Flats

A student asked me:

Why in the circle of fifths there are notes with flats instead of notes with
sharps?
Searching for a deep explanation I don't know why.

Lorenzo

Because whatever direction one chooses to travel in a traditional 12-tone Western system, whether flat or sharp, one arrives at the starting point eventually?

Or rather, the diatonical nature of staff notation requires flat-sharp equivalance in distant tonalities for easier transcription of notes?

Check out Bach's Prelude in C# major with 7 sharps in the key signature, which is written once more in Db major with 5 flats in Book I of the Wohltemperiertes Clavier. They sound exactly the same, the transposition is only done on paper.

Yet, would it have been possible to write F major as E# major conveniently? I think not.

Maybe the enharmonical equivalence of tones is exaggerated to the detriment of instruments that cannot possibly be made to perform like keyboards with a fixed tuning.

Cordially,
Ozan

----- Original Message -----
From: Lorenzo Frizzera
To: tuning@yahoogroups.com
Sent: 29 Eylül 2005 Perşembe 13:19
Subject: [tuning] Flats

A student asked me:

Why in the circle of fifths there are notes with flats instead of notes with sharps? Searching for a deep explanation I don't know why.

Lorenzo

> A student asked me:
>
> Why in the circle of fifths there are notes with flats
> instead of notes with sharps?
> Searching for a deep explanation I don't know why.

Hi Lorenzo,

If I understand your question, the answer is: Our notation
uses accidentals based on the key of C -- there are no
accidentals in that key. And because the diatonic scale is
a chain of fifths, transposing it by a fifth changes only
one note. Therefore, keys farther from C in terms of
transposition by fifths require more accidentals (one for
each transposition). If one wants notate scales with the
fewest accidentals possible, one should spell them so that
they are as close to C, in terms of transpositions by fifths,
as possible. So Bb is two fifths away from C, while A# is
ten fifths away. There may be reasons, in a piece of music,
to notate something in A#, but for a diagram would just
assume that most of the time, students will want to use Bb.

-Carl

"Carl Lumma" <clumma@yahoo.com> writes:

> If one wants notate scales with the
> fewest accidentals possible, one should spell them so that
> they are as close to C, in terms of transpositions by fifths,
> as possible. So Bb is two fifths away from C, while A# is
> ten fifths away. There may be reasons, in a piece of music,
> to notate something in A#, but for a diagram would just
> assume that most of the time, students will want to use Bb.

That of course assumes you are notating music in a tuning like 12ET,
in which Bb and A# are the same pitch. More generally: A# is the note
seven steps higher on the circle of fifths than A, while Bb is the
note seven steps lower on the circle of fifths than B:

--Bb--F---C---G---D---A---E---B---F#--C#--G#--D#--A#--

In any (just octave based) temperament other than 12ET, these are
different pitches, and the different notation reflects that. Even in
12ET, the notational difference can be important: if you want to
notate an F# major triad, you probably would not write it as F#-Bb-C#,
which notationally contains no thirds.

- Rich Holmes

> > If one wants notate scales with the
> > fewest accidentals possible, one should spell them so that
> > they are as close to C, in terms of transpositions by fifths,
> > as possible. So Bb is two fifths away from C, while A# is
> > ten fifths away. There may be reasons, in a piece of music,
> > to notate something in A#, but for a diagram would just
> > assume that most of the time, students will want to use Bb.
>
> That of course assumes you are notating music in a tuning like
> 12ET, in which Bb and A# are the same pitch.

Hi Rich,

I assumed that the context of the question was an elementary
theory book showing the circle of fifths in 12-tET.

> Even in 12ET, the notational difference can be important:

I did say that.

-Carl

--- In tuning@yahoogroups.com, "Lorenzo Frizzera"
<lorenzo.frizzera@c...> wrote:
> A student asked me:
>
> Why in the circle of fifths there are notes with flats instead of
notes with
> sharps?
> Searching for a deep explanation I don't know why.

Historically, it was a chain of fifths (first pure, then meantone) and
not a circle at all.

Hi Carl.

> And because the diatonic scale is
> a chain of fifths,

This would mean that the lydian scale is the *true* diatonic scale.

> So Bb is two fifths away from C, while A# is
> ten fifths away.

So if diatonic structures are builded from the lower to the higher pitch in
a chain of fitfths (as in lydian scale F C G D A E ) in order to use natural
notes you have to start from the flat notes.

Lorenzo

Lorenzo, I do not think that Lydian is a scale, it's a mode. in fact, it's a major scale with the fourth degree as tonic. Likewise, Aeolian or natural minor is the same scale with the 6th degree as tonic. In a strict Pythagorean diatonic major tuning, all the modes like Lydian, if I'm not mistaken, are Moments of Symmetry.

