# Question: So... who decided that 'C' is 'C'?



## suntower

I've read various articles about how the concept of tuning developed in the 17th, 18th century. But I wonder who decided that 'C' is 'C'? I assume that the first organs were the first instruments with repeatable fixed pitches, right? 

Even though they may have been lower than today, the pitches were still in the same ballpark. So... who/how did they decide the vague idea of 'A' or 'C' when they were creating the original keyboards?

Does it have something to do with the original 4' or 8' pipes?

TIA


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## Weston

And on a similar note, why in the world isn't A the "first" or simplest key - I mean the one with no sharps or flats rather than C? There must be some kind of carry over from the modes.


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## suntower

Weston said:


> And on a similar note, why in the world isn't A the "first" or simplest key


Exactly. My WAG was always that basically -everything- had -something- vaguely to do with 'organs'... something mundane like, the Flemish Plumbers Guild made tin pipes in a certain length... and the first organs were made from those pipes at the medieval version of home depot... so that became the standard.

But over the years, whenever I ask anyone... colleagues, teachers, etc... they look at me like... 'WHAT A STUPID QUESTION'.

They can tell ya what kind of fried chicken Beethoven liked, so there has to be an origin story for all this.


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## Taggart

Wiki as ever.>>>>>>>>>>>>


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## suntower

Taggart said:


> Wiki as ever.>>>>>>>>>>>>


Well.... sort of...

Maybe I missed something, but Boethius or Ptolemy used ABCDEFG... etc. but then that implies an Aeolian 'mode' as a starting point... why?

And so 'C' and solfeggio derives from -that- somehow... and then became 'major'.

But that still doesn't answer my original question... ie. why was 'A' 440? or 'C' 261ish. I mean, in the broadest sense, people as far back as 'Ptolemy' seem to have had a common idea that 'A' was a certain pitch... in the sense that 'red' is 'red' all over the place.

I understand it's a broad topic, but it seems like an interesting thing. 'Red' is (apparently) a biological universal. But few of us seem to have perfect pitch... so... how did 'C' in Turkey end up the same as 'C' in Scotland... I mean going back to Ptolemy.

OR... I'm wondering if a lot (or most) people hundreds of years ago actually -did- have something like perfect pitch?


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## Taggart

Ah. the 440hz A is part of the 1953 standard. Wiki on Concert pitch is weak pre 1800. The article on eight foot pitch gives you details on organ pipes. This article points out that early tuning were all over the shop and that's without mentioning temperament. This article covers much of the same ground and plugs a book "The Story of A" by Bruce Haynes which looks as if it was written for you.


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## millionrainbows

Taggart said:


> Ah. the 440hz A is part of the 1953 standard. Wiki on Concert pitch is weak pre 1800. The article on eight foot pitch gives you details on organ pipes. This article points out that early tuning were all over the shop and that's without mentioning temperament. This article covers much of the same ground and plugs a book "The Story of A" by Bruce Haynes which looks as if it was written for you.


Ahh, Taggart, such subtlety. That's just a nice way of saying* "Go read a book!"
*


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## suntower

Taggart said:


> Ah. the 440hz A is part of the 1953 standard. Wiki on Concert pitch is weak pre 1800. The article on eight foot pitch gives you details on organ pipes. This article points out that early tuning were all over the shop and that's without mentioning temperament. This article covers much of the same ground and plugs a book "The Story of A" by Bruce Haynes which looks as if it was written for you.


Cheers for that. That 'Eight Foot' article is interesting.

But there still seem to be gaps in all this (or I'm getting even thicker as I age.) So 'C' is an 8' pipe. And 'C' is related to 'A', which has some cosmic significance according to Boethius or Ptolemy.

Sadly, that Bruce Haynes book is $87 on Amazon which is about... (carry the 2...) $85 more than I'm willing to pay for this hidden knowledge.  I asked my library to get it from special loans... we'll see.

When I was a kid there was this weird guy with glasses on BBC who did these hour long stories where he'd link together all these seemingly unrelated science things and show they all fit together in a 'fun' way.... James something? I think this is one of those deals.

I appreciate yer taking the time. It's funny that I've spent 40 years 'studying' music and never
gotten into this crap before.


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## Taggart

Thanks for that.

The thing about C and A is that the A above middle C starts the next round of notes. Nothing exactly special about that. Then you get into musical tuning and all the physics about stopped strings, interval ratios, stacked fifths abd temperament.  Enjoy.


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## millionrainbows

suntower said:


> But there still seem to be gaps in all this (or I'm getting even thicker as I age.) So 'C' is an 8' pipe. And 'C' is related to 'A', which has some cosmic significance according to Boethius or Ptolemy...It's funny that I've spent 40 years 'studying' music and never
> gotten into this crap before.


