Problems with Ancient Musical Scales

One-sentence summary: I explain why an exponential scale was invented, and how it compares with the older scales based on ratios of integers.

This article is the continuation of the article on the mathematical nature of musical scales. In that article we have used ratios to come up with a division of an octave. We have given Latin names to musical intervals (or ratios of frequencies): octave as 2/1, tertia as 5/4, quinta as 3/2, quarta as 4/3, tone as 9/8, and semitone as 16/15.

In American English the tone, tertia, quinta and quarta intervals are familiarly known as second, third, fourth, and fifth notes of a basic scale. However, I will use the Latin names for reasons described in the previous article. (In Russian, these are more familiarly known as секунда, терция, кварта, квинта.)

When combining intervals we “add” them, but in terms of frequency ratios we multiply them. That’s because the human ear perceives frequencies on a logarithmic scale. Squaring a frequency ratio, feels like doubling a musical interval. Narrowing a musical interval corresponds to multiplying its frequency ratio by a number less than 1.

For instance, a two-octave interval has frequency ratio of 2/1 * 2/1 or 4/1. And a 7-octaves interval has a frequency ratio of (2/1)⁷ or 128/1.

(The logarithmic scale converts multiplication to addition. Before calculators were invented, men used a logarithmic slide rule to compute fast multiplications by hand.)

All approaches to tuning based on ratios are called just intonation tunings. Also, the term just tuning may also refer to the specific Ptolemaic tuning. The idea behind this name is that the musical intervals are most pleasing to listen to.

However, tuning an instrument based on ratios creates inconsistencies, because there is no single unit from which all the intervals are made up. As an analogy, take the case of mixing inches with centimeters. These units of length are incongruent. (On the other hand, inches are congruent with feet, and centimeters are congruent with meters.)

With the just tuning, for instance, a note 12 quintas away from a certain base note is dissonant with a note 7 octaves away from the same base note. The ratio between them is called the “comma of Pythagoras” and it can be calculated like this,

We can abuse the concept of “overtones” to say that the comma of Pythagoras is a distant 531,440-th overtone of the base note, translated 524,288/2 octaves down. Clearly, such note is not going to sound good together with the base note. This is the reason that this highly dissonant interval is called a “comma.”

The word “comma” came from Greek κόμμα which means “an act of cutting.” The idea behind this name is that a dissonant sound is “cutting the ear.” Although such “comma” intervals are a theoretical possibility, could one simply avoid such intervals in music? After all, why would a musical piece have a 7 octaves range? Turns out that the commas are more prevalent and are hard to avoid.

The music theorist Philalaus of Tarentum of the 5th century CE noticed that two hemitones and a comma of Pythagoras combine to a tone interval of 9/8. (A hemitone is an interval with a frequency ratio 256/243 and it is Pythagoras’ choice for a “half” of a tone).

Philalaus’ observation shows that if F# and G♭are tuned according to the Pythagorean scale, then they would clash in the Pythagorean comma dissonance. (The # and ♭notation here means raising and lowering by a hemitone). How likely is it to happen? Because some instruments are tuned with “black” keys as sharps of white keys, but other instruments are tuned with “black” keys as flats of white keys, such instruments would clash in a dissonance.

There are other notable commas that occur in just intonation tunings. The Ptolemaic comma occurs in the Ptolemaic tuning when the notes C, G, D, A, E, C are played in this ascending and descending sequence known as a “comma drift”,

Comma drift

The frequency ratios between the notes are 3/2, 3/4, 3/2, 3/4, 4/5. (For the ascending direction the ratios are greater than 1, and for the descending direction the ratios are less than 1.) Multiplied together they give the ratio of 81/80, the Ptolemaic comma. The sequence is known as comma drift, because the original base note (here C) drifts to a rogue “base” note that is dissonant with the first. This comma is also known as the Syntonic comma.

Another dissonance occurs when the Ptolemaic comma clashes with the Pythagorean comma. Suppose that one instrument plays Gb instead of F#, resulting in the Pythagorean comma 531441/524288, and another instrument plays a drifted F# resulting in the Ptolemaic comma 81/80. What’s the interval between the two notes? The interval is called “schisma,”

Schisma is the Pythagorean comma divided by the Ptolemaic comma

The syntonic comma 81/80 is also the interval between the Pythagorean tertia and the Ptolemaic tertia. From Wikipedia article on the syntonic comma:

The Pythagorean major third (81:64) and minor third (32:27) were dissonant, and this prevented musicians from using triads and chords, forcing them for centuries to write music with relatively simple texture. In late Middle Ages, musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made consonant. For instance, if the frequency of E is decreased by a syntonic comma (81:80), C-E (a major third), and E-G (a minor third) become just.

Thus we can see that the syntonic comma is important. The german physicist Helmholtz of the 19th century developed a musical notation that indicates by how many syntonic commas a Pythagorean-tuned note should be raised or lowered.

The medieval Pythagorean scale was derived by repeating quintas (multiplying 3/2 by itself) and transposing down by octaves (dividing by 2). In this way, the frequency 81/64 for a tertia was derived as (3/2)⁴ divided by 4. How could we modify this scheme, in order to get the more ear-pleasing 5/4 ratio for the tertia? If we decrease the quinta by a quarter of the syntonic comma 81/80, then a total of four quintas will contribute an overall decrease by a full syntonic comma. In numbers,

Ptolemaic tertia from four repeated quintas, each narrowed by a quarter of the syntonic comma

This exact approach was taken by musicians of the 16th century, and the result was the Quarter-comma Meantone scale. They sacrificed the perfect sounding quinta (the “perfect fifth” in American English), to get a better sounding tertia (“major third”). This scale was prevalent in the 16th and 17th centuries.

If we multiply both sides of above equation by 4, and then take a fourth root, we have a simple and revealing expression for the meantone quinta as the fourth root of 5,

Quinta in the Quarter-comma meantone scale

Thus, here we see the first example of an interval that is not tuned to a ratio of integers, but is, in fact, an “irrational” number. More on that later.

from Hacker News https://ift.tt/3ujH0zC