The discrepancy comes from the way musical intervals are named.
![Image alt text](https://math.codidact.com/uploads/wxdhLQzQDS2x1AuXhMPxwgAQ)
The names are derived from the number of "fenceposts" including the first one, where as the corresponding length of the fence is one unit shorter.
The third+third=fifth equation is thus not $$3+3=5$$ but $$(3-1) + (3-1) = (5-1)$$
The E is counted double in the naïve addition $3+3=5$ because it is counted as the end of one interval and the start of the next interval.
![Image alt text](https://math.codidact.com/uploads/awaeEUPK9MHnNHCWMAjLevq9)