1. Introduction: Knowledge of Sound Itself and Temperament (Part 2)

This article is a continuation from the Introduction (Part 1).

1.4 Twelve-Tone Equal Temperament

If you were told to adjust the [Operation 1] of "raising by one octave" repeated 7 times and the [Operation 2] of "raising by a perfect fifth" repeated 12 times so that both result in "making the original frequency 128 times higher," what would you think? [Operation 1] already results in 128 times, so no adjustment is needed; the problem is [Operation 2].

Since $${1.5^{12}=129.75\cdots>128}$$, you just need to make the number 1.5 a little smaller. In other words, you must find a number $${x}$$[Note 1] that becomes exactly 128 when raised to the 12th power. Mathematically, this means solving a 12th-degree equation, and since manual calculation is impossible, the calculation process is omitted, but the solution expressed as a decimal is as follows.

$$
\begin{aligned}
x^{12}=128 (=2^7) \\
\text{i.e.} 
x\fallingdotseq1.4983\cdots
\end{aligned}
$$

Expressing this $${x(\fallingdotseq1.4983\cdots)}$$ mathematically and accurately means "the 7th power of the 12th root of 2." The 12th root of 2 $${\sqrt[12]{2}}$$ is "a number that becomes 2 when raised to the 12th power" (expressed as a decimal, it is approximately 1.059463...).

$$
x=1.4983\cdots=(1.059463\cdots)^7 \\
x=2^{7\over12}=(\sqrt[12]{2})^7
$$

In this way, by replacing the 1.5 times part of [Operation 2] with 1.4983... times, we successfully made E# and F the same frequency, and succeeded in "closing" the octave with 12 semitones.

In fact, most pianos and other keyboard instruments on Earth today are tuned based on this idea, that is, the idea of defining "raising by a perfect fifth" as "making the frequency 1.4983... times higher." This is called "Twelve-Tone Equal Temperament" (or simply Equal Temperament) [Note 2]. It is a method of making the octave fit into 12 semitones by sacrificing just a little bit of the pure resonance of the Pythagorean perfect fifth (which comes from the clean number of exactly 1.5 times).

In other words, in twelve-tone equal temperament, "raising by 1 semitone" physically means "multiplying the frequency by $${{\sqrt[12]{2}}}$$." To put it another way, every time you go up one semitone, the frequency increases by approximately 5.9463%.

1.5 What is Just Intonation, What is Temperament

In the previous section, I wrote as if it were a "happy ending" that we succeeded in closing one octave with 12 semitones in twelve-tone equal temperament, but this is by no means something to be celebrated without reservation. We must not forget that twelve-tone equal temperament sacrifices the pure resonance of the Pythagorean perfect fifth.

The reason why the Pythagorean perfect fifth had a pure resonance was that the ratio of the frequencies of the two notes was a very simple integer ratio of "2:3". In this way, when the ratio of the frequencies of two notes is a simple integer ratio, and therefore the two notes resonate beautifully when played simultaneously, this is expressed in music theory as being "just". The "perfect fifth" we have seen so far in Pythagoras's experiments can be expressed as a "just perfect fifth".

Since twelve-tone equal temperament is based on the irrational number that is the 12th root of 2, there is in principle no such thing as a just interval there. The exceptions are the "perfect unison" (the exact same note) and the "perfect octave" (one octave apart). No matter how you define the pitch of 12 semitones, only the "perfect unison" (1:1) and "perfect octave" (1:2) will always be just (therefore, terms like "just perfect unison" or "just perfect octave" are not commonly used). In twelve-tone equal temperament, all intervals other than the perfect unison and perfect octave are non-just. In other words, surprisingly, it is impossible to play a "just perfect fifth" on your keyboard instrument—unless, of course, it has been specially tuned that way.

In fact, how to determine the relative pitch of the 12 semitones within an octave is an old yet new problem. A definitive "this is the final version" solution to this problem has not been reached even today. The mechanism or system for determining the pitch of the 12 semitones [Note 3] within an octave is called "temperament". Twelve-tone equal temperament is a type of temperament. Twelve-tone equal temperament is (one of) the historically newest temperaments and has become the overwhelming mainstream in today's music industry.

On the other hand, a temperament obtained by repeatedly stacking just perfect fifths, as in [Operation 2] of Pythagoras's experiment, is called "Pythagorean tuning". In Pythagorean tuning, you stack just perfect fifths (which are slightly wider than the perfect fifth in twelve-tone equal temperament), so the very last one (between A# and F in the previous example) becomes considerably narrower than a just perfect fifth. As we have already seen, stacking perfect fifths 12 times can cover 12 semitones, but Pythagorean tuning is a temperament that makes 11 of these 12 perfect fifths just perfect fifths and pushes the "distortion" onto the remaining one. This last perfect fifth that was sacrificed in Pythagorean tuning is called the "wolf fifth". "Wolf" refers to the animal. When this unnaturally narrow perfect fifth is played, a very strong "beating" occurs, creating a sound that can hardly be called beautiful. This "beating" was likened to the howling of a wolf.

