Why is a half step the smallest interval?
  • Kathy
    Posts: 4,979
    My Youth Schola would like to jazz things up with quarter steps, and frankly I'm having trouble explaining why they just can't do that.
  • Charles in CenCA
    Posts: 2,416
    You're joking, right?
    "Jazz it up" would be correct, if they wanna explore the mysteries of the blue note. Or they're way into musical idioms that straddle the longitudes around Bucharest and move eastward!
    Humming "KIDS!" from "Bye, Bye Birdie".....
  • Steve CollinsSteve Collins
    Posts: 1,003
    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!
  • GavinGavin
    Posts: 2,799
    IIRC, the Greeks used to use microtones. I don't care for microtones as a melodic practice, but they can make for some astounding harmony.

    As for the "why", Steve nails it. Microtones are every bit as natural as half steps, but much more acoustically distant. Go up the circle of fifths five times (6 notes), and you'll have a half-step. Go up more than 10 and you start getting microtones. Again, natural, but in general our culture tends towards pitches closer on the harmonic series (unison, octave, fifth, third)
  • rwprof
    Posts: 25
    Come to our church, and hear all the microtones you want. It's Byzantine chant.
  • Interesting! Steve makes a succinct summation of this aspect of our western musical tradition. Since your children want to sing in microtones you might do some research into some other musical cultures that use them (Arabic, Indian, Oriental) and let your children learn some examples from them. This would be an interesting study in contrasts and would teach them to appreciate the uniqueness of our tradition as well as the ethos of others. (We are told by some that quarter tones were used at some time in Byzantine and Gregorian chant, but this tradition, if it existed, has been lost.) Also, you could, of course, let your class make an original composition or melody which utilised microtones. It would be interesting to see how they sing these purposely - we hear them all the time, of course... from people who sing 'out of tune' or who slide from note to note.
  • Jevoro
    Posts: 108
    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?
  • chonakchonak
    Posts: 7,511
    "In my opinion, the pentatonic scale has its own harmonic justification, but how you fill in the minor third between the series of two notes and the series of three notes is open to many solutions. The combination of a half-tone and a whole tone is only one among many possibilities.
    It does sound natural to sing through each tetrachord, but -- between them? A whole tone? a half-tone? Raised? lowered?"
  • dad29
    Posts: 1,639
    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.'
  • francisfrancis
    Posts: 8,024
    you forgot the interval called "WAY OFF!"

    Personally, I have enough trouble getting people (and my own fingers) to sing and play the twelve chromatic half steps much less anything beyond that! I am amazed if you have exhausted the finer points of the western scale!

    When I was a student at the Peabody Conservatory (in the 70's and 80's), they had 'gone beyond' the staff and were then into graph paper and timelines and whatever sounds they could plot on the timeline. I begged them forgive me leave at that point, because I was still trying to learn to compose music on the regular staff. I had to make the masters of times past my own teachers and mimic all the styles of our bygone ancestors because no one was interested in melody and harmony anymore. I wonder where all my colleagues are now?!

    Here is an example of 21st century counterpoint using the regular staff.
  • miacoyne
    Posts: 1,805
    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.
  • Charles in CenCA
    Posts: 2,416
    Kathy, just play them "Atmospheres" by Ligeti, then ask them: "Is this what you wanna do?"
  • francisfrancis
    Posts: 8,024
    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.
  • miacoyne
    Posts: 1,805
    Very interesting. When you listen to the music and look at the arts in different periods, they really reflect the culture and minds of that period, including 21st.c. It seems that modern musicians and artists are interested in sounds and colors themselves very much, and want to be free from the tonal structures and formalities.
  • Charles in CenCA
    Posts: 2,416
    "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
    Posts: 4,979
    Interesting discussion!

    The kids were mostly just asking me to justify this statement: the half step is the smallest interval we have. I really like Steve's answer, which, if I understand correctly, is something like this. Acoustic resonances (harmonics) occur naturally in small, whole-number ratios to the struck notes. These small ratios are evident in the scales that use half step intervals.

    I think I can sell this to kids. There are two reasons: it sounds better that way, and, the other way, the math gets too complicated.

    Sells itself, actually. Thanks all.
  • Heath
    Posts: 795
    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
    Posts: 4,979
    Sometimes when singing I hear an harmonic ringing on a guitar that sits in my office. Which notes can I sing that will produce harmonics? Is there a simple harmonics chart, similar to a circle of fifths chart?
  • GavinGavin
    Posts: 2,799
    Kathy: this might help: http://en.wikipedia.org/wiki/Harmonic_series_(music)
  • Steve CollinsSteve Collins
    Posts: 1,003
    You can demonstrate, or even test this effect on a piano. Hold any key of a harmonic series down (don't play it) and then play the foundation note. The note above will vibrate sympathetically. Or, hold any key down without playing it, and then find which notes play lower will cause that note to vibrate.
  • Steve CollinsSteve Collins
    Posts: 1,003
    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".
  • WGS
    Posts: 220
    microtones:

    Coincidentally, this morning I was reading an article by Dr. Peter Wagner in the Spring 1960 issue of Caecilia (one of the two precursors of Sacred Music).

    "The time has not come for the final discussion about the fluctuation of c and b as well as of f and e. In my Neumenkunde, incident to the varying changes which certain neums in the manuscripts have assumed with Guido's line-system, I expressed the possibility that a note which lay halfway between b and c, e and f, might have been sung, and that when the tone steps were stabilized on the line-system, or staff, it was lowered or raised a trifle. That such intervals already were used in the 11th century in everyday choral praxis is clear from the comments of the theorist; this is especially true of the Montpellier manuscripts (Paleographie Musicale, VIII) which even have proper signs for them ..... If that is the case, then, from the standpoint of the earliest version, the note b would be just as incorrect as c, and e just as wrong as f."