
Why are there both sharps and flats? - ppog
https://music.stackexchange.com/questions/67046/why-are-there-both-sharps-and-flats
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seertaak
Wow -- it's amazing how bad the answers are on stackoverflow.

The first reason why there are both sharps and flats is historical. Equal
tempering produces the same pitch for, say, D# and Eb. The same was, however,
not true for some of the historical tunings prevalent at the time the very
notation became standard.

For more information about this, you can read "How Equal Temperament Destroyed
Harmony (and Why you should care)". It explains, for example, that Bach's
Wohltemperierte Klavier _did not_ , in fact, refer to the equal tempering, but
to a different tempering (whose details I can't exactly recall), but I seem to
remember something about slightly less sharp thirds. In fact to summarize that
book in a sentence -- he doesn't like overly sharp thirds... and sevenths.

There was a veritable zoo of different tunings, and each involved a different
set of compromises. So one produced lovely chords as long as you stayed in
root position, whereas another produced slightly less faithful thirds but
instead allowed you to play secondary dominants of the relative minor, and so
on.

Anyway, the point is that there was a time when playing a third up, and
playing a sixth down were _not_ the same thing (modulo an octave), and that's
why they used different symbols. But we now use equal temperament, which
_forces_ that to be true, to hell with those overly sharp thirds (as compared
to natural overtone series).

The second reason has to do with the fact that notes are not really useful _on
their own_. They're useful when considered in _relation to_ other notes. In
other words, we're really interested in intervals, rather than individual
notes. And when considering intervals, it's customary to begin at a certain
note, say, C, and count our way towards some destination note. Once you start
doing that, you arrive very naturally at the "sharp" and "flat" formulations.

~~~
MattHood0
Music teacher with degree here, the Stack Overflow answers are correct as far
as modern usage goes, with respect to common-practise tonal harmony at the
very least. Your second reason is just an alternative way of phrasing all of
the answers that cite F major as an example. Yes, in non-equal temperaments
they can be different pitches, but I highly doubt that someone asking the
difference between sharps and flats is concerned with niche subsets of early
music and contemporary classical.

~~~
seertaak
Thanks for your contribution -- I must admit I'm not a teacher, and my degree
is in another subject. I also acknowledge that the second part of my answer is
a weasel exit clause which uses the same reasoning as the stackoverflow
answers ;)

> but I highly doubt that someone asking the difference between sharps and
> flats is concerned with niche subsets of early music and contemporary
> classical

Well, the OP's question has the sentence: "If we can get away with just having
sharps (aka black notes on a piano) then why complicate things and add flats
as well?"

To my mind, this means the OP understands the idea of enharmonic equivalents,
and wants to know why it wasn't simply decreed that, e.g., "sharps it shall
be". To give an analogy from mathematics, it doesn't matter whether we add 0.5
to 1 or subtract 0.5 from 2, we still call it 1.5, not "two minus point 5" or
"one and point five". (Notwithstanding those crazy French and Germans!)

So I'm curious, given your background, what your response to this latter
question would be. Or to phrase it differently: if music notation were
invented in 2018, would it look substantially the same?

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Nnuie21
To everyone saying these answers are bad because they don't delve into the
history, equal temperament, and how D# and Eb weren't always harmonically
equivalent:

You're answering the wrong question. Yes, the question was "why are there
sharps and flats" but read the rest of the question and it becomes clear,
he/she isn't looking for the origin of sharps and flats. They are asking why
it's practical to have both sharps and flats.

The answer is: Because it's easier to write in F if you call them "A and Bb"
rather than "A and A#"

~~~
hammock
Yes. Equal temperament does not explain why there is Ab and G#. It merely
explains why they have not always been harmonically equivalent.

You, and the top SO answer, give the correct reason why. And it's not just for
writing, but reading also.

The diatonic scale, combined with the staff of lines and spaces, necessitates
both flats and sharps.

~~~
seertaak
> The diatonic scale, combined with the staff of lines and spaces,
> necessitates both flats and sharps.