A lot of this is just plain arbitrary. If you have a logical mind, follow the logic, and forget the rest (after you understand it enough to know that it is arbitrary.)

For instance, the key signature sytem is arbitrary (especially to non-pianists) because it's based on diatonic 7-note scales, and this is related to the physical layout of the keyboard (not the guitar neck, or the neck of a vocalist).

To be a scale, you have to have 7 different letter names, with no repeats. That's why there is no key of "E sharp" or "F flat."

A lot of this has to be cleared up by reading several sources.


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## Taggart

millionrainbows said:


> To be a scale, you have to have 7 different letter names, with no repeats. That's why there is no key of "E sharp" or "F flat."
> 
> A lot of this has to be cleared up by reading several sources.


7 different _consecutive_ letter names ideally with the minimum number of accidentals in the key signature. E# major would go E#, F##, G##, A#, B#,C##, D## which is enharmonically the same as F major but almost impossible to read on a staff. F flat major aka E major can be constructed on a similar basis: F flat, G flat, A flat, B double flat, C flat, D flat, E flat.


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## Guest

millionrainbows said:


> That's just a nice way of saying* "Go read a book!"*


Yup. Why come to TC when you can do it all in private?


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## millionrainbows

MacLeod said:


> Yup. Why come to TC when you can do it all in private?


I see what you're getting at, but music theory is so involved that it's going to take actually reading some books.

Besides, even if the correct information, there's always somebody who will argue with it.


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## millionrainbows

Taggart said:


> 7 different _consecutive_ letter names ideally with the minimum number of accidentals in the key signature. E# major would go E#, F##, G##, A#, B#,C##, D## which is enharmonically the same as F major but almost impossible to read on a staff. F flat major aka E major can be constructed on a similar basis: F flat, G flat, A flat, B double flat, C flat, D flat, E flat.


Okay, seven _consecutive_ letter names, and no double sharps or flats. Happy?


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## Taggart

millionrainbows said:


> Okay, seven _consecutive_ letter names, and no double sharps or flats. Happy?


Almost does it. The "ideally with the minimum number of accidentals in the key signature" is important too.That's why we prefer D flat to C#.

That then leads us to another puzzle. Why is it F# major and not G flat major? Both have 6 accidentals so they should be equivalent. The main reason AFAIK is to avoid another messy major / enharmonic minor split. Remember that for flat keys, the key signature of the relative major is found by adding 3 to the number of flats whereas for sharp keys you subtract 3 from the number of sharps; and when it goes negative, that's the number of flats for the relative minor.

So A flat (4 flats) would have a relative major of C Flat (7 flats) but it's easier to treat it as G# (8 sharps) having a relative major of B (5 sharps); so there is no A flat minor only G# minor. In the same way, D flat (5 flats) would have a relative major of F flat (8 flats) so it's easier to treat it as C# (7 sharps) having a relative major of E (4 sharps); so there is no D flat minor only C# minor.

When we get to 6 accidentals, if it was G flat, it's relative major would have 9 flats  So it's easier to treat it as F# with relative major A (3 sharps) and if we do that, why bother with G flat in the first place?

That gives us that a scale is a series of seven consecutive letter names using the minimum number of accidentals to achieve the major intervals. If there is a case where the number of accidentals is equal, then the version with sharps is preferred because it is more likely to have a simpler minor scale.

Now I'm happy.


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## millionrainbows

Taggart said:


> When we get to 6 accidentals, if it was G flat, it's relative major would have 9 flats  So it's easier to treat it as F# with relative major A (3 sharps) and if we do that, why bother with G flat in the first place?
> 
> Now I'm happy.


But there is a legitimate key of G flat: Gb-Ab-Bb-Cb-Db-Eb-Fb-Gb. It must be there to modulate to from a flat key.


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## Guest

Taggart said:


> [...] *Remember that for flat keys, the key signature of the relative major is found by adding 3 to the number of flats* whereas for sharp keys you subtract 3 from the number of sharps; and when it goes negative, that's the number of flats for the relative minor.


I'm afraid that isn't the case, Taggart. Let us apply your "rule of thumb" highlighted above, taking G minor as our demonstration. G minor is a key with two flats (B-flat and E-flat) and you suggest to find the relative major we need to add 3 _*extra*_ flats? But the relative major of G minor is B-flat (with the same two flattened notes). And to find the relative major of minor keys with sharps in the signature you say we should _*subtract*_ the number of sharps? Again, I must object, pointing out that the relative major of F# minor (3 sharps: F#, C#, G#) is A major (3 sharps: F#, C#, G#).


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## Guest

Taggart said:


> *So A flat (4 flats)* would have* a relative major of C Flat* (7 flats) but it's easier to treat it as G# (8 sharps) having a relative major of B (5 sharps); *so there is no A flat minor* only G# minor. In the same way, D flat (5 flats) would have a relative major of F flat (8 flats) so it's easier to treat it as C# (7 sharps) having a relative major of E (4 sharps); so there is no D flat minor only C# minor.


