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Iamblichos tells us that Pythagoras discovered the numerical properties of tones; how, on a string, vibration, frequency and pitch are connected. Pythagoras built further on this, claiming that the whole world is harmony and number. He coined the term "Music of the Spheres", the sound created by the planetary spheres, always present, but unheard by most people, just because it is always there.
The planetary spheres form (as claimed by Pythagoras) a celestial scale, with the pitch of each planet based on its orbital speed, and the distances between the planets are analogous to the musical intervals.
"What is this sound, so loud and sweet, that fills my ears? It is the sound which, connected at spaces which are unequal but rationally in a particular ratio, is caused by the vibration and motion of the spheres themselves, and, blending high notes with low, produce various harmonies; for such mighty motions cannot speed on their way in silence, and it is Nature's will that the outermost sphere on one side sounds lowest, and that on the other side sounds highest. Hence the uppermost path, bearing the starry sphere of heaven, which rotates at the greatest speed, moves with a high and excited sound, while that of the moon and the nethermost sphere has the lowest. For the Earth, the ninth of the spheres and static, remains fixed at one spot at the centre of the universe. But these eight spheres of which two possess the same power, produce seven sounds, a number that is the key to almost everything(...)"
(Cicero, De re publica)
Pythagoras built this scale in the so-called Lydian mode, but it was not completed to a diatonic scale until later, by Aristoxenus of Tarentum (4th century B.C.), in his "Elements of Harmony". That is one of the (few) real masterpieces of Hellenic thinking, much due to the "calculus by logarithms" he developed therein.
The most common diatonic scales today are the standard Major and minor scales which form the base for most (albeit not all) music of the last 300 years. The most simple example of it is the C-Major scale, which can be played entirely on the white keys of a piano. So let us exemplify with this.
C-Major: C - D - E - F - G - A - B - c.
In C-Major, "C" is the tonic, a sort of base for the scale. "c" is the tonic again, although one octave higher, and serving both as the end of this octave and as the beginning of the next.
There are seven steps between C and c, and since c is the second tonic, there are seven genuinely different tones in the scale: C, D, E, F, G, A, B.
Sometimes the scale is expressed as:
do - re - mi - fa - sol - la - si (or ti).
This sequence of seven tones is based on the gamut of the 11th century monk Guido d'Arezzo. He took the first syllables of the lines in a Latin hymn to John the Baptist, where the tones were rising in pitch accordingly.
Ut queant laxis
From this he got:
ut - re - mi - fa - sol - la - si
S and I in Sancte Ioanne make "si". The initial syllable, "ut", has later been changed to "do", which is easier to sing.
The diatonic scales are not only the Major and minor ones, although some dictionaries and encyclopaedias say so. The diatonic scales are the "modes", which are different ways to arrange the octave, as done in classical Greek music or medieval church music. The different ways are based on the intervals between the tones. The modern Major and minor scales are two special variants of seven existing modes.
Let us exemplify this. Consider the white keys of a piano. If you look at the C-Major scale, you have C-D-E-F-G-A-B-c, and you can see that there are black keys "in between" C/D, D/E, F/G, G/A, A/B, but none between E/F or B/c. That means that we have only semisteps between E & F and between B & c, but whole steps otherwise. This is what characterises a Major scale, the semisteps are between the 3rd and 4th tone and between the 7th and 8th.
If you begin with A (still only the white keys), you get: A-B-c-d-e-f-g-a, where the semisteps are between the 2nd and 3rd tones, and between the 5th and 6th. This is what constitutes a minor scale, which has an entirely different character.
So the modes are defined by the location of the semisteps. You can play them all on the white keys of a piano, it is just a matter of where you start.
You can get these 7 modes:
Ionian, semisteps 3/4 and 7/8, start with C.
Dorian, semisteps 2/3, and 6/7, start with D.
Phrygian, semisteps 1/2 and 5/6, start with E.
Lydian, semisteps 4/5 and 7/8, start with F.
Mixolydian, semisteps 3/4 and 6/7, start with G.
Aeolian, semisteps 2/3 and 5/6, start A.
