Posted on 07/02/2022 8:59:09 PM PDT by nickcarraway
The Greek mathematician and philosopher Pythagoras, who lived 2,500 years ago, applied his genius to music as well throughout his brilliant career, creating the Pythagorean comma as part of music theory, and his brilliance is still recognized to this day.
The Pythagorean Theorem remains one of the fundamental concepts in the realm of mathematics and is still taught in schools across the world.
The influence of the Ancient Greek thinker, who was born on the island of Samos in the year 570 BC, remains strong today in many realms—but, unfortunately, so do the mysteries surrounding the great Greek philosopher.
Pythagoras’ philosophy influenced both Plato and Aristotle, and through them, his ideas were fundamental in Western philosophy. However, a lesser-known contribution the brilliant theoretician gave to the world is the “Pythagorean comma,” the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B♯, or D♭ and C♯.
Pythagorean Comma part of essential musical theory His “comma,” which is known to music theoreticians the world over, is used to express the measurements between musical notes on a scale.
Grecian Delight supports Greece In music theory, a comma is a very small interval which is really just the difference resulting from tuning one note two different ways. The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F♯ tuned using the D-based Pythagorean tuning system and another F♯ tuned using the D-based quarter-comma meantone tuning system.
Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B7♭ and A♯ are both approximated by the same interval although they are a septimal kleisma apart.
The word “comma” came via Latin from the Greek κόμμα, from the earlier *κοπ-μα, which means “an act of cutting.”
Within the same tuning system, two enharmonically equivalent notes (such as G♯ and A♭) may have a slightly different frequency, and the interval between them is a comma. For example, in extended scales produced with five-limit tuning, an A♭ tuned as a major third below C5 and a G♯ tuned as two major thirds above C4 are not exactly the same note, as they would be in equal temperament. The interval between those notes, the diesis, is an easily audible comma (its size is more than 40 percent of a semitone).
Commas—the difference in size between two semitones Commas are often defined as the difference in size between two semitones. Each meantone temperament tuning system produces a 12-tone scale characterized by two different kinds of semitones (diatonic and chromatic) and, hence, by a comma of unique size. The same is true for Pythagorean tuning.
A Pythagorean comma is equal to the frequency ratio (1.5)12⁄27 = 531441⁄524288 ≈ 1.01364, or about 23.46 cents, which is roughly a quarter of a semitone (in between 75:74 and 74:73). The comma that musical temperaments often refer to tempering is the Pythagorean comma.
There is also the Pythagorean comma on C using Ben Johnston’s notation. The note depicted as lower on the staff (B♯+++) is slightly higher in pitch (than C♮).
Pythagorean comma (PC) is defined in Pythagorean tuning as the difference between semitones (A1 – m2), or the interval between enharmonically equivalent notes (from D♭ to C♯). The diminished second has the same width but an opposite direction (from to C♯ to D♭).
The Pythagorean comma can be also defined as the difference between a Pythagorean apotome and a Pythagorean limma (i.e., between a chromatic and a diatonic semitone, as determined in Pythagorean tuning), as the difference between twelve just perfect fifths and seven octaves, or as the difference between three Pythagorean ditones and one octave (which is the reason why the Pythagorean comma is also called a ditonic comma).
The diminished second, in Pythagorean tuning, is defined as the difference between limma and apotome. It coincides, therefore, with the opposite of a Pythagorean comma, and can be viewed as a descending Pythagorean comma (e.g. from C♯ to D♭), equal to about −23.46 cents.
As described in the introduction, the Pythagorean comma may be derived in multiple ways.
The difference between two enharmonically equivalent notes in a Pythagorean scale, such as C and B♯ (About this soundPlay (help·info)), or D♭ and C♯ (see below) is one way to derive the Pythagorean comma. Alternatively, it could be the difference between Pythagorean apotome and Pythagorean limma.
It can also be arrived at by the difference between twelve just perfect fifths and seven octaves or the difference between three Pythagorean ditones (major thirds) and one octave.
A just perfect fifth has a frequency ratio of 3:2. It is used in Pythagorean tuning, together with the octave, as a yardstick to define, with respect to a given initial note, the frequency ratio of any other note.
“Apotome” and “limma” are the two kinds of semitones defined in Pythagorean tuning. The apotome (about 113.69 cents, e.g. from C to C♯) is the chromatic semitone, or augmented unison (A1), while the limma (about 90.23 cents, e.g. from C to D♭) is the diatonic semitone, or minor second (m2).
A ditone (or major third) is an interval formed by two major tones. In Pythagorean tuning, a major tone has a size of about 203.9 cents (frequency ratio 9:8); thus, a Pythagorean ditone is about 407.8 cents.
Pythagoras’ life dogged by hardships, struggle despite brilliance Around 530 BC, the Greek philosopher traveled to Croton (today’s Crotone) in southern Italy and founded a school. Pythagoras’ schools initiates were sworn to secrecy and lived a communal, ascetic life.
This lifestyle entailed a number of dietary prohibitions, said to include vegetarianism, although some modern scholars doubt that he advocated for the diet.
Pythagoras’ mysterious personality was noticeable during his teaching. Strangely, no notes and questions were allowed in his lessons, which is why a great part of his work was lost. There is no additional information even on the renowned Pythagorean theorem.
It is also not known if Pythagoras invented this theorem on his own or with the help of his students.
He died around 495 BC in Metapontum, Lower Italy, starving himself for forty days because of his grief over the persecution of the Pythagoreans and the killing of the majority of them.
