The Ring
Theo Wanne's Proprietary Saxophone Mouthpiece Facing Curve

Genesis of 'The Ring' Equation
for a Medium Length Tenor Saxophone Mouthpiece Facing Curve

• We started by measuring the facing curves of over 50 original (never refaced) tenor saxophone mouthpieces between the sizes of 5 and 10 that played well in all registers. We used the Florida Period Otto Links of the 1950s and 1960s as they have been the standard of the Jazz industry for tenor saxophone and many of the Jazz Greats (Dexter Gordon, Stan Getz, John Coltrane, etc.) played them.
  
  
• We started plotting the facing curves using 8 points along the curve for each mouthpiece. Here, shown in graph form, are the 8 plot points for the 7* Otto Link 'Super Tone Master' curve. All points are shown in millimeters:
  
  
  
  
• We then calculated the break point on each mouthpiece. The break point is the point at which the reed very first separates from the table, and states the initial starting point of the curve.
  
  
• We then found the linear relationship of the 'Break Point' to the tip opening. We found the rate the break point increased relative to the increase in tip opening.
  
  
  
  
• The actual tip opening measurement was added as the final plot point. Adding the break point and the tip opening gave us 10 plot points in total for each of the 50 mouthpieces.
  
  
• We combined the data for each tip opening to find the average curve used by Otto Link for each tip opening in their Florida mouthpieces. Here is an example of the 7* Otto Link 'Super Tone Master' curve shown in millimeters:
  
  
  
  
• We then combined Theo's personal data gathered from years of maximizing the performance of Otto Link mouthpieces. The new combined data represented an improved curve over the original Otto link curve. Below is Theo's curve for a 7* tenor saxophone mouthpiece based off of the data taken from the actual Otto Link mouthpieces:
  
  
  
  
• Each curve (tip openings from 5 to 10) was then expressed in the form of a quadratic equation (second degree polynomial). A unique quadratic equation was created for each tip opening. Below is the curve for the 7* tip opening using our quadratic equation to calculate the facing curve data point:
  
  
  
  
• R² is a measure of how well an equation fits the data points it is modeling. R² values range from 0 (no fit) to 1.0 (a perfect fit). Although we are showing variable place holders for our coefficients, our R² values are reflective of our actually data. As the R² for the above equation is very close to 1, we have an excellent degree of accuracy in our model.
  
  
• We were then able to calculate the points for each facing curve using their individual quadratic equations. Shown below is a graph with all the facing curves (one for each tip opening size) shown together.
  
  
  
  
• Finally, we wanted to have one equation to describe all the facing curves across the different tip openings. We found that the tip opening has an accurate linear relationship to our three coefficients (F₁, F₂, and F₃) in our quadratic facing curve equations. We then created a linear equation for each coefficient so that our quadratic equation will model the change in the facing curves as the tip opening changes:
  F₁ = x² coefficient = G₁ * (Tip Opening) +G₂ (R² = 0.968)
  F₂ = x coefficient = H₁* (Tip Opening) + H₂ (R² = 0.974)
  F₃ = offset coefficient = I₁* (Tip Opening) + I2₂ (R² = 0.998)
  As all our R² values are very close to 1.0, we have excellent equations to describe the change in our facing-curve equation coefficients based on different tip opening.
  
  
• 'The Ring' or master equation, representing a single quadratic equation to calculate the facing curves for all tip opening from 5 up to 10, was then created. By simply plugging in the desired tip opening and any desired distance from the tip, The Ring equation will tell you its related facing curve plot point. This one equation will show the optimal facing curve for any tenor saxophone medium length facing curve.