Locrian, semisteps 1/2, 4/5, start B.
The Ionian became Major, and the Aeolian became minor. They have been the basis for most music composed in the west since the early 17th century, although there are many exceptions to this.
Incidentally, flamenco derives its special character from the Phrygian mode on which it is based.
An interesting observation about the number twelve and music is that, while we saw that seven is the number of tones in a diatonic scale, twelve is the number of tones, or pitches, in a chromatic scale. This is a scale based on semitones or half steps. You can play one on a piano. Start on any tone, say C, and play all the keys, black and white, until you reach c.
Then you get:
C - C sharp - D - D sharp - E - F - F sharp - G - G sharp - A - A sharp - B - (c).
So, the scale is related to both the numbers seven and twelve, to both 3 plus 4, and 3 times 4.
It seems justified to wonder why a scale with seven steps is spanning an octave, a word that means eight.
This nomenclature was created by Pythagoras or the Pythagoreans and it is still used today, although by present standards it is a little peculiar. The eighth tone is the first repeated, one octave higher, but there are only seven steps between the first and the eighth tone.
Musical intervals are named after the number of tones they "cover". For instance: a 5th (e.g. C-G) covers 5 tones, but there are only 4 steps between them; a 6th (e.g. C-A) covers 6 tones, but there are only 5 steps in between. The 8th, the octave, covers 8 tones, but only seven steps.
In this system, a 1st covers only one tone (C-C), but no step. Here the difference from contemporary thinking becomes very clear. The Greek were intellectually primitive in many ways, they could not grasp void or nothingness. They had no zero, and probably would have found no use for it if someone had shown it to them. Actually, they most probably did encounter the zero but discarded it as useless, being nothing. Today we start counting at zero, and the first step is always between zero and one. If we had created the musical nomenclature today, we certainly would have named C-C a 0th, C-D a 1st, etc. Then the octave would have become a 7th.
There are scales which do not have seven tones. One example is the various pentatonic scales that have been used, for example, in East Asia, in blues, or in folk music almost everywhere. They, however, are not representing non-octave thinking. The ordinary octave is there in the background, and the five tones that are used are selected from the seven tones of the heptatonic scale or its chromatic alternative. It is still the same base, one has just limited the use of tones to only five.
The same can be said about, for instance, Arabian or Indian scales, although instead of limiting the use to less than seven or twelve tones per octave (whole or half steps), quarter-steps, even micro-steps are sometimes added. But essentially the octave is there. It is theoretically possible to divide the octave in any number of equal steps, it still remains based on the octave. And the Pythagorean octave and its seven tones are mathematically determined and, interestingly, they are well tuned to how the human ear perceives consonance and dissonance.
Pythagoras discovered that, if a string was shortened by 2/3 he got a perfect fifth, a very consonant interval, easy for the human ear to perceive. If he then continued like that, using the new tone to create a new fifth and repeated this in twelve steps, he returned to almost the same tone as the first one. That is, he went through all the tones in the chromatic scale (of twelve tones) but spread over seven octaves. (Note here the occurrence of the numbers seven and twelve.) So when he returned to the tonic, it was seven octaves higher. He then projected these twelve tones into one and the same octave and got the chromatic scale.
This is a simplification, because his result was not exactly the same tone seven octaves higher, although it was close. There is a small difference between the 12th fifth and the 7th octave, this difference is called the Pythagorean comma. It is approximately 1/8 of a tone step (or 1/4 of the distance between two keys of a piano), and it is not without consequences. It always was a problem for tuning instruments, and it is the reason why we got the "tempered" scale. One began to divide each octave in twelve equal steps. It is a false tuning, both mathematically and perceptually, which is "only almost" right. This "almost", however, is sufficient for most people whose ability to perceive consonances is not above average, but for someone with perfect pitch this can be torture. Actually, absolute pitch is a defect for musicianship in an environment where tempered and untempered instruments are mixed. It is a situation that can become torturous, even cause health problems. One example is the combination of piano and symphony orchestra. A piano is the ultimate tempered instrument, while many of those in a symphony orchestra are not. This is unbearable for some people.