Again, I must point out inconsistencies in this part. A minor key with four flats is F minor. A-flat minor (if that is what you are referring to in the highlighted part of your quote) has *7* flats in the signature. As to there being no A-flat minor, you need only look to late Beethoven for a couple of examples (if my memory serves me correctly)!
I also refer you to *Max Reger's* slim treatise on modulation where he gives a working of modulating from C major to A-flat minor.


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## EdwardBast

TalkingHead said:


> I'm afraid that isn't the case, Taggart. Let us apply your "rule of thumb" highlighted above, taking G minor as our demonstration. G minor is a key with two flats (B-flat and E-flat) and you suggest to find the relative major we need to add 3 _*extra*_ flats? But the relative major of G minor is B-flat (with the same two flattened notes). And to find the relative major of minor keys with sharps in the signature you say we should _*subtract*_ the number of sharps? Again, I must object, pointing out that the relative major of F# minor (3 sharps: F#, C#, G#) is A major (3 sharps: F#, C#, G#).


I imagine he must have been thinking of parallel rather than relative.


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## Guest

EdwardBast said:


> I imagine he must have been thinking of* parallel rather than relative*.


Yes, I see what you mean. So, Taggart meant to say that in *major flat keys*, to find the _parallel minor_ just add 3 flats. Still, A flat minor does not have 4 flats, whichever way you look at it! In any case, worse things happen at sea.


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## tdc

Ahhhh. Thank you for clearing up that parallel/relative business. I thought my head was going to explode for a second there.


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## Guest

suntower said:


> When I was a kid there was this weird guy with glasses on BBC who did these hour long stories where he'd link together all these seemingly unrelated science things and show they all fit together in a 'fun' way.... James something?


Burke. At my school, almost as oft-talked-about the morning after as Monty Python!



millionrainbows said:


> A lot of this is just plain arbitrary.


Exactly. Didn't need a book.



millionrainbows said:


> Besides, even if the correct information, there's always somebody who will argue with it.


The whole point of TC!


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## Taggart

EdwardBast said:


> I imagine he must have been thinking of parallel rather than relative.





TalkingHead said:


> Yes, I see what you mean. So, Taggart meant to say that in *major flat keys*, to find the _parallel minor_ just add 3 flats. Still, A flat minor does not have 4 flats, whichever way you look at it! In any case, worse things happen at sea.


Quite right. I shorthanded it. My bad! I wrote:



> So A flat (4 flats) would have a relative major of C Flat (7 flats)


Two words were missing

So A flat _minor _(_major _4 flats) would have a relative major of C Flat (7 flats)

and then it makes sense. Sorry about that.


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## Taggart

millionrainbows said:


> But there is a legitimate key of G flat: Gb-Ab-Bb-Cb-Db-Eb-Fb-Gb. It must be there to modulate to from a flat key.


Quite - the entrance of the chorus in Mahler's Second Symphony for example. But you wouldn't use it in a key signature. Interesting to note that Bach used C# major rather than D flat (and C# minor) in WTC (BWV 872 and 873 in book 2).

Having said that have a look at http://www.cisdur.de/e_index.html for pieces in very odd keys.


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## Guest

Taggart said:


> Quite right. [...] My bad! [...] [...] Sorry about that.


No problem, Taggart, we all make mistakes and we got it sorted in the end and as I said, worse things happen at sea! ... But not in my class. The _pedagogue_ in me demands therefore that you write out the keys of G minor, B-flat major, F minor, A-flat major, A-flat minor, C-flat major, D-flat major, F# minor and A major 10 times each over four octaves *and* (I haven't finished, sit down!) submit to me by next Monday 17:00h multiple workings of modulations starting each working in the key of C major and arriving within a maximum of 5 chords at the tonalities listed above.


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## Ingélou

You need to take the _pedagogue_ in you to therapy, TalkingHead - he's a control freak!


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## millionrainbows

Ingélou said:


> You need to take the _pedagogue_ in you to therapy, TalkingHead - he's a control freak!


But from what I've read of his posts, he knows exactly what he's talking about.


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## Funny

Glad the parallel/relative stuff got sorted out (driving me crazy also) but I don't get the basic thesis - that G-flat major is not a legitimate key you would write a piece in, that "you wouldn't use it in a key signature." There are literally hundreds if not thousands of pieces written with G-flat as the key signature. Most if not all of the best-known composers have written movements or standalone pieces in G-flat. There's no problem with it.


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## Funny

Also, was there an answer to that question of why the keys on the keyboard weren't assigned to tones such that the key of A would be the one without sharps or flats?


