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  • 聴き比べ:古典音律(ピタゴラス音律、純正律、中全音律、ウェル・テンペラメント)と平均律



Why am I talking about the classical tunings and temperaments, in the first place?

Please listen to Sound Source 1. (Stereo earphones and headphones are recommended.)


Sound Source 1: British National Anthem (the first half) by Equal temperament


I played this in the “Equal temperament,” which is familiar to you and me. Let’s compare this with the Just intonation. I will play it, for comparison, in the following order: Equal temperament, Just intonation, Equal temperament, and Just intonation.


Sound Source 2: of British National Anthem by Equal temperament, Just intonation, Equal temperament, and Just intonation.

いかがでしょう? 純正律はピュアな感じで、平均律は、なんか人混みの中みたいにザワザワ、モヤモヤしていませんか? 楽器店に行って電子楽器でこれをやると、お店のスタッフさん達は、「平均律ってこんなに濁ってたんですか?純正律の方がずっと澄み切った響きですね」と、ビックリします。

それなら、なぜみんな純正律を使わず、世の中平均律ばかりになっているのだ? と思いませんか? そう、鍵盤楽器の調律法は、ピタゴラス音律、純正律、中全音律、ウェル・テンペラメントを経て、最後に平均律になって、以後現代まで200年近くずっと平均律です。今聴いていただいた通り、平均律はそんなに美しい響きじゃないのに。


How do you like it? Doesn’t the Just intonation sound much purer, and the Equal temperament sound zany and fuzzy (like you listen to it in a crowd)? When I demonstrate this at music shops,  the staffs always say, “I didn’t know that the Equal temperament is so muddy. The Just intonation is much clearer.”

If that’s the case, then why is it the case that no one is using Just intonation and all we have is the Equal temperament? Yes, the tuning method for keyboard instruments has started with Pythagorean tuning, and then Just intonation, Mean-tone temperament, Well temperament, and finally, Equal temperament.  The Equal temperament has been dominating for nearly 200 years since then until the present day, although, as you have just heard, the mean rule does not sound so beautiful.

 Let me tell you in order how this situation came to be. First of all, chords in Western music were not always do-mi-so from the beginning, but in the beginning, they were only do and so. Let’s play this and see the difference.


Sound Source 3: do-mi-so and do-mo chords in Just intonation


The do-so chord doesn’t have mi in the middle, so it sounds hollow and empty. The do-so chord harmonizes best when the frequency ratio is 2:3 (= 1.5). If we make a scale piling up this 2:3 frequency ratio and extending it to do-so-le-ra-mi-si… is called the Pythagorean scale. This tuning method was established prior to the Just intonation you heard in the beginning of this article. Let’s listen to it a little more.


Sound Source 4: Playing do-so-le-la-mi-si in Pythagorean tuning


It is not easy to find poplar music pieces that use the fifth chords only. In modern music, however, the fifth chords are sometimes used for exotic effects. I will give two examples. The first one is Debussy’s “The Sunken Cathedral.”


Sound Source 5: The beginning of the Sinking Cathedral in Pythagorean tuning


The second is the passage “Pines near the Catacombs” from Respighi’s symphonic poem “Pines of Rome.”


Sound Source 6: “Pines near the Catacombs” played in Pythagorean tuning


そう、ピタゴラス音律は次第に純正律に取って代わられます。なぜそうなったか? ピタゴラス音律には欠点が二つあります。1つ目はウルフの存在です。ウルフ? それは次のようなものです。


It sounds like the chanting of monks at a temple. I agree that it is not easy to differentiate the pureness of fifth chords in Equal temperament from that of the Pythagorean tuning.  So, as people gradually tend to love do-mi-so chords over do-so chords, the Pythagorean tuning gradually gives the way to the Just intonation.

Let’s see the situations when Pythagorean tuning had to give the way to Just intonation. Actually, Pythagorean tuning has two drawbacks, the first of which is the existence of Wolf. Wolf? What is it?

If you try to tune the twelve tones by the Pythagorean tuning by making a scale with a frequency ratio of 1.5, like Do-So-Re-La-Mi-Si-Fa#-Do#-So#-Mi♭-Si♭-Fa-Do, you will not get back to the proper Do. This is because no matter how many times you multiply the frequency ratio 1.5, it will never reach a power number of 2 (= 2^n = n octave difference). But we anyhow need to return to the initial “do.” This means that we have to set an abnormal fifth somewhere that is much narrower than the purely harmonized fifth. Usually, we create it at the fifth between the so# and the mi♭. This is called the “Wolf” in Pythagorean tuning.


