### Who Studied The Relationship Between Music Intervals?

Pythagorean Intervals, Section 3.2 As was mentioned in the last part of this article, Pythagoras had an interest in figuring out the notes and scales that were utilized in Greek music. He focused his attention specifically on the stringed instruments of Greek origin, specifically the lyre.

His initial insight was that when two strings of the same length, tension, and thickness are plucked, the resulting sound is identical to that of a single string of the same length, tension, and thickness. This indicates that they have the same pitch and that when performed together, they produce a pleasing (or consonant) sound.

The second thing that was discovered was that if the string’s lengths are different (while maintaining the same tension and thickness), then the string’s pitches will be different, and when performed together, they will typically sound terrible (also known as discordant).

Finally, he came to the conclusion that for specific string lengths, even though the two strings continued to have different pitches, the resulting sound was now consonant rather than discordant. The distance between two notes is referred to as an interval, which was first coined by Pythagoras. For instance, as was discussed before, when two strings have the same length, they have the same pitch, and the connection, also known as an interval, between the notes is referred to as a unison.

If one string is exactly one-half the length of the other string, its pitch will be significantly higher than the other string’s, yet when played simultaneously, they will still sound consonant. An octave is the name given to this interval. Finally, if one string is two-thirds the length of the other string, then both strings will sound consonant when played simultaneously.

This particular interval is known as a Perfect Fifth. Already, we have a really significant difference between the two. The lengths of the strings must conform to a predetermined ratio in order for these intervals to be specified. It is not possible to create an octave by shortening the length of a single string by a predetermined amount, such as 10 centimeters.

Instead, an octave is created by dividing the whole length of a string by a factor of 2. To put it another way, if the lengths of the strings are in a ratio of 2 to 1, the interval formed by the pitches of the strings will be an octave. There are a wide variety of intervals in music; each interval has its own distinctive tone, and each interval plays a unique part in the development of musical harmony.

Name | Ratio |

Unison | 1:1 |

Octave | 2:1 |

Perfect Fifth | 3:2 |

As was just pointed out, every interval possesses a unique quality. Because the two notes are identical, the unison might not seem all that intriguing to you. It’s possible that this is accurate, yet this interval is the simplest and most fundamental one there is.

- Nevertheless, it has a lot of potential when put to good use.
- Imagine a vocalist performing a simple song.
- Now picture a choir consisting of a hundred different voices, all of which are singing the same tune in unison, which means that everyone is singing at the exact same pitch.
- Actually, it is pretty challenging to get that many people sing exactly enough together, but when they do, the result can be quite stunning.

Therefore, exceptional composers are able to make efficient use of even the most basic interval. The octave is the interval that is considered to be the next most important. As was covered in the part before this one, it is responsible for defining the range of the music scale.

Because of their striking similarity in sound, two notes that are separated by an octave are always referred to by the same name. For instance, the middle C is a common starting point for works written for beginner piano. On the other hand, if you move up an octave from that point, the note you reach is still referred to be a C.

In point of fact, there are times when it is difficult to determine whether or not two notes are separated by one or two octaves. If a person has a poor voice and cannot sing a certain tune because the notes are too high for them, it is perfectly okay for that person to sing the melody an octave lower than they would normally.

The melody is repeated throughout the song in its entirety. To long last, we have arrived at the Perfect Fifth. In this instance, there is no room for debate regarding the fact that two notes separated by a Perfect Fifth constitute two distinct notes. G is the name given to the note that is a Perfect Fifth higher than Middle C on the piano.

As a result, the Perfect Fifth is the first interval that gets us into harmony, which is the subject of how the various pitches interact with one another. Once again, the Perfect Fifth possesses a characteristic that sets it apart from other intervals.

String 2 | 1/2 | 2 | 2/3 | 3/2 |

String 1 | 1 | 1 | 1 | 1 |

Ratio | 1/2 | 2/1 | 2/3 | 3/2 |

Interval | Up an octave | Down an octave | Up a Fifth | Down a Fifth |

It is also extremely crucial to keep in mind that the beginning frequency, string length, string tension, and string type do not have any bearing on the intervals. Intervals only provide information about the connection between two notes, but up until now, similar to Pythagoras, we have only defined them in terms of the lengths of strings.