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## millionrainbows

Funny said:


> Also, was there an answer to that question of why the keys on the keyboard weren't assigned to tones such that the key of A would be the one without sharps or flats?


That's a question of 'keyboard lore' which probably has more to do with the physical layout of the keyboard, which is probably connected to tetrachords or vocal lore.

I've always wondered why 'F' isn't the start point of the circle of fifths, rather than C, for this reason:

As the major scale was built on fifths, then the circle should start on F: F-C-G-D-A-E-B (all white notes=C maj), but it doesn't.

It starts on C: C-G-D-A-E-B-F# (G maj).

In fact, piano tuners recognize this fact, and usually start on F and tune their fifths.


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## Taggart

Funny said:


> Also, was there an answer to that question of why the keys on the keyboard weren't assigned to tones such that the key of A would be the one without sharps or flats?





millionrainbows said:


> That's a question of 'keyboard lore' which probably has more to do with the physical layout of the keyboard, which is probably connected to tetrachords or vocal lore.


Spot on MR but nowt to do with either keyboards or voice. Tetrachords (technically diatonic tetrachords) refer to four (consecutive) strings of a Greek lyre. You also get messed up in Greek musical systems. Since the diatonic system only had tones and semitones (corresponding to the white notes on the piano but *not* derived from them) and starts with A - top string of a lyre - it's only when you get to a mode starting on C that you get a major scale. Since C is a "neutral" key - no accidentals - it's a useful place to think of as the "beginning" of the circle of fifths. But where does a _circle _start?


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## millionrainbows

Concerning *G flat, versus F# *(same pitch, same keyboard location):

G flat is: Gb-Ab-Bb-Cb-Db-Eb-F-and back to Gb.

F sharp is: F#-G#-A#-B-C#-D#-E#-and back to F#.

*They're both 'equivalent' signatures; meaning each one has six flats or sharps; so each one is almost at the limit of its scope. *

The sharp keys, in order, go like this from C (up in fifths): C-G-D-A-E-B-F#-C#.

The next 'logical' sharp key would be up a fifth, but there is no such key as G sharp, because of the limit requirement of seven letter names (no repeats), and no double-sharps or double-flats.

As you can see: G#-A#-B#-C#-D#-E#-*F##*-back to G#. No workie, can't have a double-sharp. Besides, these pitches are already covered by the key of A flat.

Try the other direction, in fourths, counter-clockwise around the 'circle of fourths.'

The flat keys, in order, go like this, from C (up in fourths): C-F-Bb-Eb-Ab-Db-Gb-Cb.

The next 'logical' flat key would be up a fourth from Cb, but there is no such key as F flat, because of the limit requirement of seven letter names (no repeats), and no double-sharps or double-flats.

Check it out: Fb-Gb-Ab-*Bbb*-Cb-Db-Eb- and back to Fb. No workie, can't have a double-flat. Besides, these pitches are already covered by the key of E major, a sharp key.

So why use G flat rather than F sharp, or vice versa?

It largely depends on musical context. If you are already playing in the flat key of D flat, and want to modulate up a fourth, then G flat is the obvious choice.

If you are playing in B major, and want to modulate a fifth, then F sharp is the logical choice.


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## millionrainbows

Taggart said:


> Spot on MR but nowt to do with either keyboards or voice. Tetrachords (technically diatonic tetrachords) refer to four (consecutive) strings of a Greek lyre. You also get messed up in Greek musical systems. Since the diatonic system only had tones and semitones (corresponding to the white notes on the piano but *not* derived from them) and starts with A - top string of a lyre - it's only when you get to a mode starting on C that you get a major scale. Since C is a "neutral" key - no accidentals - it's a useful place to think of as the "beginning" of the circle of fifths. But where does a _circle _start?


Yes, I think that's got it, Taggart. I should have figured this was traceable to the Greeks.

Think of the circle of fifths this way: C as the central node, with (clockwise) sharp keys in that direction, and (counter-clockwise) flat keys accumulating in that direction.

It forms our trinity of subdominant/tonic/dominant: F-C-G.


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## Guest

suntower said:


> C.
> 
> When I was a kid there was this weird guy with glasses on BBC who did these hour long stories where he'd link together all these seemingly unrelated science things and show they all fit together in a 'fun' way.... James something? I think this is one of those deals.


The show was "Connections" with James Burke. I never missed an episode.

That A is not the primal note and scale allows us to deduce that whoever invented the notes and scales, they were not Canadian. Otherwise, it would went:

"Hey, why don't we play something, eh?"

"Okay. What should we play, eh?"

"Something in an easy key with no sharps and flats, eh?

"Okay. What should we call this key, eh?"

"That sounds good to me, eh?"

Well, you get the idea.

