Chapter 4: Appendix 4.2

Mathematical representations of portamento curves

Technical Definitions:

• Rate of change: the slope of the graph or the slope of the tangent to a curve if the graph is curved

• Exponential: graphs in the form y = aebx

• Polynomial: graphs in the form y = ax2 = bx + c

• Negative/Positive graph: the dominant term in the graph’s equation is negative (N.B. this does not mean that the rate of change is increasing or decreasing)

• Growth/Decay: how the rate of change of an exponential differs as x increases (as the graph goes to the right)

Linear:

Negative linear: the graph is increasing1

Technical definition: constant rate of change

Visually: a straight line increasing

Equivalent portamento type: usually fast ascending slide with no slowing or quickening of pitch change between notes

Positive linear: the graph is decreasing2

Technical definition: constant rate of change

Visually: a straight line decreasing

Equivalent portamento type: usually fast descending slide with no slowing or quickening of pitch change between notes

Exponential:

Technical name: negative exponential decay

Visually: increasing pitch, with the rate of change initially steep then decaying to a shallow rate of change3

Equivalent portamento type: ascending slide which starts fast and then slows as it approaches destination note

Technical name: positive exponential growth

Visually: increasing pitch, with the rate of change initially shallow then growing and becoming very steep4

Equivalent portamento type: ascending slide which starts slow and then quickens as it approaches destination note

Technical name: positive exponential decay

Visually: decreasing pitch, with the rate of change initially steep by decaying until it is shallow5

Equivalent portamento type: descending slide which starts fast and then slows as it approaches destination note

Technical name: negative exponential growth

Visually: decreasing pitch with the rate of change initially shallow then growing until it is very steep6

Equivalent portamento type: descending slide which starts slow and then quickens as it approaches destination note