Because the master mathematician Pythagoras says so! The ancient Greeks came up with the pentatonic scale - just 5 notes - that are mathematically related. This actually comes about through the harmonic series created by any vibrating source, air column or string. As this series extends into the 4th octave, most of the notes in our scale are included, in their mathematical purity. Those pitches, brought back down to the original octave, undergo some adjustment in order to blend within that octave. All other octaves contain those same notes, each one of which is twice the frequency upward, half the frequency downward. So, in chords, some combinations sound better than others because their component notes are more closely related to the harmonic series. Notes included in more "colorful" chords are less related, or from further up the harmonic series. In order to have quarter-tones relative to the 12 notes in the octave, you would need to look at (or be able to audibly discern) notes in the 6th and 7th octaves of the harmonic series. Even taking the 12 notes that we have, and mixing them in uncommon ways, was tried early in the 20th century - 12-tone or atonal music. It was an experiment that pretty much failed. All the composers who had tried to compose within that system gave up on it because they had to feed themselves and their families! IOW, nobody liked it - except maybe the hardcore academicians!

Come to our church, and hear all the microtones you want. It's Byzantine chant.

D'après mon avis, la pentatonie a ses raisons d'être vraiment harmoniques, mais la manière dont vous remplissez les petites tierces entre la suite de deux tons et la suite des trois tons, est très variable et donne lieu à plein de possibilités dont les solutions combinant des demi-tons et des tons entiers sont une solution parmi d'autres.
Cela sonne naturel de procéder par tertra-cordes, mais entre les tetra-cordes?

Meiner Meinung nach hat die Pentatonik ihre harmonische Daseinsberechtigung. Die Art jedoch, die kleinen Terzen zwischen der Zwei-ton und Drei-tonfolge zu füllen kann sehr unterschiedliche Lösungen finden, und dei Kombination von Halb-ton und Ganz-ton ist hier doch nur eine unter vielen (Lösungen).
Es klingt natürlich, von Tetrakord zu Tetrakord zu singen... aber zwischen den Tetrakorden? Ganz-ton? Halb-ton? erhöht nach oben, vertieft nach unten?

Kathy, it's NOT the smallest interval. I know several singers in choirs who consistently demonstrate that there ARE smaller intervals. We call it 'flat' or 'sharp.'

So Francis , your piece is not based on 12 tone series. ( I kind of think 12 tone music is interesting to analyze. I forgot the name of the mathematician, who was a chairman of mathematics depts. in Princeton Uni. long time ago who wrote lots of music based on numbers. I'm not sure whether he actually liked the sounds.) Does this piece end on a certain key? Just curious. I just had a piano stundent who composed (improvised) music with wondering chords, no certain tonality. And she also composed music that ends with dominant chord, not resolved. She said she liked the the feeling of suspension. I guess music in 21 c. have so many possibilities.

Kathy, I still don't understand what they mean by 'Jazz up' with quarter tones. is it for chanting? Are they secure with singing half steps? Sometimes I found children come up with strange ideas when they have trouble with something, especially in music. If they do, I would tell them only if they are secure with singing half steps then they can try singing quarter steps, otherwise it will be pretty confusing.

If they have a trouble with half and whole steps, ( my adult schola still do) you can have them listen to half steps and whole steps on the piano, and have them identify them, both melodically and harmonically.
Then you can have them sing those intervals, and move on to singing Major 3rd and minor 3rd after a few weeks of practice.(Just a few minutes at a time)
I also recently discovered (from this forum I believe) that half steps in scale occur after the solfege names that end with 'i' ; "mi"and "ti." ( I think music schools miss out many things. never learned that in school!) I'm not sure, how it happened, but simple trick like that can help people to remember the half steps also.

No 12 tone.

This is my take, straight to the point.

That's like lining up all the colors next to each other and calling it art.
Most of us call that an artist pallet.
You use THAT to create art.

That's like lining up all the elements in nature.
Most of us call that the periodic table.
You use THOSE to make raw materials.

"It seems that modern musicians and artists....want to be free from the tonal structures and formalities."

Actually, miacoyne, I believe a clarification between modern and contemporaneous needs insertion into your contention.
As the perceived need for tonality moved from obfuscation into obsolescense roughly at the turn of the 19th to 20th centuries, that vacuum was filled with all kinds of constructs, from the sublime to the ridiculous. What all of these constructs adhered to was a sort of code of intellectualism, which then required professors and students to study them well passed the expiration dates of their aesthetic worthiness. Now that the last of these, minimalism, seems to finally be in its death throes (now that Glass is back to doing commercial filmwork), I would say that we're in a period where the contemporaneous composers don't feel a need to be free from archaic or foreign forms; rather, they seem to be in a modality of syncretism, embracing and incorporating such forms into many compelling new works. Tavener and Part have been cited here hundreds of times, as well as McMillan of late. Coriliagno and Adams (just take in "Oppenheimer" if it's documentary shows up on PBS) on the orchestral/operatic side of things represent this new ideal, IMO.

Kathy,

It seems another way to convey this would be just to show the frequencies associated with each note (on the piano) and point out that if someone sings the frequencies in between, that they fall in the "no-man's land" of micro-tonality.

For instance, we tune our A above middle C to 440hz. G# (I believe) is at 415hz. What if someone sings 420? or 430? It's not A and it's not G# . . .

You could also (if you have the ability) have them hold a note, and then you could start in unison with them and slowly slide up to the 1/2 step above, singing the microtones in between.

Let us know how it goes!

Kathy: this might help: http://en.wikipedia.org/wiki/Harmonic_series_(music)

Addendum:

This is also a very good way to demonstrate my original post. It will show the relative sympathetic strength of any note of the harmonic series to its played fundamental. The further away from the fundamental, the less sympathy. You'll find that even the 1/2-steps have very little resonance. Therefore, micro steps will seem virtually unrelated.

The concept of music is the relativity of pitches. If you "construct" a system which looks correct in the simple math, but the end results lacks this relativity, it is not going to pass for "music".