I was reading through some of Ralph Morgan's notes on this topic a while back. Many of his notes and papers are very technical, but here is the gist of it from what I understand:
A-440 means that an extremely finite amount of a "material", when set in vibratory motion, will produce the pitch level of A. This material can be a metal bar, wire, wood, air, etc. When set in vibratory motion it vibrates at 440 cycles per second. Imagine a metal bar as on a xylophone, when struck it will vibrate. If it is properly constructed with a precise amount of metal, it will vibrate at 440 cycles per second. If it has too much metal, it will vibrate slower thus being flat. If not enough metal, it will vibrate faster thus being sharp.
On a saxophone, the "material" set in motion is the column of air. The air set in motion is the cubic volume of air inside the instrument; however, you also need to include the amount of air inside the mouthpiece…all the way to the tip of the mouthpiece.
For a saxophone to be in tune, the tone must be produced by an exact and finite cubit volume of air. In the design formulas for saxophones, those cubic volumes are measured in the thousandths of cubic centimeters, so small they are not visible to the naked eye. If the volume of air in the mouthpiece is too small, the air will vibrate too fast. If the volume of air in the mouthpiece is too large, then it will vibrate too slow. In either case, the instrument will be out of tune.
Assuming the C-Melody saxophone was designed using the same design formulas as other saxophones, then it would make sense that only a properly designed C-Melody mouthpiece would tune well. The problem with many modern C-Melody mouthpieces is they are simply cut-down tenor pieces or modified alto pieces with no consideration to the interior cubic volume of air…thus the cause of many intonation problems with the instrument.
I will try to put together a more detailed blog article someday on the topic, but hopefully, this makes a bit of sense.
On another topic which I will not elaborate on much right now is that using reeds designed for other mouthpieces can cause problems as well…using bass clarinet reeds on a tenor mouthpiece, or tenor reeds on alto, etc. Again, more on this in a future blog article, but the basic principal is: The slope of the vamp of a reed is designed to be symmetrical with the facing curve of a mouthpiece. If you are using the incorrect reed, it is not vibrating as efficiently as it should.
So this brings us to my shameless plug on the Morgan C-Melody Mouthpiece…
The following is text from the Morgan Mouthpieces Website:
Quite possibly the best C-Melody Tenor Saxophone Mouthpiece ever made. This mouthpiece was designed by our Master Crafter, Erik Greiffenhagen and was approved by Ralph Morgan. This is not just a shortened tenor mouthpiece like other companies offer. This mouthpiece has an interior specifically designed for the C-Melody Saxophone. It has a large round chamber with low baffle. It is designed to use modern tenor saxophone reeds and standard size tenor ligatures. The material is the Morgan Mouthpiece Company's 100% pure rubber formula, which is the same as was used by manufacturers in the 1920s-1950s. The facing curve, interior chamber and baffle are all shaped by hand. It is a true hand-crafted pure hard rubber C-Melody Saxophone Mouthpiece. If you are a serious C-Melody player, then the Morgan C-Melody Tenor Saxophone Mouthpiece is for you.
These are fairly expensive mouthpieces; however, if you are a serious C-Melody player, then it is certainly worth giving it a try.
https://www.morganmouthpieces.com/products/morgan-c-melody-tenor-saxophone-mouthpiece
Ralph Morgan left us tons of research and articles which I hope to get out on the website. Check the Blog every now and then for more articles:
https://www.morganmouthpieces.com/blogs/news