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I'm trying figure out how get the frequencies of pitches from the pentatonic scales. To start, I build a chromatic scale tuned in 440 starting on D (67 semitones below A440), using this equation.

2 -67/12 x 440hz = 9.177hz

So I can build a chromatic frequency table using a ratio to find semitones: 21/12 = 1.0595

    9.1770 * 1.0595 = 9.7227
    9.7227 * 1.0595 = 10.3009
   10.3009
   10.9134
   11.5623
   12.2499
     .
     .
     .
  415.3047
  440.0000
  466.1638
  493.8833
  523.2511
  554.3653
  587.3295
  622.2540

My question is how to extract the pentatonic (major and minor) frequencies from my chromatic table?

I need this for all these scales: C • D • E • F • G • A • B

  • I'm a bit unclear on what you're asking - do you want the frequencies of CDEFGAB (google it), a formula for finding the frequencies of the various notes in a scale given a starting pitch, or something else? – Josiah Feb 23 '16 at 15:25
  • how extract the pentatonic (major and Minor) frequencies from my chromatic table?for example how get the Pentatonic frequencies from Minor F ? – user2721828 Feb 23 '16 at 15:31
  • I've edited your question somewhat. If I've missed something important, feel free to roll back or re-edit. – Josiah Feb 23 '16 at 17:17
2

Using Equal Temperament

There are a few things to consider when building a pentatonic scale. It looks like you used the formula for equal temperament in building your table, which is

fn = f0 * (21/12)n

Where f0 is a known frequency (usually A440) and n is the number of halfsteps you are from f0. (See here for more info.) | Google can do this math for you, just change the 1 in the search bar to number of semitones you are from A440.

From this you can generate a table, as you have done. I'd suggest adding a column to your table with what note names you are referring to, like

A | 440.0000

A# | 466.1638

From there you can easily figure out what the frequencies of the notes are if you know their names. Relative to "normal" major scales, the major pentatonic scale is made up of tones 1, 2, 3, 5, 6. In A these would be A, B, C#, E, F#. Relative to the minor scale, a minor pentatonic uses 1, 3, 4, 5, 7, which correlate to A, C, D, E, G. Just look those up on your table, and you have the frequencies.

The formula can also be used with a standard unit of length, so you could determine string length, pipe length, etc.


Using Pythagorean Tuning

However, you might not want to use equal temperament for working out a pentatonic scale. After all, the pentatonic scale doesn't necessarily have much to do with the 11-tone scale equal temperament describes.

Pythagoras based his fifths on perfect 3:2 relationships. A pentatonic scale can be easily built knowing only the relationships between fifths and octaves, which are 2:1 relationships. For more information, look here.

So starting from A 440,

  • E = 440 * 3 / 2 = 660
  • B = 660 * 3 / 2 = 990
  • F# = 990 * 3 / 2 = 1485
  • C# = 1485 * 3 / 2 = 2227.5

Now we know all the frequencies of the pitches in a pentatonic scale, it's just a matter of dividing by 2 until we get to the right octave. All the frequencies need to end up between 440 and 880.

  • A = 440
  • B = 990 / 2 = 495
  • C# = 2227.5 / 4 = 556.875
  • E = 660
  • F# = 1485 / 2 = 742.5

Ratios for all notes can be computed by simply using 1 as your starting pitch:

  • A = 1
  • B = 9/8
  • C# = 81/64
  • E = 3/2
  • F# = 27/16

The minor scale can be built similarly by going up a fifth to E and then down fifths through D, G, C. Any other scale can be built simply by picking a different starting note. Using this method, all frequencies can easily be worked out exactly on paper, but it really doesn't work well for scales other than pentatonic scales. For why, read up on the development of temperament.

  • this not make sense for me, the major pentatonic frequencies in D is 9.1770 11.5623 13.7500 18.3540 23.1247 30.8677 36.7081 46.2493 61.7354 73.4162 97.9989 123.4708 146.8324 195.9977 246.9417 329.6276 391.9954 493.8833 659.2551 783.9909 does not match with your results – user2721828 Feb 23 '16 at 19:44
  • 2
    @user2721828 - I think there might be something off with your list. It only shows the frequencies corresponding to the pitches D F# B. The notes of the major pentatonic scale in D should be D E F# A B. From here the frequencies of the lowest possible set of those notes on the piano is D: 36.7081 E: 41.2034 F#: 46.2493 A: 55.0000 B: 61.7354 – Josiah Feb 23 '16 at 20:19
  • By pythagorean tuning, you'd get D: 36.6666 E: 41.15625 F#: 46.40625 A: 55 B: 61.875 (all exact except for D, a repeating decimal), which you can see are slightly different as the intervals aren't tempered at all. They are still quite close though. – Josiah Feb 23 '16 at 20:27

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