Cordially,
Ozan
----- Original Message -----
From: Lorenzo Frizzera
To: tuning@yahoogroups.com
Sent: 30 Eylül 2005 Cuma 14:58
Subject: Re: [tuning] Re: Flats

Hi Carl.

> And because the diatonic scale is
> a chain of fifths,

This would mean that the lydian scale is the *true* diatonic scale.

> So Bb is two fifths away from C, while A# is
> ten fifths away.

So if diatonic structures are builded from the lower to the higher pitch in
a chain of fitfths (as in lydian scale F C G D A E ) in order to use natural
notes you have to start from the flat notes.

Lorenzo

--- In tuning@yahoogroups.com, "Lorenzo Frizzera"
<lorenzo.frizzera@c...> wrote:
> A student asked me:
>
> Why in the circle of fifths there are notes with flats instead of
notes with
> sharps?
> Searching for a deep explanation I don't know why.
>
> Lorenzo

Well, it is history and convention.

Very roughly, a sharp is a chromatic semitone above the corresponding
natural and a diatonic semitone below the next highest one.

A flat (not Ab!) is a chromatic semitone below the corresponding
natural and a diatonic semitone above the next lowest one.

Through much of musical history the diatonic and chromatic semitones
were thought of and played and heard as different intervals. Only in
standard keyboard instruments (tuned in a circulating manner), or some
fretted string instruments, are the flats and the sharps always the
same 5 notes.

This identity *allows* the circle of fifths to be constructed. So, if
you have a circle of fifths, you might as well write G# as Ab. There
is no deep explanation why some notes in the circle are printed as
flats and some as naturals, only convenience.

