The mess

This page shows how we have arrived at the point of today's musical system. To understand the need for my system, you will need to really appreciate how hopelessly inadequate the existing system is for the purpose of representing music.

Beauty and the beast

A musical system should reflect the elegance and beauty of the music it depicts. Here is an excerpt from one of my favourite classical pieces: Chopin's Etude Op 10 # 3 in E Major.

This is an unholy mess. A total disaster. Just look at it! Here is the actual Chopin study played by Murray Perahia Have a listen, put it in the background while you read on. Ask yourself how such beautiful music can have such a hideous representation.

In this page I am going to dig into the internals of musical theory. I'm going to show that today's conventional musical system is a botch. I contend that any attempt at musical representation is going to be a botch, and given that this is the case, we would do well to choose the most elegant and simple botch that represents music accurately enough to infer the original musical meaning.

A brief musical history lesson

From the origins of music, one common feature always emerges: there are seven notes in the scale. This is known as a diatonic scale.

external image Pythagoras_with_bells.png
Pythagoras was maybe the first person to figure out some mathematical theory of ' what sounds good '.

He worked everything using the numbers 2 and 3. so take a bell and call it F. A bell {half/twice/etc} the size gives F an octave {higher/lower/etc}. But a bell 3/2 times the size gives a new note -- not an F (a C actually -- a perfect fifth up from F). Now do the same thing until you have 7 distinct notes, these we place in order and label A B C D E F G. (the actual order of generation was F -> C -> G -> D -> A -> E -> B).

Of course, people had figured out seven note scales well before this, but this was the first attempt at a tuning system.

But it is a very limited musical system. When we sing we hit all sorts of notes that can't be described using this mathematical framework. In fact nobody has produced a successful theory for representing music, just as nobody has produced a Grand Unified Theory of physics that successfully represents the world around us. My intuition is that no one will. The universe is playful. The analogy is far reaching. Our laws of physics are useful, with then we can build cars, mobile telephones and fusion reactors. But one cannot understand existence through these laws. In the same way musical theory can give some useful tools, but it will never reach the root.

Our ears and brains like ratios. But we are not limited to 2 and 3. Plenty more prime numbers are available. 5, 7, 11, ... when we sing freely in an unconstrained way, free from all conditioning, singing as the bird sings, there is no musical system that can represent this with complete accuracy. And consequently no amount of studying musical theory will open the gateway to musical fluency.

Forget Pythagoras for a moment, let's start with the earliest written Western music.

Gregorian Chant

The diatonic (7-note) scale was fine for basic melodies. it was fine for the Gregorian monks to sing their plainchant.

external image b-bruggs-d7-99r-kyriepaschale.jpg

Can you see? This is the origin of modern musical written representation. Notes on a stave. This representation is perfect for this music. ie diatonic music that does not change key, ie does not modulate. Just keeps using the same seven notes, like church bellringers. Look how beautiful the representation is, how clean. Compare that with the Chopin study above.

What has happened since then is that music has evolved, and this basic framework has struggled badly to cope with this evolution. It is as if the child has grown to become a man, but is still squeezing into the same old tricycle.

Let's look at the first step towards this disaster.


Let's say you are playing your Gregorian chant, your seven notes are: C D E F G A B. The most common pattern is to revolve around C -- to start and finish on this note. But you could equally well start and finish on any of the other notes. This is known as playing in different modes. There are seven modes.

But how about you pick one of these notes, say G, and construct a new scale starting at this note, but of the same form as the original scale. you will find you have a new set of notes, most of them are either spot-on or very close to one of your existing seven notes. G A B C D E. Only one is substantially out. It lies some way between F and G. So you can think 'we have raised the F', and it has now become F#. So we are still using each of our seven symbols once.

And we can botch our musical stave to make a note that the F is sharpened now, to indicate we are in the key of G.

external image G-Major-Key-Signature.jpg
But already the notation is lying. All of those other notes were not identical. The 'D' in a C-Major scale is not necessarily the same as the 'D' in a G-Major scale. A singer will sound these notes differently, if they sound the same, it will be artificial. On a piano keyboard it is the same note, but you may have noticed that simple melodies sound a little bit bland on a piano keyboard. Even the beautiful richness of tone cannot make up for this. It is a compromise.

That's why I threw that bit in about Pythagoras above. You can only make so many church bells, you can only fit so many keys on a piano, so a huge amount of effort went into constructing botches, systems using a finite number of notes that let you play with more and more musical freedom. All sorts of crazy contraptions came out. Remember, this is before the time of the piano keyboard. If you are singing, F# will be at a different pitch to Gb.

The Well Tempered Klavier

Well Temperament is a process of fudging all of the note values so that for example F# and Gb are pushed together into the same note. it is also known as ' circular temperament ' because it brings the keyboard ( in terms of harmony ) into the form of the circle -- you can modulate forwards and backwards to your hearts content, and you will still be only using the same 12 basic notes.

