palimpsests

36 millimeters == 36mm != 36MM

yes, I am aware that conventional guitars have fundamental issues with intonation in equal temperament systems.

this has not prevented it from being a versatile instrument that is quite capable of being played “enough” in tune with ensembles of other instruments, such that the vast majority of people hear zero problems.

how is what we hear in music less relevant than whether or not a given instrument is not perfectly in tune, mathematically speaking, if that variation in tuning is imperceptible to human hearing?

the unit of measurement is called a “cent”.

there are 100 cents between each 1/2 step in 12-tone equal temperament.

the progression you are referring to is I-IV-bIII-bVI, where are all these chords are “power chords” i.e. dyads comprised of a root and fifth.

it is an awesome progression but violates absolutely nothing in music theory.

we can find examples of similar progressions across jazz and classical music, most of which was composed by folks who have mastered western tonal harmony.

relative to mathematical perfection, of course it’s all an approximation when a human instrumentalist is involved. that’s the nature of our physical reality.

the most important element here is how it sounds to our ears. not how closely it tracks to an equation.

how would you use this approach in practical application?

i haven’t met many working musicians who had much difficulty learning the relationships between different keys, how they connect to the circle of fifths (fourths), and key signatures.

i get that it can seem overwhelming and non-intuitive, but it’s really not that complicated once you spend time playing and practicing music that illuminates these relationships (like playing ii-V-I progressions in every key, going around the circle of fifths). very little memorization involved; moreso muscle memory and an accumulation of applied theory in context.

most of the musicians i know are jazz players, where being able to play in any key is a critical aspect of mastering the genre. all the classical musicians i know are professional orchestral musicians, and they don’t seem to have any difficulty either.

it depends on where it is on the fretboard.

the higher you go in frequency, the physical distance between each interval on the fretboard becomes smaller. so if a +/- 5 cent adjustment is 0.5 mm at the first “fret” after the nut, it will be something like be 0.1 mm when you are at the 7th “fret” location (i.e where the interval of perfect fifth, relative to the open string, is played).

wind and brass players adjust intonation via embouchure all the time…

likely 1/100,000 at most. more likely 1/1,000,000

the sandman (*) intervals aren’t coming from microtonal tuning… it’s dynamically modulated detuning in equal temperament, just as you say. it’s an extremely common type of modulation, especially if there are synthesizers involved.

* what an incredible show!

so slightly that it can be on the range of 0-5 cents, provided the instrument is sufficiently constructed and the player is sufficiently skilled.

this is why a guitar using equal temperment can play consistently in-tune with itself as well as with other instruments tuned in the same system. it’s not about perfection according to some abstract mathematical model.

the last reason you gave is exactly why open strings sound different. check out zero-fret guitars.

skilled instrumentalists are quite capable of consistently reproducing intervals in a given tuning system. particularly thirds in just intonation. it’s not an approximation. it’s one of the reasons we spend so much time learning ear training in conservatory.

including multiple methods for the user to detune oscillators is quite common on modern synthesizers, advanced or otherwise. it’s almost never a fixed amount of detuning.

one of those methods is called a “chorus” effect. this is extremely common across effect platforms and is not limited to synthesizers / keyboard-type instruments.