Pythagoras lived about 475 BC.  In his youth he traveled to India and Egypt in search of knowledge.  When he returned to Greece he established a new philosophical school and taught mathematics.  Pythagoras' extended his mathematical knowledge to designing musical instruments based upon ratios.  In the image at the left we see an 8 x 8 matrix of lambdoma ratios.  The ratios have been simplified to the least common denominator.

The logarithmic nature of frequency harmonics is illustrated when we analyze the matrix.  For any fundamental note with a ratio of 1/1, the next higher octave note is the ratio of 2/1 and the next octave lower note is ratio 1/2.  So twice the fundamental note is an octave higher, and half the fundamental note is an octave lower.  Similarly, the ratio 3/1 is three times the fundamental frequency and represents the third octave above, while 1/3 is the third octave below.  

When comparing the ratios of the lambdoma to the notes of a piano, we can see they do not line up.  Standard notes are artificially forced into a twelve tone per octave format.  The concept of keys such as C major or F minor do not apply to the lambdoma scale.  Lambdoma ratios center around a fundamental frequency and all the other notes are played relative to it.  It is a completely different paradigm for music.

One can play similar songs on the lambdoma as with equal tempered tuning.  Yet, they are only similar, they are not the same.  And since there are more possible notes in the first octave with a lambdoma, it offers more possibilities for composers.  But the best feature of a lambdoma composition is that every note is in tune with every other note.  Of course, since the tunings are different, not all compositions in equal tempered tuning are possible on the lambdoma.  It isn't that one system is better than the other, but that they offer two useful paradigms, thus giving composers more options.