J S Bach (above) was one of the first to play around on this new musical technology.

Back to that hideous looking Chopin score at the top of the page, clearly Chopin is modulating all the way round the circle. Round and round. And the notation is hopelessly unable to keep up.

12-TET -- today's tuning system

Ultimately, someone figured out a mathematical construction, a method of tuning these 12 notes so that every key is equivalent to every other key.
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Basically to get from C to C#, you multiply its frequency by the 12th root of 2. And from C# to D, the same. etc. Do that 12 times, and you have multiplied your original pitch by (12th root of 2) to the power of 12, otherwise known as 2. ie you have doubled the pitch. ie you have gone up an octave. You are back on C again, just an octave higher.

Much musical beauty is lost by playing in this system, all of the crispness of for example Vivaldi would be lost in this system. None of the intervals are perfect ratios. Nothing sounds perfect. But then nothing sounds horrible. It is a decent compromise. This is the tuning system that modern musical instruments are tuned to.

However, if you listen to a rendition of Beethoven's violin Concerto, I can guarantee you that the violinist (here David Oistrakh) will be playing free from any of these tuning systems, using the violin as an extension of his voice, while many of the fixed pitch instruments (like the oboe) will just have to approximate as best they can in their fixed 12-TET tuning. I throw one more in: A Beethoven sonata for violin and piano. If you analyse the pitch of each note, I guarantee you will find that the violinist is not playing in 12-TET. But the pianist has no choice.

Wake up!

If you haven't actually fallen asleep by this point then well done! This is the kicker coming up. Get ready for the logic.

So, we can express any music we wish ( albeit not completely accurately ) on a modern piano keyboard, which uses 12 notes.

So why not construct a musical system based on 12, rather than the existing one, which is based on 7.

Musical tuning systems have made a quantum leap from seven ( the diatonic scale ) to 12 (12-TET).

It is time for the system of conceptualisation to also make this quantum leap. If one leg jumps over the crevasse, the other must also make the jump, otherwise the result is not going to be pretty.

remainder of this page needs to be tidied up / rewritten / moved around the website...

However, this system is not an accurate representation of music, any more than newton's theory of gravity is an accurate representation of the universe around us.

later research ( 15th century ? ) showed that music is basically a play of ratio, play a string together with a string 2/3 of its length, and you will sound a very beautiful interval, known as ' perfect fifth '. all of music can be examined in this way: ratios. however, this leads quickly to uncharted ground. It becomes practically impossible to ( maybe read that as impossible to practically ) express a complex musical passage in writing. and impossible to accurately represent it using instruments such as piano, guitar, etc that only have a finite number of notes.

various tuning systems emerged to attempt to provide a good compromise that can express a musical utterance tolerably using a finite number of notes. meanwhile the system of written representation did not advance from its origins in the Pythagorean tuning system. instead of creating new systems of representation, the existing system has been repeatedly botched, in much the same way as an old typewriter keyboard has been repeatedly botched into the modern computer keyboard you see today.
finally a breakthrough came: 12-TET. a tuning system that allows a tolerable approximation of pretty much any musical concept. I haven't heard any music that stimulates me that cannot be expressed reasonably well on a modern piano. that is what modern piano is tuned to, by the way. and any other modern fixed pitch instrument. and tuning forks etc. just A is fixed at 440Hz, and all the other notes are derived from this frequency.

yet still the written notation is based on the Pythagorean tuning system.

my musical system is based on 12-TET. just 12 basic building blocks. it makes sense. I will attempt to cover this ground from several angles, so I will be repeating myself, so that you can build a neural network of the task in hand.

I am following on from the work of Evelyn Fletcher-Copp, I think I may have mentioned her before, she is an inspiration! her methods for teaching music to children are wonderful, it's really sad to look up from her book and see the blind leading the blind everywhere ( there are few schools that teach her methods though ).

but I am going further. I'm actually reworking the musical system. the fact ( as far as I can see ) is that musical expression doesn't come from the theory, any more than the universe works around our so-called laws of physics. so my theoretical basis is as bare and minimal as possible.

everything came out of joint a few hundred years back with 12-TET tuning, this permitted modulation around the circle ( infinite modulation as opposed to finite, a circle as opposed to a line ). but the notation still reflects the old tuning systems.

ie in the notation F# and Gb are distinct, but on a piano keyboard they are the same note

in my system I represent them with the same symbol

now they are actually different pitches if you sing them. because the modern tuning system is a general compromise on everything, it is just approximating. and both of these notes can be approximated by the same black note on the piano between F and G.

but the piano can successfully approximate a piece of music that contains both F# and Gb, and the alert musician can be aware from the context which is which -- if they then go away and and sing the melody, they would have produce these notes correctly (ie distinct from one another).

