As a starting point, it should be pointed out that the Tartini tones produced do follow a pattern that is similar to the Harmonic series; it isn't a perfect fit, however... in fact, some notes are off by as much as 49 cents (50 cents is a quarter tone--get it: 100 cents = one half tone; one-hundred one-hundredths (i.e: 1) = one whole of the West's smallest intervals, which are half-tones) and most are off by more than 5 cents (the smallest perceivable difference between tones when they are played consecutively rather than simultaneously--because when consecutive they don't produce interference beats).
The amount off from the harmonic series, however, is actually insignificant at it's base level; the Perfect fourth is almost as far off as the Tritone--although the tritone does have the greatest difference at 49 cents. Where it gets hairy is when you look at the 2nd-, 3rd-, and 4th-order Tartini tones. The Perfect intervals' higher orders continue to produce low octaves, just repeating itsself over and over again; however, the Tritone produces 4 different tones in the first 4 orders--where the perfect intervals, with higher-order Tartini's included, continue to only resonate in 2 tones which blend together well, the Tritone produces 4 tones which, because they sound simultaneously, clash horribly. I haven't investigated the other intervals yet (i.e. the 2nd's, 3rd's, 6th's, and 7th's), but I will do so shortly and examine how they may add to the consonance or dissonance that is produced in their respective intervals.
Secondly: other research I've done since the last update. This has more to do with music history and development than physics. Eventually, hopefully, I'll be tying the history and the physics together--music has become what it is for a reasons both cultural and scientific. It's hardly arbitrary.
First I'll be posting some random incohesive notes I've taken which I'll tie together later. Then I have some information which I have already begun connecting related to Hexachords, Tetrachords, Solmization, and the rebirth of flats and sharps (accidentals) in the Middle Ages.
So here are random notes:
- Semitic music/theory most likely influenced both Arabian and Greek music.
- Arabian theory was very much influenced by the science of music. Al-Farabi wrote texts in the 800's AD motivated by both mathematical predictions and his aesthetic knowledge of music as a performer.
- Al-Farabi describes fretted string-instruments which utilized quarter tones.
- Greek music was more influenced by aesthetic, and traditional/cultural practice rather than mathematics/science of music. They did, however, understand the science and math and had knowledge of the overtone series--tradition would not allow them to develop many new music styles though.
- Earliest records of melodic music are from 3rd millenium BC in Mesopotamia
- A lyre (stringed instrument) found in Mesopotamia was tuned to a heptatonic (7-tone--just like western) scale. I don't know if it used half tones or quarter tones or what the scale quality was. But, it did divide the octave seven times.
- Hindu music theorists divided the octave into 22 "Shruti"--these are just barely larger than a quarter tone, which would have 24 divisions to the octave.
- In 1953, research done by Constantin Brailoiu suggests that Pentatonic scales (5 tones to the octave with intervals of a Major 2nd or Minor 3rd between consecutive tones) can be found in the indigenous music of all 5 continents.
- East-Asian theorists list pentatonic scales which contain intervals of almost-semi-tones (remember a semi- or half-tone is a Minor 2nd)(perhaps "Shruti"?), whole tones (remember a whole tone is a Major 2nd), and minor thirds. (In fact, if you play a regular, Western pentatonic scale, you'll most likely think it sounds "Asian").
- Some pentatonic scales actualy divide up the octave entirely equally rather than in variations of Minor 2nd's, Major 2nd's, and Minor 3rd's. This is common practice in some Southeast-Asian, Asian/Pacific Island (Java), and African cultures. In Javanese these are called "Slendro" modes ('mode' is just another word for 'scale').
Now then ("now then"--what a stupid, oxymoronic phrase...), notes which I have begun piecing together: These notes relate to the development of music, specifically in the 1000's to 1200's AD. Later, I will go further back, concentrating on the Greeks in the 1000's BC to 100's AD, and the Mesopotamians in the 2- and 3-thousands BC. Eventually, I will also be connecting Eastern musics, Semitic music, and perhaps even African music, to modern physics. It is my belief that music has developed the way it has because of the reaction of acoustic physics with our neurology. For some reason, almost all cultures have decided to divide up the octave into 12 tones; some cultures have chosen 24 tones, some 22, 5, or some other number--but all of these divisions have a reason for occuring (especially 12, since it is the most common). It isn't human nature to do things arbitrarily... especially for 4- or 5-thousand years.
Hexachords:
The word "hexachord" comes from the greek words "hexa"--meaning 'six'-- and "chorda"--meanging 'string'. So 'hexachord' literally means "six-string". But, a hexachord doesn't have six strings; a 'hexachord' is a scale which consists of 6 notes--specifically: the first 6 notes of the Western diatonic ("regular") major scale. The reason it is named "six-string" is because in Ancient Greece scales were played on instruments called 'lyres'; lyres were stringed, but unfretted, instruments (like a harp), and so each string could only be assigned one note. If a musician wanted to play 6 different notes, he would need a lyre with six strings. As I've said, I'll be posting more on lyre's, tetrachords (Grecian four-note scales), Grecian music, kitharas, auloses, and the monochord--which is a one-stringed instrument used to study the physics of sound, and is not a scale. (All these "chords" start to get confusing, don't they? Especially when you add in Major and Minor and other qualities of chords which are neither stringed instruments nor scales...)
