, same difference....

does that help anything at all? is it just what you anyway knew? does it arouse 100 new questiones? too complicated? too simple?... i cannt know, as i dont know you well enaugh...

one other thing: download the overtone analyzers free version, change it to display logaritmic vs equel and look at what it does to the distance of overtones, and to the keys to the left...

this is only about the logarythm part of your question..
i would like to answer to the other, but have to do something else now..
it is a good training for my english...
but your other question has even mor different ways it can be put and answered, hard to decide where to start..

David

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Permalink Reply by David Stuetzel on March 3, 2011 at 3:06pm

unfortunately the site changed the spaces betwen numbers, and that changes the meaning...
i still leav it there uncorrected, as i dont have time now, but i will trie to put it right another time.
some of it cannt be undertood like this unfortunately...

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Permalink Reply by David Stuetzel on March 3, 2011 at 3:08pm

if you give me an email adress i can send what i wrote as a pdf, and it will show things as i ment them to be...
mine is dstutzel "at" yahoo.de

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Permalink Reply by Skye Løfvander on August 18, 2011 at 5:06pm

... as a supplement to Davids fine explaination: The illustration shows an arrangement of the overtones in a logarithmic spiral (in this case it is Archimedan, but that is also logarithmic). All even numbers are octaves of a more fundamental tone:
1-2-4-8-16-... Prime and octaves of prime
3-6-12-24-... Perfect fifth and octaves of p.f.
5-10-20-... Just major third and octaves thereof.

One spiral winding, 360 degrees, corresponds with one octave (=1200 cent). So you may logaritmically translate any cent value to an angle. Example: The axis of 5-10-20 comes from the interval 4:5.
So the cent value is log(5/4) x 1200 : log2 =386.3 cent. (whereas the value of the 12 TET major third is 400 cent ~60 degrees)
To translate 386.3 to circle degrees we only need to multiply with 0.3 (360:1200). 386.3 x 0.3 = 115.89 degrees.
This provides you a very direct way to see the deviances between for instance 12 TET tunings and harmonics.

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Permalink Reply by Aindrias Hirt on December 5, 2019 at 3:34pm

Well, it's been 8+ years since you posted this. I thought that I'd answer it anyway.

It isn't the logarithmic scale, but the natural scale/harmonic or overtone series. If you measure it, it isn't logarithmic, it's linear. Really. It just seems logarithmic because that's how our senses work as animals. Our bodies are sensitive to things based upon our size and therefore what threatens us most. Mice hear high-pitched sounds; whales hear low-pitched sounds; we're in between. So you'll notice that the decibel scale is written along the same lines.

So your body interprets the natural scale/harmonic/overtone series as being logarithmic when in fact if you measure it, it's linear. If you measure the diatonic scale, which is octave equivalent, you'll see that it's exponential. Since our senses are logarithmic, we perceive it as being linear. However, the spacing of the notes within the octaves is not consistent. It also explains why the cycle-per-second (Hertz) between the notes increases as you get higher.

The two systems combined around c.1260+ (look at the Rutland Psalter p.98r where a king is tuning his harp to a natural trumpet playing the natural scale). From the natural scale, we get the major and minor modes, triadic harmony and V-I cadence. With the diatonic scale, we have a scale that seems linear and where you can have instruments with different ranges playing together in different octaves. The natural scale is constrained to a gamut of notes. So with the acceptance of the natural scale, you have 10-stringed frame harps quickly evolving into 30-stringed Gothic harps.

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