And remember that in the days of Ptolemy and those cats, music and astronomy were the same discipline and they believed that the planets "sang" as they orbited because they were scooting along their own crystalline spheres. When you rub your finger around the rim of the crystal glass, what happens? The glass produces a beautiful clear tone. So they surmised that the spheres produced the most beautiful, purest tones ever--if only we could hear them.

So began the quest to find these notes. Now, I'll leave it to you to discover the rest.


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## isorhythm

millionrainbows said:


> That's a question of 'keyboard lore' which probably has more to do with the physical layout of the keyboard, which is probably connected to tetrachords or vocal lore.
> 
> I've always wondered why 'F' isn't the start point of the circle of fifths, rather than C, for this reason:
> 
> As the major scale was built on fifths, then the circle should start on F: F-C-G-D-A-E-B (all white notes=C maj), but it doesn't.
> 
> It starts on C: C-G-D-A-E-B-F# (G maj).
> 
> In fact, piano tuners recognize this fact, and usually start on F and tune their fifths.


I don't think the diatonic major scale really comes from the circle of fifths, which is an ugly fiction made possible by equal temperament.

The diatonic scale can be derived from the tonic, subdominant, and dominant triads, giving you just-tuned intervals of:

1:1
9:8
5:4
4:3
3:2
5:3
15:8
2:1

Now assume some kind of tempered tuning system that reconciles our diatonic scales and our fifths. Take any pitch start moving in fifths in both directions. We find that we can go up five fifths and get only notes in our diatonic scale, but we can only go down one fifth before we hit a note that is not in our scale (Bb, if we're in C).

Charles Rosen in _The Classical Style_ writes that this fundamental asymmetry is the basis of all tonal music.


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## millionrainbows

isorhythm said:


> I don't think the diatonic major scale really comes from the circle of fifths, which is an ugly fiction made possible by equal temperament.


I don't see it that way. First, you have to account for the 12-note octave division, and that was derived from Pythagoras' stack of fifths; and in order to 'close the spiral', or Pythagoran comma, you have to flatten that last fifth, then spread out the error to the other fifths.



isorhythm said:


> The diatonic scale can be derived from the tonic, subdominant, and dominant triads, giving you just-tuned intervals of:
> 
> 1:1
> 9:8
> 5:4
> 4:3
> 3:2
> 5:3
> 15:8
> 2:1
> 
> Now assume some kind of tempered tuning system that reconciles our diatonic scales and our fifths. *Take any pitch start moving in fifths in both directions.* We find that we can go up five fifths and get only notes in our diatonic scale, but* we can only go down one fifth before we hit a note that is not in our scale* (Bb, if we're in C).


I agree with the 'going up in fifths' part of that statement, but going down (counter-clockwise) makes no sense. You should go down counter-clockwise in* fourths, *not fifths, because that's the inversion or compliment of the fifth.



isorhythm said:


> Charles Rosen in _The Classical Style_ writes that this fundamental asymmetry is the basis of all tonal music.


I agree with Rosen's statement, but not by the 'proof' you offered. A fifth is *7* semitones, and its compliment, the fourth, is *5* semitones. This is the "asymmetry" spoken of. Neither 7 nor 5 is divisible into 12; a larger numerator is required: 5x12=60, 7x12=84. This 'outside the 12' factor is what makes tonality 'travel' to other keys.

All the other 6 intervals are *symmetric:* 12/1=*12* (m2), 12/2=*6* (tritone), 12/3=*4* (M3), and 12/4=*3* (m3). (bold indicates number of semitones in each interval)


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## millionrainbows

If you're after *true symmetry,* try the tritone, like Bartok and other modernists did. It is 6 semitones, which divides the octave perfectly in half (2/12=6).

So, yes Rosen is correct; tonality is based on an inherent asymmetry of the diatonic scale, not the symmetry of the chromatic scale.


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## Guest

Aha, time for a little quiz! In common practice harmony, why is the tritone sometimes allowed and at other times forbidden?
Answers on a postcard please, to arrive no later than 19:00 tomorrow!


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## tdc

isorhythm said:


> We find that we can go up five fifths and get only notes in our diatonic scale, but we can only go down one fifth before we hit a note that is not in our scale (Bb, if we're in C).


While I admit to not fully "groking" this conversation (to use MR terminology) I just want to point out that Bb is not a fifth down from C. On the circle of fifths you go down by 4ths twice to arrive at that note.


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## Mahlerian

tdc said:


> While I admit to not fully "groking" this conversation (to use MR terminology) I just want to point out that Bb is not a fifth down from C. On the circle of fifths you go down by 4ths twice to arrive at that note.


It's two fifths down from C (or two fourths up): C-F-Bf


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## tdc

Mahlerian said:


> It's two fifths down from C (or two fourths up): C-F-Bf


You're correct. By "down" I was just referring to moving backwards (or counter-clockwise) on the circle of 5ths.


