The mesures of the tones (a theorical study)

Ok... let's work a bit for those who wish to learn the nature of the musical pure tones. The chart I give here has been taken from my works for my opera... so it's not a 'ready thing' to explain... but it will help more than just writing text.

* * *

Briefly... you got two axes starting from a D.

The horizxontal axis shows the cycle of fifths... the position of the cycle has been choosen to work well with the open strings of a string section... you got C, G, D, A, E all in the same chart line.

The vertical axis shows an already more complexe cycle of pure major thirds. Like in Db, F, A, C#, E#.

* * *

For each note appears in descending order : the frequency ratio, the note name and the cent value.

* * *

There a multitude of grids and information to retreive from that chart for I postulate that probably all music (played with intervals) can be understood through that chart... but let me do my post-doctorate on the subhect before going further ! eheh The information shown in that chart is the symetry of twelve pure tones at the distance of one comma (G#(569), G#(590), Ab(610) and Ab(631)). Then again, it's only one information that is read here !

* * *

So the first step here, is to memorize the cent values of all the most centered ratios from D#(25/24 - 71c) to around Eb(27/25 - 133c). Memorizing is helpful for any transpositions... even though I'll teach you some tricks eventually ! eheh

* * *

Second thing to analyse is drawing the shape of the chords (like a major chords D-F#-A making a L for example). You will see that everywhere you draw the same figure, you'll get a major chord... it's a the for anychords... even jazz chords that can be seen as pure as they can be. You'll be able to analyse the uniqueness of the dominant 7 chords with that chart... etc.

Have fun !

Chart - fifths by thirds cycles.pdf

Doo-ba!

D#(25/24 - 71c) is grooving real hard.

Plus, I can do nice drawings with a Calt/Balt polychord!

Hey, why not apply mathematical operations on notes!

G#(569) substracted of Ab(631) would give a (-62) note.

Oh damn, do you realize?!

There are negative notes!

Wait, let's do more...

We could do "functional" songs!

F(x)=3arcsin(x/3)

x being every note

F(x) being the pitch of this note

So... we could reduce our piece dy "deriving" it (sorry, I don't do English maths)... that would give us F'(x)=1/(sqrt(9-x^2))

Maybe I could take this for my "B" section...

Yeah. Doo-bop!

Awesome. But I have a question. You explained earlier how the ratio is the beginning tone in Hz values under the final tone in Hz. For this chart, are you saying that they all are starting from D? It looks like that, but I want to be sure.

OK, so another 'before we get started' sort of question. How did all of this start, anyway? You find different ways to get to notes using the ratios, ending up with different tones for the same notes, but, from what has been explained to me, it sounds like we know what the notes are because of the Hz, and we know the Hz because of the ratio, and we know ratio because of the Hz. Where does this begin? Hope I am making sense. I just want to have it strait in my mind before going on. Perhaps a little history.

Thanks.

Colin Thomson

Well... things are probably more simple than you are imagining them now... the ratio is the same thing as the value in Hz... except that to create a ratio... you need at least two sounds... If you have more sound, then you create a chord and you can notate it like 5:3:2 (a major chord). But I have to confess that I would need to revise these chords stuff before explaining furthur for I never use them.

To answer your question... everything in pure tones harmony (or Just Intonation) start from a reference tonic (effectively D in my chart). Just as all musicians tune themselves on the A of the first oboe in an orchestra... it's the oboist that determine the pitch of the ensemble.

So the ratio aren't tied to specific pitches... but to a relation between them... besides... the Hz value expresses a precise et defined pitch that will create relations only if at least one other pitch is heard.

For the history stuff... I don't know exactly what you want to know... a pitch history would be something awfully long to make here ! ehhe

* * *

I shall explain the difference between harmonicity and symphonicity eventually when you think you understand the chart given there. :blush:

OK, I think I am beginning to understand. The chart helps a lot.

