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#2242971 - 03/08/14 02:11 AM How to measure cents in just intonation  
Joined: Mar 2014
Posts: 491
Parks Offline
Full Member
Parks  Offline
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Joined: Mar 2014
Posts: 491
Northern CA
So, the 'cent' measuring system was started by Helmholtz, right? A nice and clean division of the octave. But how can other temperaments be measured in cents? Can you explain the math of why a 'just' major third is 386.31 cents? How do you get 386.31 from 5/4?


Michael

"Genius is nothing more than an extraordinary capacity for patience."
Leonardo da Vinci
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#2242976 - 03/08/14 02:44 AM Re: How to measure cents in just intonation [Re: Parks]  
Joined: Feb 2011
Posts: 2,511
Withindale Offline
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Withindale  Offline
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Joined: Feb 2011
Posts: 2,511
Suffolk, England


Ian Russell
Schiedmayer & Soehne, 1925 Model 14, 140cm
Ibach, 1905 F-IV, 235cm
#2243046 - 03/08/14 10:16 AM Re: How to measure cents in just intonation [Re: Parks]  
Joined: Mar 2014
Posts: 2
Smythe Dakota Offline
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Smythe Dakota  Offline
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Joined: Mar 2014
Posts: 2
Measuring in cents is "clean" because it converts ratios to differences.

If you start with low A at 27.5 hz and go up an octave at a time, you get

27.5
55
110
220
440
etc.

Each A is double the frequency of the previous A.

The human ear hears each octave as being the same "size" as the previous octave. If we arbitrarily assign to low A the value of 0 cents, then we have

0000
1200
2400
3600
4800
etc.

Clean!

What mathematical device converts ratios to differences in this manner? Logarithms (ooh). To convert a ratio to cents, you take the base-2 logarithm of the ratio, and multiply by 1200.

You probably don't have base-2 logarithms on your calculator, but you probably do have base-10 logarithms. So you'd have to find the base-10 log of the ratio, then divide by the base-10 log of 2, then multiply by 1200.

In Excel, it's even easier, because you can specify the base of the log. If, in cell A1, you type

=5/4

and, in cell B1, you type

=1200*log(A1,2)

then bingo -- you get your 386 cents.

Change the ratio in cell A1, and the cents value in cell B1 changes with it. In this way you can easily convert any "just" (or other) ratio to cents.

Bill Smythe

#2244058 - 03/10/14 05:36 AM Re: How to measure cents in just intonation [Re: Parks]  
Joined: Mar 2008
Posts: 9,230
Olek Offline
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Olek  Offline
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Posts: 9,230
France
perfect simple demonstration , it is better to do the sample with a spreadsheet, certainly, but using an old slide rule may show what happens more clearly



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