\subsection{Gafurius}

Franchinus Gafurius (1451\EnDash{}1522) was a personal friend of Leonardo da Vinci and a famous musician in Italy in his day. His \Title{Theorica musicae} (Milan, 1492)\footnote{Gafurius} contains a woodcut showing various 'Pythagorian experiments'. One of these 'experiments' involves water filled cups; it is uncertain whether these cups are made of glass.

The discovery that musically pleasing combinations of notes have simple numerical relationship is ascribed to Pythagoras (569\EnDash{}475 B.C.E.), and certainly at least belongs to the School he founded. For example, the relationship of 'Do' to the 'Do' above it (as in ``Do Re Me Fa So La Ti Do'') is 1 to 2 (and is called an 'octave'), meaning this can be achieved by doubling the frequency, or making the string half as long (all other variables remaining unchanged), and so on. This much has been verified by modern physics, and it may be argued that this observation marks the beginnings of Western Science as we know it: quantifying experience with numerical relationships.

Gefarius text accompanying the woodcut reads as follows:

\begin{quote}
After truly obtaining these sounds Pythagoras himself investigated by a number of methods whether a system encompassing all harmonies consisted in these ratios, and he attached to equal strings weights derived from the same numbers and ratios. Then by adapting the length of various pipes so as to conform to particular dimensions and experimenting with them variously he upheld his absolute trustworthiness in these matters. Often too he measured vessels of equal weight, filled them with water according to some ratio and struck them with a rod. Also, by striking bronze bells of differing weights with a bronze or iron rod he concluded that musical intervals of this kind are formed and answer each other harmoniously through the ratios themselves. Indeed the effect whereby the length and thickness of strings of bronze and gut bring notes of music together in harmony as a result of the lead taken by actual ratios is more assured when derived from this source. And by using these methods he discovered a rule which takes its name from its character, not because the rule itself by which we measure sounds and their magnitudes is in the wood or bronze, but because an investigation of this kind is so firmly based that it cannot cause any investigator to come to an erroneous conclusion.\ldots

%why not ldotsplus
The diagram below demonstrates the forms of these specific proportions and the intervals that are derived from them. 
\end{quote}

%??? FIGURE
\begin{figure}[p]
%\begin{leftfullpage}
\includegraphics[width=\textwidth, bb=0 0 127mm 179mm]{gefarius001.jpg}
\caption[Gefarius' \Title{Theorica musicae}]{Woodcut from Gefarius' \Title{Theorica musicae}}
\label{Gefarius001}
%\end{leftfullpage}
\end{figure}

In the upper left panel of this woodcut we see the event which legend says gave Pythagoras his insight. According to the legend, Pythagoras looks on as various blacksmiths pound the anvil, observing that different notes are produced depending on the weights of the hammers. Unfortunately this wouldn't actually work\EmDash{}a hammer that is twice as heavy or twice the size of another won't produce a note an octave different\EmDash{}the physics is more complex in this situation. (``IUBAL'' which occurs in this panel is Latin for ``Jubal'', who was considered the inventor of music.)

In the upper right panel we see bells and musical cups. With regards to the bells, there is no one characteristic to which numbers can be assigned which will give the Pythagorean musical intervals: a bell that weighs twice as much as another will not necessarily be an octave apart. A bell that has twice the volume as another will not necessarily be an octave apart. A bell whose rim is twice the diameter, or circumference, as another will not necessarily be an octave apart. (All other factors being equal).

The same is true of the cups in the same upper right panel. (I invite the reader to take essentially identical drinking glasses from their kitchen cupboard and see if any amount of water in them can result in a pitch difference of an octave. It can't be done.) The image would suggest that the cups are transparent, but how else could the artist show the differing levels of water? The text does not say they are of glass, simply that they are 'vessels' (Latin: \textit{ciatos}).

The lower left hand image suggests that the Pythagorean Ratios apply to the tension of strings on a zither-like instrument. Vincenzo Galileo (the famous Galileo's father) demonstrated that this is false\EmDash{}to achieve the Pythagorean musical ratio of 1 to 2 (the octave) it is actually necessary for the string tension to be 1 to 4.

The lower right hand image is the only unequivocally correct one: all other factors being equal: two pipes\EmDash{}one twice as long as the other\EmDash{}will have musical notes an octave apart.