Source 7: Playing a Wolf: So # and Mi Flat chord in Pythagorean tuning

どうです? 酷いでしょう? とても音楽に使えません。これをウルフと言います。狼の鳴き声のように非音楽的な和音という意味です。この「五度を積み重ねてもドがドに戻らない矛盾」を一手に引き受ける、ウルフというかわいそうな和音は、あとで純正律や中全音律にも出て来ますので、名前を覚えておいてください。ウルフです。


What do you think? It’s terrible, isn’t it? You can’t use this chord for your composition or arrangement. The meaning of wolf is a non-musical chord like a wolf’s cry. Please remember the name of this poor chord, Wolf, because it will appear later in the just intonation and the mean-tone temperament as well.

 Now, the second drawback of the Pythagorean tuning, which was actually conceived to eliminate, is that, in the Pythagorean tuning, “do” and “mi” don’t harmonize nicely.


Sound Source 8: Playing a do-mi chord in the Pythagorean tuning

そんなに酷い和音ですか? とおっしゃるかもしれませんが、実際の曲で比較した方がわかりやすいと思いますので、たとえばキリスト教で歌われる「神ともにいまして」という歌があります。これを純正律で、続けてピタゴラス音律で弾いてみます。

Maybe you didn’t feel so strange at this do-mi chord.  But that is because a chord containing only “do” and “mi” is too simple. If you hear real music with full of do-mi-so, fa-ra-do, and so-si-re, then you will notice the big difference. Next example is a hymn “God be with you. I will play it in the just intonation, followed by the Pythagorean tuning.


Sound Source 9: “God be with you” played with the Just intonation and the Pythagorean tuning.


では純正律には欠点はないのでしょうか? あります。先ほども言いましたが純正律にもウルフがあるのです。しかもウルフの位置を変えられるピタゴラス音律と違って、純正律のウルフは理論上レとラから成る和音に生じて、これは動かせません。弾いてみましょう。

You can clearly see the difference even when played on a piano tone, which has few overtones.

So, is there any flaw in the just intonation? Yes, there are. As I mentioned before, the just intonation also has a wolf (actually, wolves, if you want to play with many # or flats). And unlike the Pythagorean tuning, where the position of the wolf can be changed, the wolf in the just intonation is theoretically fixed on chords consisting of the “re” and “la,” and this wolf cannot be moved. Let’s play it.


Source 10: Playing a Wolf: re-la chord in the just intonation.



Wow, that’s terrible. The re-la chord is frequently found in C major and F major, and is even the main chord of D major and D minor. Mozart would never have been satisfied with this.

This is where the mean-tone temperament came in. The English word “mean-tone” is also often used as a Japanese word. For the details of how it works, please refer to No. 21-23 of “Kyuuri”. Let’s see what happens when we play Mozart’s favorite pieces in D major and D minor using the just intonation. The first is a passage from the beginning of the third movement of the piano sonata (Köchel number 576).


Sound Source 11: Compare a passage from the third movement of the Sonata in K.576 in the just intonation and in the mean-tone temperament.


Wow, they are totally different. Next is the beginning of the piano concerto in D minor (Köchel number 466).


Sound Source 12: Comparison of the beginning of the concerto K.466 in just intonation and in mean-tone temperament.



Yes, they are totally different. We went to the trouble of improving the Pythagorean tuning to the just intonation, but that was not enough to play Mozart, a composer of the late 18th century, and we can see that the mean-tone temperament was necessary. This is what Mozart meant by the mean-tone that he loved so much.

In this way, with the mean-tone temperament, all of Mozart’s pieces can be played beautifully, and a recital can be held with Mozart alone. And at the end of the recital, you decide to play an encore. I wrote in “Kyuuri” that it would be a big problem if you played Chopin’s “Farewell”, a composer from the first half of the 19th century, saying, “I will play a farewell piece for the last time, ladies and gentlemen.” Let’s do it.


Sound 13: Play the beginning and the end of Mozart’s Turkish March by mean-tone temperament. Continue to play the beginning of Chopin’s “Farewell.”


Wow, you just ruined a beautiful Mozart recital. The same thing happened if you play an encore of a piece by Beethoven.


Source 14: The beginning of the second movement of Beethoven’s ‘Pathétique’ in Mean-tone temperament.



This is also terrible. This tells that if you want to play Beethoven or Chopin freely, even Mean-tone temperament is not good enough.

This is why the Well temperament was invented.  There are many different types. For now, let’s just confirm that you can play Chopin and Beethoven comfortably with the Well temperament. Of the many, the Werckmeister III temperament and the Kirnberger III temperament are the most famous, but let’s play here in the Kirnberger III temperament, a pupil of Bach.


Tone 15: Play the beginning of the Farewell and the beginning of the second movement of the Pathetique in Kirnberger III.



Yes, it’s all right. Thus, by means of the Well temperament, you can now play any chord in any key without any inconvenience.

 In fact, the difference between the Werckmeister III temperament, the Kirnberger III temperament and Equal temperament is very subtle. Nevertheless, I wonder, “Can we just do the Equal temperament and forget about the rest?” I think. I’ll talk about that another time, so that’s all for today. See you next time.




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