I'm new to this forum and have recently started taking Randy Potter's course. One of the reading assignments is in Reblitz's second edition on page 214 and deals with partials. Initially, Reglitz says that all fifths are narrow, and all major thirds, fourths and sixths are wide. On page 215, this is restated saying that Fourths and Fifths are narrow. so my question is: Are fourths narrow or wide? Can anyone clarify?

If (major) fifths are narrow, fourths have to be wide.

...because an octave is a fourth plus a fifth.

Fifths have to be narrow because of another complication Pythagoras discovered. An octave above A440 is A880, a ratio of 2:1, or 2^1 (two to the first power). Over the span of seven octaves, the highest note's frequency is 2^7 that of the lowest, or 128:1.

A fifth has frequencies in the ratio 3:2. Seven octaves contains 12 fifths. The ratio of the highest note to the lowest, calculated from fifths, is (3/2)^12, or 129.7 and change. This difference between 129.7 and 128 is the "Pythagorean comma".

So to make fifths fit into octaves over the span of a piano, they have to be narrow.

--Cy--

Thanks for your replies. I understand now.

My book's in the car so I can't check it but I suspect there may be some confusion between "narrow" and "slow". Fourths and fifths both beat slow (while thirds and sixths are fast), but fourths are wide and fifths are narrow.

What is important to know is how to determine whether to raise or lower a note when it is close. If the note a fifth below the note is beating too fast and the note a fourth below it is beating too slow, the note is flat, and vice versa. Once you know all these things, you are well on the way to an exciting career as a key holder for a pipe organ tuner! Add hammer technique and you can tune pianos!