Now, in Pythagoras's experiment, E# and F did not become the same note, with E# being a slightly higher note, and the difference between these two notes is called the "Pythagorean comma". Calculated from the perspective of twelve-tone equal temperament, this is a deviation of about 0.2346 semitones (that is, a little less than a quarter of a semitone). In addition, a unit [Note 4] that further divides one semitone of twelve-tone equal temperament into 100 equal parts is commonly used, called "cent". Using this unit, the Pythagorean comma is 23.46 cents.

Also, while twelve-tone equal temperament is uniform everywhere and thus only exists in one type, there are 12 types of Pythagorean tuning. This is because there are 12 patterns for which of the 12 perfect fifths is to be sacrificed.

1.6 The Emergence of Just Intonation

So far, intervals such as "perfect unison," "perfect octave," and "perfect fifth" have appeared. In Europe, until the Middle Ages, music was created centering on these intervals (more accurately, including the "perfect fourth," which is the inversion of the perfect fifth), but from the early modern period to the modern era, third intervals such as the newly emphasized "major third" began to gain importance (incidentally, this overlaps with the transition from melody-oriented music to harmony-oriented music). You have likely heard the terms "major/minor" (or the German version "dur/moll"). As I will explain in detail later, without the idea of emphasizing thirds, the concept of distinguishing between major and minor keys would not have emerged.

Now, to put it simply, a "major third" is the interval from C to E on the white keys of a piano. Let's look at the frequency ratios of C and E in both twelve-tone equal temperament and Pythagorean tuning. Incidentally, a major third corresponds to 4 semitones.

• In twelve-tone equal temperament: $${(\sqrt[12]{2})^4\fallingdotseq1.260\cdots}$$ (cannot be written as a fraction because it is an irrational number)

• In Pythagorean tuning, one simply takes the pure perfect fifth four times (C⇒G⇒D⇒A⇒E) and then descends two octaves:
  $${1.5^4×0.5^2=1.265625}$$ (written as the simplest fraction, $${\dfrac{81}{64}}$$)

It is clear just by looking at the numbers that the sound of a major third is not very beautiful, not only in twelve-tone equal temperament based on irrational numbers but also in Pythagorean tuning based on pure perfect fifths. The condition for making the sound of two different notes beautiful—that is, the condition for an interval to be pure—is that the ratio of the frequencies of the two notes must be a simple integer ratio. To review, a (pure) perfect unison is 1:1, a (pure) perfect octave is 1:2, and a pure perfect fifth is 2:3. Compared to these, the major third in Pythagorean tuning [Note 5] is 64:81, which is a difficult number to call a simple integer ratio.

So, if we were to artificially set a "pure major third", what should we do? Could we adjust the 64:81 ratio just a little to make it a cleaner ratio? Those with a good intuition may have noticed that if you lower the 81 slightly to 80, you get 64:80, or 4:5. Happily, we have obtained a simple integer ratio without deviating too far from the major thirds of twelve-tone equal temperament or Pythagorean tuning. From this, the frequency ratio of a pure major third is defined as 4:5.

4:5, when written as a decimal, is 1.25 times. This is a significantly lower number than the 1.260... of twelve-tone equal temperament or the 1.265625 of Pythagorean tuning. The difference between the major third of twelve-tone equal temperament and a pure major third is 14 cents (0.14 semitones) (the pure major third is 14 cents lower).

Now, by defining the pure major third, a new temperament, by the name of "just intonation", finally makes its appearance. Historically, the theory of just intonation was already established by the great 2nd-century Greek scholar Ptolemy, but it is said that just intonation was put into practical use in the late 15th century by the Spaniard Bartolomé Ramos. That is more than 2,000 years after the establishment of Pythagorean tuning. In terms of number theory, it took over 2,000 years to introduce the new prime number "5" into a system that had relied solely on the two prime numbers "2" and "3".

1.7 Diatonic scale in just intonation

Twelve-tone equal temperament essentially has only one type of "weapon": the "twelfth root of two." Pythagorean tuning has only one type of weapon as well: the "pure perfect fifth (2:3)." On the other hand, just intonation has two types of weapons: the "pure perfect fifth (2:3)" and the "pure major third (4:5)." Let's use these two weapons to define the diatonic scale of C Major, the so-called do-re-mi-fa-sol-la-ti-do.