Another ancient account states that Pythagoras was killed for political reasons, as followers of his opponents Cylon and Ninon had attacked the Pythagoreans during a meeting.
Following his death, the Greek philosopher’s house was made into a sanctuary of Demeter, and the road that led to it became a sanctuary of the Muses—an apt transformation, as the great man was responsible for explaining the workings behind the most ephemeral of all the arts, namely that of music.
and all the kids in his class called him A square, which pissed him off cause he was more into triangles.
then they would chant B square
finally C you square
or at least that is what I was told once
I tried to follow this distinction beteen notes which are chromatically they same yet differnt but I just couldn’t. it was all greek to me.
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Lack of commas is a grammar crime resulting in a long sentence.
One of his students was the important but often overlooked Treble Clef.
I used a Commodore 64 to tune my musical instruments when I was a kid. I had learned the basics about how one note an octave higher than a second note had twice the frequency. So I calculated the frequency of each note in between to be 1/12th the difference in a logarithmic scale and ruined my sense of pitch. No-one taught me about the comma or any of the finer details of scales.
Imagine the Pythagorean tuning system as a spiral and the 12 tone Equal Tempered tuning as a circle. With the former you can never arrive back at your starting point by stacking consecutive fifths. With the latter each 5th has been flattened a little allowing you to arrive back at your starting point. This great adjustment allowed music to be transposed to all keys and made all keys usable
“I tried to follow this distinction beteen notes which are chromatically they same yet differnt but I just couldn’t. it was all greek to me.”
Pitch is like real numbers. Between 1 and 2, there are an infinite number of numbers, like 1.1 and 1.02976 etc. Between C and C#, there are an infinite number of possible pitches (you hear it when a blues singer goes a little flat on a note and it sounds good or when a guitar player “bends” a note by pushing on a string while it is vibrating).
So the pitches on a piano keyboard are particular spots on that continuous line of possible pitches that have been picked out for common use because of their versatility in different keys.
The term “well-tempered” (Bach’s The Well-Tempered Clavier is probably the most famous use of the term) is a selection of pitches for the notes in a scale so that the same instrument can play compositions in different keys and not sound stupid. Pianos are usually tuned to a well-tempered scale. The intervals between notes in a well-tempered scheme are set just so and they are a little different from note to note. There are a lot of different “well-temperment” schemes and each sounds a little tiny bit different. The Pythagorean temperament discussed in the article is one example.
However, the result of that is that different keys for compositions have a different feel as the interval between the notes changes just a little depending on what key you are in.
Get a pencil and paper, look up the ratios of the overtone series, and it will make sense.
I agree.
Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B7♭ and A♯ are both approximated by the same interval although they are a septimal kleisma apart.
“Uh…can ya’ hum it”?——-Glen Campbell.
This thread is useless without pitchers.
People with music degrees can probably understand this much better than me. I’m a musician, I play or sing the note by the pitch apparent to me. I do understand that the math of music does not always agree with the sound of pitches that music comes from. I think the article explains the way around the inexact math of music. I don’t worry about it. When I play the piano I play the notes the tuner gave me and don’t have to think about what it would sound like in a different key. Different keys however do make the same musical piece sound different, that is mostly because of the compromises made with musical math. I play an organ that has a transposer on it. The organ uses a computer to come up with frequencies of each note and can be tuned (programmed) to different pitches. The organ is different than the piano, that is an electronic organ in that the computer recomputes notes based on the frequency of only one octave. Pipe organs on the other hand are individually tuned, each pipe in a pipe organ of perhaps thousands of pipes must be tuned to sound right with the pipes around it. Sounds like a simple task but it isn’t so simple, tuning an organ is a much more complicated task than tuning a piano.
Tuning forks used to be used to tune pianos now a little computer listens to each string and tells you when it is in tune, the better instruments even help you tune the multiple strings of a piano note slightly off of each other to make them blend without beats. Doing this takes care of the same piece played in different keys, for the most part anyway and makes it sound more natural and correct while to some very good ears may make the piano sound out of tune. It can be a real science. I’m glad I just have to play.
BTTT for the Pythagorean comma!
Having said that: I thought I was well-versed in music theory, and I did understand how Pythagoras linked mathematics and music, but I have never heard of this!
As a guitar player and luthier, I see the relationship between the ratios in frequencies and their corresponding position on the fret board. As anyone who plays the guitar would know, twelve frets is twelve semitones which is an octave. The ratio in frequencies between octaves is 1:2; interestingly enough, the twelfth fret is halfway between the nut and bridge. The fretted note at the twelfth fret should sound the same as the harmonic - one octave above the open string; that's how a guitar's intonation is checked.
In 1978 I took music theory as an elective in high school. My teacher was brilliant - played the piano, was a clarinetist in the musician's union who would accompany touring musicians, and spent his retirement building violins. He took a small group of us and taught us to tune pianos. Mind you, in those days, it was done completely by ear, so you had to be able to hear what he heard. You could have used a Conn Strobotuner, but no serious piano tuner would have ever considered using one - it was sacrilegious! We would tune middle C with a tuning fork using sympathetic vibrations. Then, we would tune the A-flat below middle C by playing the middle C, letting it ring, and then play the A-flat and listening for the C harmonic two octaves above middle C - it was in tune when you could hear eight beats a second. Using equal temperament, we tuned the octave between middle C and the C# below it, then we would tune the octaves - as we went up the keyboard, the octaves were slightly sharp, and as went down, the octaves were slightly flat. It's been over forty years and I only did it once, so some of you might know better than I, but it was an experience!
LOL. Celebrate obscurity!
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