and in exactly the same way my musical system approximates. it uses 12 pitch classes, and every musical note maps onto whichever pitch class approximates it best. it is a mathematical marvel that this is possible. the piano has used it. the notation should also use it. and in exactly the same way a skilled musician would realise that the same symbol that matches F# will now match Gb as we move a complete revolution clockwise around the ring. ie the actual note can be inferred from the context.

this means that we can throw away the ludicrous note naming system in use today.

for example, in the existing notation, the scale of C comprises C D E F G A B.

for a start, why does it not begin on A? this is straight away illogical and confusing for a beginner.

next, let us say we modulate to G. now the scale comprises G A B C D E F#. so conceptually we are viewing this scale from the point of view of C. as a modified C- scale ( only the F is modified ).

even at this point there is a problem. it is only in the Pythagorean tuning system that ' only one note is modified '. this musical notation appears to come from this tuning system. in Just Intonation ( or in the tuning system that is natural to our voice ) most of these notes are going to be modified. Only the G and the C are guaranteed to remain the same. [C, G] is a perfect fifth, so in our new key of G, [G, C] will be a perfect fourth, all of the other notes will be different.

so it is a very misleading notation. it makes us think these notes are the same. but it is a false representation. if you are singing, the 'D' which is the {second note / super tonic} of C Major is going to be at a different pitch from the 'D' which is a { fifth note / dominant} in G Major.

but if you are on a piano they will be the same, because the piano is tuned with 12-TET, which is a set of 12 equally spaced pitch classes, which can approximate pretty much any musical interval to a decent degree, but doesn't get anything spot-on. ( you may remember that simple tunes like three blind mice sound particularly bland when played on a piano, as opposed to being sung by someone who is musically in tune with their own voice )

now if the same symbol eg 'D' is not specifying a unique pitch, why not just do away with all of the sharps and flats, and revert to notational system based around the 12 notes on a piano?

it is unbelievable to me that nobody appears to have done this. No amount of digging has uncovered any such work. It is such a simple step forwards.

I will say everything multiple times, coming from different angles, so that you can understand what it is I'm doing.

So in the conventional system, 'D' is a placeholder, an approximation, it is up to the musicality of the musician to find the appropriate pitch for it. if she is a singer, she will pitch it differently depending on whether she is singing in C or G. so the conventional system requires interpretation.

so consider a system comprising simply the 12 pitch classes found on a piano. pretty much any music, classical or modern, Western or Eastern, past or present, can be approximated on the piano so well that a musician hearing the recital could go away and play the music as it was originally intended on a non-fixed-pitch instrument, such as a violin, or their voice.

my musical system accomplishes the same. it requires interpretation. I think you'll find ANY system of conceptualisation is going to require interpretation. Nobody has cracked the mathematics of why music sounds beautiful. Maybe nobody ever will. Maybe this is a good thing!

so the existing system is a compromise. any system is going to be a compromise. And if we are going to compromise, we should make it as clean and simple as possible.

the existing system carries with it a theoretical basis that was founded on Pythagorean tuning system. it carries outdated logic from the Greeks. ( when, in Pythagorean tuning system, you modulate C->G, introducing the F#, where is that F# coming from? Can you see that to a Greek, there is no disparity -- all of the other notes match up, only one note is added -- today's musical system will make perfect sense -- a 'D' in C major is the same note, the same pitch, identical to a 'D' in G Major, that is why it uses the same symbol. but this tuning system was found to be wanting, to be a polar mathematical description of music, just as Newton's laws of physics have since been found to be a poor approximation of physical reality. ) it is not suitable working basis. just as the Greeks refuse to accept irrational numbers, and their mathematical system of the rationals have since been superseded, it is high time to supersede the musical system

notice also that in the conventional musical system, if we are playing a piece of music in G, notationally we must think in C, and write everything in terms of C, modified. the actual staves themselves are intended to depict a C scale. you then modify these staves by adding a # into the key signature over the F note. this suggests the F has been ' sharpened ' or raised. this is conceptually outdated. For a Greek, using Pythagorean tuning system, this is true. All of the other notes are the same. Only the F has been raised. but look at the frequency readouts from a trained singer, and you will see that ALMOST EVERY note has in fact been modified.

Now, it is a simple musical device to hint at other keys, two briefly modulate for a few bars, or even a fraction of the bar. and rather than writing a key signature for such a small interval, notes are flattened, sharpened or naturalised. this makes for the world's biggest headache, attempting to decipher this mess.

as we modulate further, the scale becomes a more and more 'bent out of shape' C-scale, until all of the notes have been modified (C# Major = C# D# E# F# G# A# B#) and then if we wish to go further, some note must be doubly modified. so G# major will have an F##. some composers have kept going to the point of four # signs. at this point the music score looks an unholy mess, a forest of symbols. open up a Rachmaninov piano concerto on any page.

I cannot express succinctly the shortcomings of such a complicated and botched musical system. The best I can do is to present succinctly a musical system that will effortlessly express musical ideas in such a way that a child can easily assimilate it.