Careful -- you're in danger begging the question. See my remark regarding
mathematical summation. Intervals are nothing other than a pitch distance.
Note names are the absolute value of a pitch. I'm not at all convinced that
it's not possible to design a notational system that does away with the
sharps/flats and yet retains the compactness optimality w.o. the ionic scale.

~~~
hammock
The critical thing here is that the diatonic scale (of which ionian is one) is
whole, whole, half, whole, whole, whole, half. There is no symmetry and the
pitch distances are not consistent. How would your proposed novel notation
system handle that?

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yesenadam
Gee.. page full of very low quality/BS answers there! Hadn't seen Music
StackExchange before.. super non-impressive. The 2 top-voted answers are
absolute without-a-clue nonsense, people-who-don't-know rambling aimlessly,
not explaining anything.

I am also no music historian, don't know all the history, but:

Not until the third-top-voted answer, the one beginning "Historically,
keyboards didn't always work that way" is it mentioned that..keyboards didn't
always work that way. But they go on "Our musical notation is older than
enharmonic equivalency that you get with "well-tempered" keyboards" \-
evidently confusing well temperament with equal temperament.

Which is understandable - it seems everyone's 'educated' with that
misconception; I certainly was. (Classical then jazz pianist) Even in my 30s I
remember reading a book from the 1880s talking in detail about the specific
emotional qualities of the different keys, thinking they were just imagining
things, deluded. Then I learnt that no, Bach's "well-tempered" actually wasn't
our equal temperament, with each note 2^(1/12) higher than the next.[0] Every
key (C major, E minor etc) actually sounded different, the intervals were
different etc. So that 1880s book wasn't crazy at all. Apparently that's about
when equal temperament was brought in everywhere (mid to late 19th C), and the
old "different keys sound different" was abolished.[1] Many composers
complained about the loss of..key personality, didn't want the new system. Now
all the keys sounded exactly the same. (Well, not to me, I've got perfect
pitch. But to most people they do now.) It seems a strangely huge cultural
forgetting, that somehow I never heard about all that, during decades in
various music worlds.

So before the equal temperament system of e.g. middle C is 3 semitones above
A, so 220 x 2^(3/12) Hz, many different systems of ratios were used to compute
the note frequencies, producing lots of different tunings, 'temperaments'.

The problem in all this is the Pythagorean comma, the gap resulting from the
awkward fact that when trying to construct a scale from octaves (2x the
frequency) and 5ths (3/2x), 2^x=3^y has no positive integer solution. With
continued fractions they worked out that 2^7~(3/2)^12 (128~129.7463..) is a
good approximation, which is, in short, why pianos have 12 notes per octave,
and going up 12 5ths takes you up 7 octaves. With equal temperament, the gap
is spread evenly among the notes, only now none of the intervals have simple
ratios of frequencies like they did before (except the octave). There's now
nothing perfect about a "perfect 4th" or "perfect 5th".

Anyway..going _up_ in 5ths you get C,G,D,A,E,B,F#..

Going down you get C,F,Bb,Eb,Ab,Db,Gb..

Gb and F# on modern pianos are (just different names for) the same note, but
in the pre-equal temperament days, _going up a 5th_ wasnt just Freq x 2^(7/12)
but multiplication by some ratio, most simply 3/2\. And then the note you got
going up (F#) was different to that you got going down (Gb). Some keyboards
had black notes split in half, one playing the sharp, one the flat.

But I won't go into more historical detail, because I don't know it and I'm
rambling aimlessly enough already. And you could fill a book with the answer
to the Q.

[0] well, pedantic piano tuners won't agree, but that's the theory.

[1] Some instruments had been equal temperamentish for centuries, e.g. guitar
I think, (it using the same frets for different keys) but others, like pianos,
not until then.

~~~
tropo
It seems that we could get our perfect ratios now that we have computers.

As the computer plays, it uses a psychoacoustic model to determine which notes
would have significant local tonal impact in the listener's mind. (the
automated choices can be overridden as desired) Both past and future notes are
considered. Exact ratios are used to determine every note frequency, using
nearby significant notes as references.

You'd get nice integer ratios all throughout the music. The pitch standard
would slowly vary, such that you could start in A440 and end up in A337. In
most music, the pitch wouldn't change all that much, since it is something of
a random walk by tiny amounts. There would be the occasional piece of music
that causes a continuous unidirectional change in the pitch standard,
requiring a few tweaks if that is undesired behavior.

~~~
m00g00
Note that this is already kinda the case for fretless instruments, such as the
violin. What you are suggesting to be done by a computer would be done by the
player themselves. Obviously deciding exactly what tuning of each note to land
on for any part of a song relies on musical intuition which computers are
notoriously bad at. But when decided to be appropriate, on the violin perfect
5ths and 4th intervals would be actually (to the limits of the players ability
and ear) perfect ratios. The major 3rd itself, which is decidedly enharmonic
in equal temperament would sound cleaner and less "beaty".

I've always thought Bach would love the keyboard tools we have today. Simply
the ability to switch tunings on the fly, rather than stopping to laboriously
change gears on your harpsi/clavichord. I imagine the ability to switch to
whatever temperament/tuning you want at the press of a button would have
masterfully been taken advantage of by him (to say nothing of arbitrary
sound/timbre for each voice/key-range etc. I think he would have loved
Switched-On Bach).

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bhnmmhmd
It was my question too! Thanks for clarification.

