Steve Strogatz illlogical Bait & Switch covers up Noncommutative phase 2...

Math Professor Steve Strogatz NEGLECTS to mention the pitch (do-fa-sol) as root tonic 1 and fa as Perfect Fourth for his below description of Pythagorean music harmonics - WHY?!! Because the example he gives of the Perfect Fourth is actually ALSO still SOL as the Perfect Fifth only up to the octave do - thereby covering up the change in direction of time-frequency!! This change in direction is also noncommutative phase.

"This sense of awe goes way back in the history of mathematics. According to legend, Pythagoras felt it around 500 BCE when he and his disciples discovered that music was governed by the ratios of whole numbers. For instance, imagine plucking a guitar string. As the string vibrates, it emits a certain note. Now put your finger on a fret exactly halfway up the string and pluck it again. The vibrating part of the string is now half as long as it used to be - a ratio of 1 to 2 - and it sound precisely an octave higher than the original note (the musical distance from one "do" to the next in the do-re-mi-fa-sol-lat-ti-do scale). If instead the vibrating string is 2/3 of its original length, the note it makes goes up by a fifth (the interval from do to sol; think of the first two notes of the Star Wars theme). And if the vibrating part is 3/4 as long as it was before, the note goes up by a fourth (the interval between the first two notes of "Here comes the Bride")."

p. xiv, Infinite Powers by Steve Strogatz