In just intonation, we first define the C-E-G triad. Since C:E is a pure major third, it is 4:5, and since C:G is a pure perfect fifth, it is 2:3. Therefore, if we write the C-E-G triad together as a ratio, it becomes C:E:G = 4:5:6. We have converted the 2:3 ratio of C:G to 4:6. A triad whose frequency ratio is in the form of 4:5:6 is called a "pure major triad". As an example, in a pure major triad with A=440Hz as the root, the frequencies are (A, C#, E) = (440Hz, 550Hz, 660Hz).

As I will discuss in detail later when describing the function of chords, there is a concept called primary triads. The primary triads in C Major are the three major triads C-E-G, F-A-C, and G-B-D (they have the functional names tonic, subdominant, and dominant, respectively, but you don't need to memorize those now). If you master these primary triads, you can cover all seven notes of C-D-E-F-G-A-B without any excess or deficiency. Just intonation is created by stacking three pure major triads (F-A-C, C-E-G, and G-B-D) in the order of subdominant, tonic, and dominant.

Let's look at the frequencies specifically. First, we define the frequency of the lowest note, F, as 176Hz.

• Since F:A:C = 4:5:6, (F, A, C) = (176Hz, 220Hz, 264Hz)

• Since C:E:G = 4:5:6, (C, E, G) = (264Hz, 330Hz, 396Hz)

• Since G:B:D = 4:5:6, (G, B, D) = (396Hz, 495Hz, 594Hz)

Of these, if we raise F and A by one octave (i.e., double the frequency) and lower D by one octave (i.e., halve the frequency), all notes will fit within one octave. Summarized, it looks like this:

Note Name / Frequency
C / 264Hz
D / 297Hz
E / 330Hz
F / 352Hz
G / 396Hz
A / 440Hz
B / 495Hz

The C Major just intonation created in this way has the following characteristics. Let's indicate the advantages with ○ and the disadvantages with ×.

[○] All primary triads (C-E-G, F-A-C, G-B-D) are pure major triads (4:5:6) (self-evident from the definition)
[○] The two minor triads, A-C-E and E-G-B, also become pure minor triads (10:12:15)
[×] The D-F-A minor triad is not pure (D:F:A = 27:32:40)
[×] Because it is a temperament created specifically for C Major, if you modulate to a key far from C Major, extremely unpleasant sounds appear, making it unusable
[×] Notes other than C-D-E-F-G-A-B (the 5 black keys: C#, D#, F#, G#, A#) cannot be clearly defined
[×] Two types of whole tones are created: the major whole tone (C:D = F:G = A:B = 8:9) and the minor whole tone (D:E = G:A = 9:10), resulting in inconsistency

In particular, from the perspective of today's music, the fact that modulation is not freely possible is the fatal flaw of just intonation. In music after the late 19th century, when dramatic effects through modulation became emphasized, it became rare for just intonation to be used as is [Note 6].

Column: How many types of Just Intonation are there? ~ Just Intonation in Minor Keys

I mentioned that "there is only one type of 12-tone equal temperament" and "there are 12 types of Pythagorean tuning depending on where you place the wolf fifth." So, how many types of Just Intonation are there? While there are various ways to think about this, I take the simple position that "there are 24 types." Why 24? To explain that, I must explain how to create Just Intonation for minor keys.

Just Intonation for C major could be obtained by making all the primary triads of C major (tonic, subdominant, and dominant) into pure major triads. So, for its relative key, A minor, is it not okay to use the same temperament obtained for C major? In my opinion, it is not.

Just Intonation is a temperament created from primary triads. Therefore, for A minor, the primary triads of A minor—namely (in order of tonic, subdominant, and dominant) A-C-E, D-F-A, and E-G-B—must all be made into pure minor triads (10:12:15). The D-F-A chord in C major was not pure (27:32:40). In A minor, this minor triad must be made pure. In the Just Intonation for C major mentioned earlier, D was 297Hz, but by retuning this lower to D = 293.333...Hz, we can make D:F:A = 10:12:15, and with this, Just Intonation for A minor is obtained.

In this way, since two types of Just Intonation, major and minor, can be created for each of the 12 notes, a total of 24 types of Just Intonation can be created.

By the way, a pure minor third, when written as a ratio, is 5:6, and as a decimal, it is 1.2, but the minor third in 12-tone equal temperament (= 3 semitones) is (the 12th root of 2) cubed = 1.189.... The difference between a pure minor third and a 12-tone equal temperament minor third is 16 cents (0.16 semitones) (the pure minor third is 16 cents higher).