- It has been discovered that the frequency of a string will increase by a factor of two if you take the length of the string and reduce it by a factor of two.
- Therefore, the ratios for the intervals are the same regardless of whether we are thinking about the lengths of the strings or their pitches.
- Intervals are incredibly significant in music, so why are we spending so much time talking about them in a class that’s supposed to be about physics? The fact that these simple consonant intervals are determined by ratios of relatively tiny integers is, from a scholarly point of view, by far the most essential aspect of those simple consonant intervals.

Pythagoras found this to be a really fascinating development. Take into consideration the fact that Pythagoras had just completed the development of mathematics, in particular integers. In addition to this, he had recently finished developing arithmetic.

- But this was only a theoretical consideration.
- Pythagoras did not have any means of knowing whether or not mathematics will be beneficial for anything in the future.
- After further investigation, it was discovered that numbers and ratios were able to adequately express the consonant musical intervals.
- In point of fact, this also brings to light the profound relationship that exists between the experience of humans and abstract mathematics.

This interval sounds nice (consonant), whereas that interval sounds unpleasant (dissonant), seems to be a judgment that can only be made by the listener, yet it can really be predicted using abstract mathematics. It would appear that our subjective reactions to the outside environment adhere, at times, even to mathematical principles.

To get back to intervals for a moment, we may describe a new operation by saying that we can combine intervals. For instance, we could raise the pitch by an interval equal to an octave, or we might lower it by an interval equal to a perfect fifth. Increasing the pitch by an octave in this context translates to doubling the frequency by a factor of 2.

Taking away a fifth of something requires dividing the total by three and two. It turns out that 2/(3/2) equals 4/3 when added up. Therefore, by merging several intervals, we have created a whole new interval that we will refer to as the Perfect Fourth.

- The ratio of four thirds to three fourths is the definition of the perfect fourth.
- To summarize: Octaves may be generated using the ratios of 1/2 and 2/1.
- Fifths are obtained using the ratios 2/3 and 3/2.
- Fourths may be calculated using the 3/4 and 4/3 ratios.
- Take note that the only integers involved in the ratios presented above are 1, 2, 3, and 4.

It will be beneficial to practice writing down ALL ratios that use these numbers, so let’s do that now.

1/1 | 1/2 | 1/3 | 1/4 |

2/1 | 2/2 | 2/3 | 2/4 |

3/1 | 3/2 | 3/3 | 3/4 |

4/1 | 4/2 | 4/3 | 4/4 |

Now, your mission is to figure out which time period each of these relates to. As an illustration, falling down by a Perfect Fifth corresponds to the fraction 2/3. Going up by an octave TWICE is what increasing by 4/1 means. In other words, moving up one octave requires you to multiply by 2, therefore going up two octaves requires you to multiply by 2 twice, which results in the equation 2×2 = 4.

At this stage, we are able to pose the question, why do we stop at 4? Why don’t you just use ratios of 5, 6, 7, and 8 for the intervals? The solution to this problem is that we can make advantage of bigger numbers. Pythagoras, on the other hand, stopped at the fourth because he understood that he already possessed all of the intervals that he need to construct the musical scale, which was his primary objective.

In addition, Pythagoras was a man who placed a high value on simplicity. He sought to provide the musical with the most straightforward explanation possible. He was of the opinion that the simplest explanation must be the one that was right. This has remained one of the most fundamental beliefs held by physicists and scientists right up until the present day.

Pythagoras was in fact so pleased with these intervals that he referred to them as “Perfect,” namely the Perfect Fifth and the Perfect Fourth. (The unison and the octave are likewise perfect; however, as was said earlier, because they are so perfect, they are not truly thought of as being a new note.) First, we will organize what we have learned in a slightly different way, which will make the process of creating the scale a little bit simpler.

Next, we will discuss how to generate a musical scale from these intervals, which will be covered in a later portion of this lesson. Beginning with the note D, the table that follows provides an illustration of the link between the notes on the scale, intervals, and frequencies:

Name of note | Name of interval | Ratio of interval | Decimal equivalent | Frequency of note. |

D | Octave | 2/1 | 2.000 | 587 Hz |

A | Perfect Fifth | 3/2 | 1.500 | 440 Hz |

G | Perfect Fourth | 4/3 | 1.333 | 391 Hz |

D | Unison | 1/1 | 1.000 | 293 Hz |

These are the notes that the strings of the Greek lyre were tuned to, despite the fact that we do not yet have enough notes for a complete musical scale. However, we do not yet have enough notes. In addition, since Gregorian chant was mostly focused on fourths and fifths, these notes were utilized the majority of the time.

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