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## Guest

TalkingHead said:


> Aha, *time for a little quiz*! In common practice harmony, why is the tritone *sometimes allowed and at other times forbidden*? Answers on a postcard please, to arrive no later than 19:00 tomorrow!


No takers? Come on, have a go! 
Little clue: it's about "spelling"? (E.g. "horse" and "hoarse" are spelt differently but sound the same - ditto with the tritone).


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## Guest

millionrainbows said:


> [...] The next 'logical' flat key would be up a fourth from Cb, *but there is no such key as F flat*, because of the limit requirement of seven letter names (no repeats), and no double-sharps or double-flats. [...]


Well, *Max Reger* thinks that there *is* such a key! In his slim treatise "Modulation", he gives workings from C major to F-flat major, A minor to F-flat minor, and A minor to F-flat major! 
And according to Wikipedia, "_The Finale of *Bruckner's Symphony No. 4* employs enharmonic E for F-flat, but its Coda *employs F-flat directly*, with a phrygian cadence through F-flat onto the tonic._"
I'll have to check that last point with the score and Horton and/or Simpson.


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## Guest

TalkingHead said:


> No takers? Come on, have a go!
> Little clue: it's about "spelling"? (E.g. "horse" and "hoarse" are spelt differently but sound the same - ditto with the tritone).


Oh alright then, I can see you're all dying to have the life-changing answer to my question.
MillionRainbows is right that the tritone comprises six semitones whether up (B-F ascending) or down (B-F descending), but the former makes a *permitted* melodic (horizontal) interval of a *diminished 5th* and the latter a *forbidden* melodic interval of an *augmented 4th*.
What's more, if you have a jump of a diminished 5th, the next note should be within that 'envelope', so to speak.
OK, we can now all get back to our mundane lives ... Phew!


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## Taggart

TalkingHead said:


> ... tritone comprises six semitones whether up (B-F ascending) or down (B-F descending), but the former makes a *permitted* melodic (horizontal) interval of a *diminished 5th* and the latter a *forbidden* melodic interval of an *augmented 4th*.


Technically the inversion of a diminished fifth is an augmented fourth.

You will meet tritones as diminished fifths in diminished 7 chords and in dominant seventh chords you will also meet them as augmented fourths in augmented sixth chords (usually called Italian sixth, French sixth, and German sixth). The Italian sixth is enharmonically equivalent to an incomplete dominant seventh. All of these are acceptable and all can be resolved in common practice harmony.


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## Guest

Taggart said:


> Technically the inversion of a diminished fifth is an augmented fourth.
> 
> You will meet tritones as diminished fifths in diminished 7 chords and in dominant seventh chords you will also meet them as augmented fourths in augmented sixth chords (usually called Italian sixth, French sixth, and German sixth). The Italian sixth is enharmonically equivalent to an incomplete dominant seventh. All of these are acceptable and all can be resolved in common practice harmony.


This is not what I am saying, Taggart. You are talking "vertically" (harmony) and I am talking about their horizontal aspect (melody). In CP melodic practice, augmented intervals are forbidden. What I thought of interest is that "spelling" what is aurally the same can be viewed as forbidden or permitted.


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## Guest

It is the same with an augmented 2nd (let's say E-flat to F#) which is more likely to be heard (in isolation) as a minor 3rd. In *writing* a CP melody, leaps of the augmented 2nd are forbidden, whilst leaps of the minor third are not. Such errors of _écriture_ are easily remedied with an opportune accidental or note change here and there.


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## Taggart

TalkingHead said:


> Aha, time for a little quiz! In common practice *harmony*, why is the tritone sometimes allowed and at other times forbidden?
> Answers on a postcard please, to arrive no later than 19:00 tomorrow!





TalkingHead said:


> Taggart said:
> 
> 
> 
> Technically the inversion of a diminished fifth is an augmented fourth.
> 
> You will meet tritones as diminished fifths in diminished 7 chords and in dominant seventh chords you will also meet them as augmented fourths in augmented sixth chords (usually called Italian sixth, French sixth, and German sixth). The Italian sixth is enharmonically equivalent to an incomplete dominant seventh. All of these are acceptable and all can be resolved in common practice *harmony*.
> 
> 
> 
> This is not what I am saying, Taggart. You are talking "vertically" (*harmony*) and I am talking about their horizontal aspect (*melody*). In CP melodic practice, augmented intervals are forbidden. What I thought of interest is that "spelling" what is aurally the same can be viewed as forbidden or permitted.
Click to expand...

Umm .........

..................................................


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## Guest

Ah, I see what you mean! Still, worse things happen at sea, eh Taggart? But not in your class, right? Am I to write out the various Augmented 6ths (Italian, French & German) in all keys (_à la _Max Reger, for the love of Vishnu!) for four voices on two staves or four? By tomorrow, 17:00?


