Thirds and 6ths need to be adjusted by the performer(s) to have no waves. They should be “perfectly in tune” to the chord.
A perfect third is flat (as compared to a piano third) to be in tune compared to a piano third. Piano thirds are extremely sharp with 7 vibrations a second.
Orchestras, choirs, and bands need to teach the performers to “listen" and adjust”. To "listen and adjust" is essential while also phrasing using proper dynamics, tempo, and so many other things. But visualizing the pitch is an important part of performing well.
The human element is such an important part of this process.
Electronic chromatic temperament orientated tuners help the performer get close to the right pitch. But fine tuning is so important to make the waves smooth out.
Instrumentalists and vocalists are expected to "lock on" to what the group is doing, even if they have gone sharp or flat to the world pitch. Performers are expected to compensate and adjust to each other. It is common for flutes and clarinets (woodwinds) to "lock on" to the pitch and compensate. The same thing for strings and brass.
Fixed Pitch Percussion has a problem. The bells in the orchestra, band and choirs are "set".
They are tuned in the factory for chromatic “equal temperament” tuning.
This can be a major problem when the tubular bell is sounded while the musical group has gone sharp or flat.
The same problem is prevalent with pianos, organs and harps. They are all tuned with a chromatic world standard temperament. And some musical organizations like to play sharp. This is a real problem.
This open and "in tune" sound with "pure" intervals sound amazing. And in a cathedral this is breathtaking. The Gregorian Chant in ancient days was performed in this fashion. It was "in tune" and glorious to hear. The smooth sounds in a performance hall (cathedral) that had over 6 seconds of echo made this sound "heavenly".
The next example shows this very sound with modern music performed "in tune" with an amazing "live" echo over over 6 seconds.
The modern piano tuning is tuned so that all of the music sounds accurate in any key. This modern accuracy permits music in all the the key signatures and intervals to be in a modern temperament.
No key signature sounds different than another with this system. It is "equal".
Historically this was not the case for the harpsichords and organs.
Early temperaments were quite astonishing to the modern ear. The chords had different sounds and made smooth and other times harsh sounds that would offend many modern listeners. Some intervals would be smooth with no beats.
French Horns didn't have valves and had to use "crooks" and hand embellishments to change to the correct notes and keys for years. Valves were beginning to be manufactured for french horns in the mid 1800s.
Early music had a very open sound because the harpsichord was tuned to the key that was being played for the concert. Changing the key was almost impossible to have a nice open sound. Composers in early instrumental music used very few key signatures. Many sharps or flats made it difficult for the harpsichord to fit in with the instruments if the key signature was complicated.
This was the piano tuners paradise. Sometimes harpsichords were tuned at intermissions or between numbers to sound "open" and acceptable to elite music connoisseurs like Kings and Queens.
Bach was offended that composers were making their music in the "popular" keys so that the harpsichord would sound acceptable in a concert. So he made an attempt to make all of the keys available in his "Well Tempered Clavier". He composed in all of the keys. And I'm sure the tuning community and audiences were astonished at this brazen and brash idea.
Bach carried the day and eventually keyboard temperaments evolved into todays modern "equal temperament".
Overview of Piano Tuning with an A Tuning Fork
Most pianos are tuned using 523.3 C above middle C, not 440 A.
The piano is tuned to one note. In many cases it's the 523.3 C above middle C.
Then the entire piano is tuned to that one note. The formula used is a well tempered temperament that makes all of the 4ths a bit stretched and the 5ths compressed.
To the normal musician this lack of precision is not heard. It is so slight it is not noticed. It almost creates a slow vibrato in the sound.
These imperfections are unacceptable in orchestras, choirs, bands, and other live performance groups in modern times.
Ancient tunings are hard to listen to in todays environment.