Chapter III: What is an Interval?

An Interval is the difference between a tone and the next tone; a jump from one sonic frequency to another. We all know that only certain intervals sound pleasing, while others sound unpleasing and dissonant. In this chapter we will explore how mathematics, the language of nature, plays a fundamental role in whether we interpret an interval as pleasing or not.

In the previous chapter we discussed the natural phenomenon of a single tone to produce so called harmonics. Harmonics occur everywhere around us; it’s just that we are not used to pay attention to them. One man, who really paid attention, and who went into modern history as the first one to write about them, was the Greek philosopher Pythagoras.

Pythagoras discovered that the harmonics of a sound arise from precise mathematical ratios, because only precise multiples of waves can vibrate synchronously with the fundamental vibration. He attributed the mathematical purity of the ratios, as the reason why our brains interpret these combination of sounds as pleasant, namely because they have a mathematically harmonious relationship to one another.

We define a ‘mathematically pure ratio’, as a ratio with small natural numbers, such as 3:2, 9:4, 4:3, etc. These ratios, when written as a decimal number, have only a few digits or the same digit repeating infinitely behind the dot, in our example above these would respectively be 1.5, 2.25 and 1.33333333… The ratio 43:39 is not considered pure, because the numbers are not small, and its decimal representation (1.1025641…) has many different decimals behind the dot.

As we can see in previous chapter’s Figure 1, the second harmonic has the purest ratio (2:1), and the purity of the ratios decreases with each cascading harmonic, namely 3:2, 4:3, 5:4 and so on. The resulting frequencies of these harmonics form the base of the world’s musical scales, precisely because the mathematical purity of their ratios makes the notes sound pleasant to combine into music. The known variances in the scales of different traditions (western, Indian, Chinese, etc.) is only a matter of how many harmonics each culture still considered pleasant enough to include in their music.

The notes of the modern western music and their harmonic correspondence are listed as follows, in order of decreasing mathematical purity:

 

Note

Harmonic

Interval

Ratio

Do

1st

Fundamental

1:1

Octaves of Do

2nd, 4th, 8th, ...

 Octave

 2:1, 4:1, 8:1, ...

 Sol

3rd, 6th, 12th, ...

 Fifth

3:2, 6:4, ...

 Fa

Interval from 3rd to 4th

 Fourth

 4:3

 Mi

 5th, 10th, ...

 Major Third

 5:4, 10:4, ...

 Sol#

Symmetry of Mi

 Minor Sixth

 8:5

 Re#

 Interval from 5th to 6th

Minor Third

 6:5

 La

 Symmetry of Re#

Major Sixth

 5:3

 La#

 7th, 14th, ...

 Minor Seventh

 7:4 or 9:5*

 Re

 Symmetry of La#, 9th

 Major Second

 10:9, 9:8

 Fa#

 11th

 Tritone

 11:8

 Ti

 15th

 Major Seventh

 15:8

Do#

 Symmetry of Ti

 Minor Second

 16:15

Table 1: The notes of the musical scale with associated harmonic and interval name with respect to Do
*The 9:5 alternative of the minor seventh creates a symmetric major second with ratio 10:9

 

Interestingly enough, the development of western music has been incorporating the tones of the harmonic scale into the musical scale in a gradual manner, starting with the purest of intervals: the octave, the fifth and the fourth; and then adding intervals of decreasing mathematical purity successively as the culture and the music developed.

We know based on oral tradition and ancient musical records, that the Greeks and Romans sang songs primarily in unison. Their melodic instruments were mostly single-toned, wind instruments and stringed liras and harps with only a few different notes, which were usually not played at the same time. Not many musicians ventured into polyphony, except a few futuristic pioneers, who were skilled at playing 2 flutes at the same time, tuned a fifth apart.

During the middle Ages, very rudimentary musical notation transcripts started to emerge in the monasteries. These early versions of our modern musical notation system only signaled when the tone went upwards or downwards, and if the jump was small or large. The musical notation system simply didn’t need to be very complex, because only a few intervals were even allowed!

The music of this period is still alive today thanks to the existing monasteries that continued the medieval musical practices unchanged throughout the centuries. If you listen to Gregorian Chanting, for instance, you can still experience the musical simplicity of employing only a handful of intervals: the octave, the fifth, the fourth and the occasional major second sung downwards to signalize the end of the song.

With time more and more intervals were allowed to be part of the musical scale, for example the third (which gave rise to the differentiation of minor and major scales) and its symmetrical counterpart, the sixth. Later, the minor seventh was relaxed to incorporate the major seventh, and their symmetrical counterparts, the minor and major second were also accepted to use only occasionally in popular music. The tritone however, was still taboo well after these other intervals were included. This interval, also known as the diminished fifth, was even referred to as the ‘devil’s interval’ and avoided at all cost in music!

 

A Side Note on the Sacred Numerology of Intervals:

Note how the following symmetrical pairs of notes add up always to the sacred number 9:

The Fundamental and the Octave: 1 + 8 = 9
The Fifth and the Fourth: 5 + 4 = 9
The Major Third and the Minor Sixth: 3 + 6 = 9
The Major Sixth and the Minor Third: 6 + 3 = 9
The Major Second and the Minor Seventh: 2 + 7 = 9
The Major Seventh and the Minor Second: 7 + 2 = 9

Only the Tritone remains pairless and doesn't add up to the sacred number. Is this perhaps why it was considered demonical, besides its obvious dissonance?

Also, the multiplication of the ratios of the above mentioned symmetrical pairs gives always 2. For example the Fifth and the Fourth: 3/2 * 4/3 = 2 and so on.

The Sacred Numbers 12 and 1 are represented by the 12 semitones and the Fundamental (the 12 Apostles and Jesus) and they add to form the sacred number 13, the Holy Completion.

The two most pure intervals, the Octave and the Fifth, are the 12th and 7th semitone, respectively,  counting from the Fundamental, which are very sacred numbers.


Even today, the last 3 notes from Table 1 above are considered dissonances, as they don’t sound pleasant to the ear when played together in a chord. These notes are located beyond the appearance of the third octave, in a region known as the 4th-order harmonics. Today the western musical scale uses only the notes depicted in Table 1 and no more. This is why you see only 12 keys per octave in a piano. These 12 notes represent all of the notes in the first 3 harmonic orders, plus only a selected few from the next order. This means there are other notes in the very same harmonic order, from which the minor second, major seventh and tritone come, which western music has neglected but eastern music embraced.

Which tones sound pleasant when played together and which don’t is the topic of the next chapter: Harmony.

 

A Side Note on the Relevance of the Octave:

The octave is a very special harmonic. Not only is it the first overtone of any sound, but mathematically seen it is always the doubling in frequency of the fundamental. An octave recreates an exact replica of the original tone in a higher order. We can apply this knowledge to make extrasonic vibrations hearable. For instance, we can produce a high enough octave of the frequency of rotation of the Earth that we can hear its actual tone! Producing hearable tones of naturally occurring constant frequencies in the universe is the the topic we deal with in the Tones of the Solar System section of this website.

 

 

Read More: What is Harmony?

Audio Samples: Intervals

Open in new window