1.8 (Reference) Other Temperaments

In my opinion, the only temperaments important for understanding today's popular music theory are the three I have already touched upon: Pythagorean tuning, Just Intonation, and 12-tone equal temperament. However, in the history of Western classical music, 12-tone equal temperament did not appear immediately after Just Intonation; numerous improved temperaments were developed in between. I will briefly explain the representative ones. The explanation in this section is intentionally incomplete (because I have only summarized it to the extent of my understanding). For more detailed information, please consult other books.

Meantone Temperament: Developed in the 16th century. A temperament where the perfect fifth is taken narrower than pure so that when four perfect fifths are stacked as C -> G -> D -> A -> E, the interval from C to E becomes a "pure major third + 2 octaves." Mathematically, this perfect fifth is the "fourth root of 5" (the fourth root of 5), which is approximately 1.4953... in decimal. This is, of course, narrower than pure (exactly 1.5 in decimal) and even narrower than the perfect fifth of 12-tone equal temperament (1.4983...). Although the pure perfect fifth was abandoned, the range of modulation was slightly expanded compared to Just Intonation. However, keys where the wolf occurs remain. Because the width of the whole tone becomes the midpoint between the major whole tone (8:9) and minor whole tone (9:10) of Just Intonation, it was named "meantone" temperament.

Kirnberger III Temperament: Developed in the 18th century. A temperament devised by skillfully combining the pure perfect fifths of Pythagorean tuning and the narrow perfect fifths of meantone temperament so that the wolf in Pythagorean or meantone temperament becomes less noticeable. Depending on the key, there are those where major thirds sound pure and those where perfect fifths sound pure, making it easy to create dramatic effects through modulation.

Werckmeister III Temperament: Almost the same as Kirnberger III, but the order of combining pure perfect fifths and narrow meantone perfect fifths is different. The pure major thirds found in Kirnberger III are lost in all keys, making it a temperament closer to 12-tone equal temperament.

1.9 Summary of Temperaments

Setting aside the improved temperaments after meantone, let's summarize the three basic temperaments for understanding popular music in a table (you can also view the Google Spreadsheet). I have also added some information not mentioned in the main text here.

To repeat, as a trend of history, most popular music is made in 12-tone equal temperament, but 12-tone equal temperament is not an all-purpose temperament. All temperaments have the nature of "robbing Peter to pay Paul." I believe one should have the awareness that most of the music flowing from modern televisions, radios, and the internet is music thatsacrifices pure resonance. I think that music theory regarding "pitch" is most sharpened in the field of "jazz" in popular music (I will delve into actual jazz music theory from Chapter 4 onwards), but that rich and profound world of sound is built upon this noble sacrifice.

Also, from Chapter 2 onwards, I would like to proceed with explanations based on 12-tone equal temperament in principle, but the concepts of Pythagorean tuning and Just Intonation can also be applied there. In that sense, I believe that having a minimum of correct knowledge about temperament is not a wasted detour, but necessary "basic physical strength" for thinking about the theory of pitch in popular music. Thank you for staying with me this far. See you in the next chapter.

Footnotes

[Note 1] To be mathematically precise, x must be a "positive number" here. "Negative numbers" and "imaginary numbers" are excluded. This is because frequencies have no meaning unless they are positive numbers.

[Note 2] By the way, the great German composer J.S. Bach has a work called "The Well-Tempered Clavier," but it is said that this "well-tempered" is a mistranslation. In J.S. Bach's time, 12-tone equal temperament had not yet been put into practical use, so please do not misunderstand.

[Note 3] Temperaments that divide the octave into finer parts than 12 semitones have also been devised (such as 53-tone equal temperament), but they are not covered here.

[Note 4] Of course, to express it mathematically strictly, it means dividing equally in a geometric progression based on frequency. Using the unit called cents, a semitone in 12-tone equal temperament is 100 cents, a perfect fifth is 7 semitones, so 700 cents, and one octave is 12 semitones, so 1200 cents. If you view 1 cent as a ratio of frequencies, it becomes the "1200th root of 2" (the 1200th root of 2), which is approximately 0.05778% when expressed as a percentage.

[Note 5] Strictly speaking, this applies under the assumption that the process of deriving a major third from a perfect fifth and a perfect octave does not include the wolf fifth.

[Note 6] Although rare, it is not extinct. In the world of modern popular music, for example, Enya from Ireland is famous for using just intonation. Additionally, Hiroki Tamaki (1943-2012) is cited as an important figure who advocated for the revival of just intonation in Japan. As a practical matter, when trying to create major triads using only the human ear, such as in a cappella performances, one naturally gravitates toward pure major triads. Furthermore, instruments that utilize the harmonic series, such as brass instruments, inevitably make it easier to produce pure intervals rather than those based on twelve-tone equal temperament.

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