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## isorhythm

Augmented intervals do occur in common practice melody. I can think of a few places where the harmonic minor scale shows up in the melody and that contains an augmented second. There are probably others.


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## millionrainbows

Alas, these tritones! But there is hope! They can be resolved into a flat seventh/major third!

It's invertible, thus the saying, "Once a tritone, always a tritone!" 

...and "The only good tritone is a resolved tritone!"

"Those damned tritones! All they ever did was pave the way for vagrant harmonies, flat-nine dominants, and endlessly cycling sequences of V7-I's and the eventual weakening of tonality!"


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## suntower

So, through the magic of Inter-Library Loan, I got Bruce Haynes' book. And what a page-turner it is. zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz. I've read tax instructions that were more gripping.

There's rather a lot about cornetts and organs and Venice v. Northern German organs... but here's what I got out of it...

The whole deal is this:
1. Originally, all instruments were tuned to their choir. There was a certain tessitura to the voices of each well-known choir and they would tune the sackbutts and organ pipes to suit. (Apparently the Roman choirs sang higher than German choirs.)

2. 'A' is 'A' because (I -think-) there is a vaguely 2 octave range of the male choir. 'A' was the average lowest note of the choir singers so the 'A' pipe of the organ was tuned to that. And that's around 415. So basically most things were sung with that 'A' as the 'tonic' for a long time. This kinda makes sense to me as a general baritone voice goes down to (modern) G-ish... which I suppose was 'A' back then.

3. Also, apparently there developed a certain technique for manufacturing 'cornetts' in Venice which became 'standard' (I guess because choirs in Italy tended to be pitched up to a minor 3rd higher than Northern German choirs?) And so that became another de-facto tuning standard. This tended to be higher than the vocal standard 'A' which caused confusion, but since it -was- more 'standard' ie. repeatable, it gradually began to take over since one cannot transpose natural horns. And that is close to the 440-465 in use today. So choirs and organists transposed.

4. There are many discussions about the timbre of various frequencies. ie. they were very aware of the difference in tone between a choir or instruments playing/singing @ 415 vs. at 440. And they seemed to prefer the lower versions... thinking they had more 'resonance'.

Or this could all be rubbish.  But it was an interesting journey. Not sure it's worth slogging through all that detail, but... interesting.


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## millionrainbows

suntower said:


> There's rather a lot about cornetts and organs and Venice v. Northern German organs... but here's what I got out of it...
> 
> Or this could all be rubbish. But it was an interesting journey. Not sure it's worth slogging through all that detail, but... interesting.


No, not at all. This is a very informative and interesting summation. These subjects are also touched on in* How Equal Temperament Ruined Harmony (and Why You Should Care)*. Bach was known to carry several keyboards around, for playing with choral groups or with violins.

The lower tuning standard may have been somewhat a necessity as well as being resonant, especially with string tension on violins. Tell us more if you've got it.

But yeah, the major third is the interval that suffered most from the stacked fifths/12 note octave division, and choirs would tend to want to sing those major thirds flatter. Our ET maj 3rd is a full 14 cents flat~! Violin players did this, too...in fact the book shows alternate fingering charts for violinists, which changed according to what key they were playing in.

I've noticed lately, when I hear organ music, that some of those old cathedral organs are tuned rather idiosyncratically. Theree's probably some experts on that subject around here.


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## Che2007

How an old organ is tuned depends on quite a few factors. Sometimes there is a tradition on a particular organ. Sometimes they are trying for an accurate representation of how an older organ would have been tuned. Sometimes it depends on the repertoire that is going to be played on the organ. Often you will hear meantime organs (where the syntonic comma is spread across the 5ths to give perfect Major 3rds). Sometimes you will hear a well temperament (Where the diatonic comma is spread unevenly across the 5ths so you have variable qualities of 3rds). Sometimes you will hear 12EDO and it will sound strange because of the timbre of the instrument. It is all very variable between instruments.

As for the question of why C is C: when letter names where given to the greek pitches that Boethius had transmitted C landed is a position where the major scale mode (ionian) started. There are a few reasons that major has historically been given prevalence over minor. 1) it is invariant i.e. it does not require some scale degrees to be sharpened to function tonally. 2) The major triad appears as overtones 4,5,6 in the harmonic series so these intervals have been seen as directly intelligible. 3) Often in the baroque period a minor piece would be concluded with a major chord (picardy third) as though the satisfaction of a major triad somehow gave completion to the piece.

Why we have a particular reference pitch is basically down to shared convenience. The greeks started naming pitches in reference to their relative position within the greater and lesser perfect systems. These systems really are there to encompass the pitch range a performance would entail so each pitch name gets associated with a particular area of pitch (really in reference to the human voice). From there the route to finally agreeing what precise pitch a note name refers to is a pretty clear path. It is about being able to communicate to a performer what you want!

