Blackwood

Blackwood [3] provides detailed intonation annotations to musical examples, illustrating various issues associated with just intonation. Blackwood's tuning theory is hampered by the fact that his ``interval size convention'' restricts him to frequency ratios in the range $\left[1,2\right)$. This approach works fine as long as any two pitches in the same register are tuned to less than a doubling apart. Unfortunately this is patently not the case for useful pitch sets, since they need to contain pitches such as C3 and B3. In his own words, ``the interval size convention, although helpful in revealing interconnections among diatonic intervals, cannot be expected to produce similar insights in the case of the chromatic intervals'' [3, 58]. (In Blackwood's vocabulary, diatonic intervals are those that are present in the pitches of a major scale, while chromatic intervals are all others.) Blackwood's inability to deal with the fact that tunings of pitches in a single register are not confined to a doubling causes errors such as tuning C3 near B3 rather than B2.