The organ tuner Werckmeister, for example, talks of D# even when it
would seem to make more sense to say Eb. But it makes no difference to
him since he uses circular tuning with 12 notes.

~~~T~~~

Hi Lorenzo,

> Hi Carl.
>
> > And because the diatonic scale is a chain of fifths,
>
> This would mean that the lydian scale is the *true* diatonic
> scale.

I would say, "This would mean that the lydian mode is the
*true* mode of the diatonic scale."

I don't see why, though. Just knowing that a scale is made of
a chain of fifths doesn't suggest to me what mode would be
preferred.

> > So Bb is two fifths away from C, while A# is ten fifths
> > away.
>
> So if diatonic structures are builded from the lower to
> the higher pitch in a chain of fitfths (as in lydian scale
> F C G D A E ) in order to use natural notes you have to
> start from the flat notes.

The ionian mode was preferred for some reason, and then
the notation was based around it (it was assigned to the
naturals) because it was preferred. Of course in real
history these evolve simultaneously. But it's convenient
to explain it as one causing the other.

So maybe your question is: Why was the ionian mode
preferred? This is indeed a deep question that has no
consensus answer that I know of. One reasonable answer
that occurs to me is: it is the only mode of the diatonic
scale in which a I-IV-V-I progression is possible.

-Carl

Hi Ozan,

> Lorenzo, I do not think that Lydian is a scale, it's a mode. in
> fact, it's a major scale with the fourth degree as tonic.
> Likewise, Aeolian or natural minor is the same scale with the
> 6th degree as tonic. In a strict Pythagorean diatonic major
> tuning, all the modes like Lydian, if I'm not mistaken, are
> Moments of Symmetry.

Hi Ozan,

You might see that I've made a similar response to Lorenzo.
But one thing: to the best of my understanding of the MOS
concept as Erv Wilson uses it, MOS give you only scales,
and do not determine their modes.

-Carl

Lorenzo Frizzera wrote:

> Hi Carl.
> > >>And because the diatonic scale is
>>a chain of fifths,
> > > This would mean that the lydian scale is the *true* diatonic scale.

Just because it starts on the "left"? Try the middle of the chain, and you are on save historical ground.

Our note naming scheme was built on a classification of Gregorian chants by interval content. It so happenend that all of them, or the simpler ones at least, could be derived from the same, almost(!) fixed collection of pitches, with four different final notes a tone, semitone and tone apart, and two different tessituras: either all above the final or with the final up a fourth from the lowest note. That's four finals with two different compasses for each one, or eight modes (differently from others on this list, I think of modes as ways of making music out of available tones, so the meaning extends beyond scale rotations to typical tessituras, places for turns in the melody, interval jumps and whatever else may be needed to give form to a tune in a given culture).

The lowest note to be considered is a fourth below the lowest of the four finals. They called it A and numbered the others alphabetically. When they stopped at G and restarted at the next octave, this scheme was almost up to date (~900 AD). It gives you the English (for example) note names. The final, to return to the issue of the "true" diatonic scale, is D. This mode has been called Hypodorian in the late Renaissance, but in the Middle Ages it was just the second mode (the first has all its notes in the octave above the final, D).

Further Complications Introducing a Bracelet of Fifths and Why Flats Look Like a B

Notes were referred to not by letter names and an octave number, but by these "fixed" names and their possible positions in a different, movable, system, the hexachord. The hexachord is a collection of two ditones (two Pythagorean thirds: ut-re-mi, fa-sol-la) joined by a semitone ("mi contra fa, diabolus in musica"). You'll recognize this as the origin of (for example) the Italian note names. They derive from the first syllables of the phrases of a chant in the first mode; ut is a C that approaches D. Originally, these notes were

C-ut, D-re, E-mi, F-fa, G-sol, and A-la.

Since some chants modulated, the hexachord could be transposed a fifth/fourth in either direction:

F-ut, G-re, A-mi, B-fa, C-sol, D-la and

G-ut, A-re, B-mi, C-fa, D-sol, E-la.

The hexachord on C doesn't touch B. The two transpositions do, and they give it noticibly different interpretations: as fa, it is a fourth above F, as mi, it is a third above G (and I am wondering whether it is clearer or more misleading to emphasize that it is a major third - there is no choice; a third at this position in the hexachord is always major). These hexachords were called mollis and durus (source of the German names for major and minor), and the notes themselves B rotundum and B quadratum. To differentiate them in tablatures, the b was written either round ("b") or square ("h", at least after tablatures were printed with letters), and eventually, they were called that, too (at least in Germany). And of course, the two Bs are the origin of the flat and natural, probably also the sharp sign.

Polyphonic music gradually introduced E-fa (with a B-fa-ut a fourth below) and other altered pitches. They were written without accidentals, however, but probably practised, at least analyzed, with the hexachord syllables - nobody was thinking about the circle of fifths. They took there cues from the note names: a D in the middle octave was re-sol-la; you arrived there with one solmisation syllable, but you might reinterpret - "modulate" - and continue in two different ways that might make your Es and Bs flat or natural. Extending the chain of fifths in the flat direction meant making a fa note the new ut, extending it towards the sharps meant making a sol note the new ut.

This is a different and more complicated story than the other answers you got, and I tried to make it short. You might want to have a closer look at topics like Gamut, Hexachord, or musica ficta.

klaus

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> The ionian mode was preferred for some reason, and then
> the notation was based around it (it was assigned to the
> naturals) because it was preferred. Of course in real
> history these evolve simultaneously. But it's convenient
> to explain it as one causing the other.

This is completely untrue, Carl. Our modern notation was "assigned to
the naturals" long before anyone preferred the ionian mode. I don't see
what the "convenience" you refer to is.

> So maybe your question is: Why was the ionian mode
> preferred? This is indeed a deep question that has no
> consensus answer that I know of.

I have an answer, involving the tritone of course, but the question is
completely irrelevant to this discussion, and I'm very puzzled that you
bring it up here, Carl.

Moment of symmetry require two distinct steps, no? Then, do you agree that a Pythagorean major scale is a MOS? Granted, there need not be any tonic for a MOS scale, so all the Greek modes depend on a single MOS scale?

Cordially,
Ozan
----- Original Message -----
From: Carl Lumma
To: tuning@yahoogroups.com
Sent: 30 Eylül 2005 Cuma 20:35
Subject: [tuning] Re: Flats

Hi Ozan,

> Lorenzo, I do not think that Lydian is a scale, it's a mode. in
> fact, it's a major scale with the fourth degree as tonic.
> Likewise, Aeolian or natural minor is the same scale with the
> 6th degree as tonic. In a strict Pythagorean diatonic major
> tuning, all the modes like Lydian, if I'm not mistaken, are
> Moments of Symmetry.

Hi Ozan,

You might see that I've made a similar response to Lorenzo.
But one thing: to the best of my understanding of the MOS
concept as Erv Wilson uses it, MOS give you only scales,
and do not determine their modes.

-Carl

> I don't see what the "convenience" you refer to is.

That's ok: I wasn't explaining it to you.

> > So maybe your question is: Why was the ionian mode
> > preferred? This is indeed a deep question that has no
> > consensus answer that I know of.
>
> I have an answer, involving the tritone of course, but
> the question is completely irrelevant to this discussion,
> and I'm very puzzled that you bring it up here, Carl.

Puzzled, eh? Did you miss the part where I ask Lorenzo
what it is he's asking? I'm not sure what he's asking,
are you? His original question seemingly has nothing to
do with the Lydian mode being the true one, etc. etc.

-Carl

> Moment of symmetry require two distinct steps, no?

Yes. (Note: there is some debate about whether certain
scales such as the octatonic scale, where some interval
classes come in only one size, count as MOS. This is
a minor point here.)

>Then, do you agree that a Pythagorean major scale is a MOS?

Yes.

>Granted, there need not be any tonic for a MOS scale, so all
>the Greek modes depend on a single MOS scale?

If you mean the "church" or "greek" modes of the diatonic scale,
then yes.

-Carl