Hope this helps!


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## millionrainbows

Here is another question: why are there only seven letter-names for notes, yet there are twelve pitches?

That must be one of the implicit structural features of tonality.


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## Taggart

millionrainbows said:


> Here is another question: why are there only seven letter-names for notes, yet there are twelve pitches?
> 
> That must be one of the implicit structural features of tonality.


Going up in fifths (the 3:2 ratio) gives 6 of the 7 notes. The other is the fourth - the 4:3 ratio. That's enough to give you the modes on the white keys. The idea of twelve pitches comes later when accidentals were added to the modes and also when you transpose a mode to a different starting note where you need accidentals to preserve the intervals.


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## Mahlerian

Additionally, it wouldn't have been considered true that there were only 12 pitches before equal temperament, and you have things like Nicola Vicentino's keyboard that included separate keys for D# and E-flat.


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## Guest

Mahlerian said:


> Additionally, it wouldn't have been considered true that there were only 12 pitches before equal temperament, and you have things like Nicola Vicentino's keyboard that included *separate keys for D# and E-flat*.


I'll have to dig the right source out, but there is a Haydn string quartet where in the cello part there is an E-flat and a bar or two later a D#.


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## tdc

Taggart said:


> Going up in fifths (the 3:2 ratio) gives 6 of the 7 notes. The other is the fourth - the 4:3 ratio.


It seems when I visualize what you are talking about, one would have to be a sharp 4 (or flat 5), unless I'm missing something here.

ie. C-G-D-A-E-B-F (5th,5th,5th,5th,5th, flat 5)

*edit -* perhaps you are referring to the fact that the last note is a fourth away from the first?


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## Taggart

This is nothing to do with note names. If the original string is of length 1, then a string of length 3/2 will sound a fifth up. This is generally felt to be the most consonant interval. Going up in fifths means increasing the length of the string by 3/2 at each stage i.e.

3/2 3/2^2 3/2^3 etc using ^ to mean raised to the power.

This gives the sequence 

1 3/2 9/4 27/8 81/16 243/32

except for the first two, the rest go beyond the octave (doubling the string length). We can bring them back within an octave by dividing by 2 (repeatedly) .This gives

1 3/2 9/8 27/16 81/64 243/128

putting them in numerical order we have 

1 9/8 81/64 3/2 27/16 243/128

The gap between 81/64 and 3/2 is larger than the rest so we fill that gap with a fourth - 4/3 ratio and add the octave to give a (C major scale) of

1 9/8 81/64 4/3 3/2 27/16 243/128 2

This all based on ratios of string lengths and is made up of fifths, fourths and the octave.

Notice that a tone gap is in the ratio 9/8 (1.125) and a semi-tone gap is in the ratio 256/243 (1.05). Two semi-tones would mean increasing the length of a semi-tone by 1.05 - about 1.11. This is not quite the same as a tone which is why you can get different values for E flat and D# in non-equal temperament.

So the statement "going up in fifths" means adjusting your string length to produce notes which are (basically) a fifth apart and then filling in with the fourth (4/3 ratio) and octave (doubling). You then label the resulting pitches with names or solfege or whatever.

The difference between tone and two semitones then explains why we don't (strictly) have 12 pitches.


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## millionrainbows

tdc said:


> It seems when I visualize what you are talking about, one would have to be a sharp 4 (or flat 5), unless I'm missing something here.
> 
> ie. C-G-D-A-E-B-F (5th,5th,5th,5th,5th, flat 5)
> 
> *edit -* perhaps you are referring to the fact that the last note is a fourth away from the first?


Yes. he's talking about a fourth from the first note, C, in the other direction. If you wanted all white notes, you could start on F: F-C-G-D-A-E-B. That gives you a Lydian scale, though. But that's what George Russell used in his Lydian Chromatic Concept. He maintains that the Lydian scale is more natural and symmetric, the aug 4 being a 'leading tone' to C, the dominant station.

Pentatonic scales are more consonant than diatonic scales because they are also built up by fifths, only they stop at C: C-D-E-G-A. It is more consonant not only because it has fewer notes, but also because it does not have the fourth degree F.

So the "F-C-G" formula of IV-I-V has an inherent tension, because of that F.

This guy says you can hear the difference:


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## millionrainbows

millionrainbows said:


> Here is another question: why are there only seven letter-names for notes, yet there are twelve pitches?
> 
> That must be one of the implicit structural features of tonality.





Mahlerian said:


> Additionally, it wouldn't have been considered true that there were only 12 pitches before equal temperament, and you have things like Nicola Vicentino's keyboard that included separate keys for D# and E-flat.


That's the main reason I'm here: to spread lies and untruths! :lol:


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