Colin Thomson

:musicwhistle: I'll come up with more as soon as Time allows me to work. :P

I look forward to it.

Colin Thomson

Well, meanwhile... try to look for the patterns of all the chords you know inside the grid... look at their shape, analyse them, analyse their relations... try out to find how transposition mechanisms work and all... in that grid is the key to understanding tonality, atonality and microtonality !... if you master it... you master harmonicity and symphonicity (verticality and horizontality of music pitches).

There's a new version of the chart !... about harmonicity and symphonicity... I explain in my personnal and non-official nor perfect definitions :

Harmonicity : is the quality of two or more pitches heard at the same time in terms of beating (sonance : located between pure consonance (unisson) and pure disonance (white noise)). These pitches show in that chart are those with the less beatings and since the interval is calculated from the tonic (1/1) the interval of pure fourth (4/3) stands like a dissonance... but in practice it can be used as a consonnance in chords that aren't in root position (first inversion).

Symphonicity : by that term we refer to what the Greeks from Antiquity where using to say 'the sounds that goes well one beside the other'. So it's the sound that you can put one after the other (horizontaly) as in a melody writing.

* * *

Note that the more the symphonicity is complexe, the less complexe is usually the harmonicity... and the reverse is also true. In occidental music with a very rich harmony we can a very 'childish' use of the symphonicity comparing to the Indian or Arab music... or even Antique music that was only for one voice.

* * *

Of course... if you write a melody with the symphonicity stuff... don't forget that with a new note... a new tonic (1/1) is defined... thus if you melody is D, F, A... so D is the first tonic... then F becomes the new (1/1), and so on. I'll explain more later for now I'll be late to work !! eheh :D

Cheers and have fun !

Harmonicity and Symphonicity.pdf

Dunael, I am confused why are the pitches D# and Eb have different ratios. Are they enharmonic in the language of reading music and not in the language of science, or are the numbers(71-112) different for some other reason? Please enlighten me I find this subject very interesting but am having a hard time understanding it.

Well, this is far more avanced theory of music than you study at university. But if you read any good music theoricians, they will say exactly the same thing as I. Music theory that is teached at school is has been simplified a good deal and is based only on the results of long researches by theoricians of the 16th-19th century about egal temperament. The dream (that now has become a kind of rule-without-wit) of the egal temperament was actually to be able to make fit a broader range of tonalities through a minimum of keys (12 actually)... but any non-keyboard player of any talent will tell you that they never play exactly on the same pitch depending of the notes context... and this famous 'context regulated by the ear of the instrumentalist' has it's own rules which are demonstrated in my charts.

So that's why if you play a G (498) and want a pure major third below you need the Eb... but if you pass after that from G to it's pure major third higher which is B (498+386 (M3)) = 884 and if you wish to modulate in B sharp after that for example.... then you need the major third of B which isn't the Eb but really the D# (for 884+386=1270... to which you got to remove one octave (1200c) to understand more easily and get 70... which is the 71c)... of course... that process would need very far modulations... but Listz and Wagner have done so (and others).

Just to remember so you understand better... a pure third (the only one that will really sound just) has a value of 386c and this is physical... it's a question of harmonic distance in the nature of the sound. :P

Hope it help... it's not something to be grasped instantly... so don't worry if it's still a bit hard to get ! ;)

Can you give examples of where Liszt and Wagner (and others) have done this?

I just want to hear more of it put into practicality.

Thanks.

Colin Thomson

Well, any examples of foreign modulations is good... I'll try to devise some example as soon as I can ;-) I'm going to hear the Pierrot Lunaire tonight ! :)

Dunael, I am confused why are the pitches D# and Eb have different ratios. Are they enharmonic in the language of reading music and not in the language of science, or are the numbers(71-112) different for some other reason? Please enlighten me I find this subject very interesting but am having a hard time understanding it.

Eb and D# are NOT the same note, EVER, except for the godamned piano :huh: