How to get frequencies for the major and minor Pentatonic scales

Question

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: 2^1/12 = 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

Answer 1

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

f_n = f₀ * (2^1/12)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.