**In the following few paragraphs, I'll be using some music notation which I haven't explained before--this notation relates to the octaves of notes, but doesn't use subscripted numerals, this is called "Helmholtz Pitch Notation". Pay attention to the comma (,) and prime (') symbols which may be placed after a note name. Here's how it works: The lowest C is notated as [C,,] (capitalized, 2 commas), the next higher octave: [C,] (capitalized, one comma), next: [C] (capitalized, no marks), then: [c] (lower case, no marks), then: [c'] (lower case, 1 prime), then: [c''] (lower case, 2 primes), and finally: [c'''] (lower case, 3 primes). So, to recap and simplify: [C,, C, C c c' c'' c''']. All 12 notes between [C,,] and [C,] get two commas. I.e: the note "D" which is just higher than [C,,] is notated: [D,,] . So the entire octave (excluding notes with flats or sharps for the sake of simplicity) would be [C,, D,, E,, F,, G,, A,, B,, C,] next octave: [C, D, E, F, G, A, B, C] then: [C D E F G A B c] etc. Sorry for the ridiculously confusing punctuation. **Note that in actual notation, no brackets were used. I'm just using them to hopefully make it somewhat less confusing which letter receive which punctuation.
**Keep in mind that music in the time period of the hexachord's invention was primarily for writing melodies with no accompaniment or background harmonies-- no chords, counterpoints, etc. Most musics in the Dark Ages were either Alleluias (one-line church "songs") or monks' chantings.
So a hexachord is a scale of 6 notes, the first 6 notes of the major scale, and was first introduced into written music theory in the early 1000's AD by Guido of Arezzo. Guido decided to start on the note [G], and from there created the first hexachord: [G A B c d e]. He then jumped to the fourth note of that scale, and created a 2nd hexachord: [c d e f g a], and then jumped to the fourth note of that scale, and made a third and final hexachord: [f g a bflat' c' d'] , lastly, he took these same 3 scales an octave higher and added them to the mix: [g a b c' d' e'], [c' d' e' f' g' a'], and [f' g' a' bflat'' c'' d'']
At this point, I will point out 2 things:
1) These 3 complimentary scales, all used to construct one single melody, contain in them a B and a Bb despite being constructed from complimentary major scales and despite their supposedly being entirely consonant to form a melody which would please the ear.
2) Some of the notes in these 3 complimentary, yet separate, 6-tone scales occur in more than one scale. [g], for example, occurs in the [c]-hexachord, the [f]-hexachord, and the octave of the [G]-hexachord. [d] occurs in both the [G]-hexachord and the [c]-hexachord. If you look, you'll realize that several notes occur in 2 or 3 scales.
By having both the B and the Bb in the 3 (6 counting the octaves) usable scales, Guido opened up the door for "mutation" or in modern terminology "modulation"--this is the process by which a melody and with it it's harmony change keys (remember, the key is the 8-tone scale that the song is based in). This opened up a world of new possibilities to composers as they no longer had to use just 8 tones to create a song, and was certainly a great step forward in the development towards modern Western music.
One more thing (and inadvertently perhaps the most important thing) that Guido of Arezzo did was that he "Solmized" the music; he called each note by a syllable, rather than by a letter (ut re mi fa sol la si ut). By doing this, he made the music entirely relative--a concept which in modern music is very common, but in medieval music was unheard of due to variances in intonations (I'll talk about the specifics of that in another post). Now, Guido didn't entirely solmize the music--there were still letters associated with the notes--but what he did do was use solmization to distinguish between say, the [c] in the [c]-hexachord and the [c] in the [G]-hexachord. I won't go into the details of how he went about doing this, though, because the main point here is that he made his scales solmized; this will alter the way Western music progresses considerably.
Over the next 100 or 200 years, from 1000 AD to 1100 or 1200 AD, polyphonism started to gain a larger role in Western music. And most western music had adopted the use of solmized hexachords as the basis for writing melodies. When solmized notes and polyphonism tried to combine, however, a problem occured. In the harmonies, there was a need to place a note a perfect fifth above and below the B's and Bb's; however the note a perfect fifth above B is F# (remember # means "sharp"... sharp means "plus one half step"), and the note a perfect fifth below Bb is Eb. If you look at the 3 hexachords, you will notice very quickly that there are no Eb's or F#'s. At first, composers used what was called "musica ficta" or "musica falsa"; in other words, they 'invented' notes (Eb and F# were known notes at the time, they just didn't fit the scales--hexachords-- in use). Over the years, this became a very common practice since it was musically necessary to produce real harmonies. In modern music, we call these kinds of notes--notes which aren't in the 'proper' scale--"accidentals", and they are commonplace in all music except for the most basic elementary studies.
As you can see, due to hexachords and solmization, Western music made a huge advancement towards its modern state.
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