Continuing the previous article on tuning, it is time to look at some of the theory and science behind how we play in tune. In this issue, we will discuss the harmonic series and some his­torical aspects of tuning.

To fully under­stand “how” to tune, it is always good to dis­cuss the prin­ciples behind the practical subject of tuning. I have seen many musi­cians who have great difficulty in grasping the “why” of tuning — and, con­sequently, “how” to play in tune. This should be of concern to all musicians — but, with many recorder players, what happens is that they assume that using a correct fingering will automatically produce a note played in tune. These players don’t consider pitch to be important, which is very unfortunate.

For this reason, I’ve decided to explain a number of issues relating to only one subject: tuning, which does not have to be difficult or complicated. Besides being a musician, I am also an engineer, so I’ve also decided to employ musical, visual and mathematical arguments. Thus, people of various interests should be able to understand “how” and “why” we tune each note as we do.

Do not worry if you do not under­stand a particular concept. The impor­tant thing is the sound that is produced when you play your instrument. I will provide ways for you to apply concepts in your recorder study. Please contact me for help if you have questions.

Why A=440 as a reference?

Sound is the result of vibra­tions that travel through the air to our ears. These vibra­tions through a medium produce a sound wave. Sound travels in the air by moving its mole­cules, so it is considered a mechanical sound wave.

In a stringed instrument, the string vibrates at a speed that is in proportion to the string tension. More tension produces faster vibrations and higher pitch. This is in inverse proportion to the string’s length and diameter: the longer in length or larger it is in diameter, the slower and lower it will be in wave and pitch.

In a wind instrument, it is the air that vibrates in inverse proportion to the length of the instrument (larger is slower and lower) and in proportion to the velocity of the air inside the instrument (higher velocity and higher pitch). This speed — or, more accurately, the number of cycles per second — we call frequency.

Nowadays, most instruments are pitched so that the note A is tuned at 440Hz or 440 cycles per second, also called 440Hz pitch. This is not a very old convention, because it was only defined after World War II. A few orchestras tune their instruments at 442Hz, or even higher at 445Hz, which produces a slightly brighter sound. Usually in Baroque music calling for “low pitch”, we use A=415Hz tuning, sounding exactly one half­-step below the modern tuning convention.

These are not the only possible standards for tuning. During the his­tory of Western music, there is evi­dence of pitch ranging from 380Hz to 502Hz, a variation that is greater than an interval of a perfect fourth. However, for two instruments to play together, it is necessary to define a tuning convention. Usually recorders use a pitch of 440 or 415Hz.

The harmonic series

Each mechanical sound wave also produces what we call harmonics or overtones. Overtones are simultaneous sounds, with a frequency that is a mul­tiple of the fundamental frequency. A specific pattern of harmonics defines the timbre of an instrument.

As an example, imagine a sound with a frequency of 100Hz. Its over­tones have frequencies of 200, 300, 400, 500Hz and so on. In a stringed instru­ment, that is very present and visible, because we can split the string in 2, 3, 4 or 5 equal parts, which allows us to hear the sound of each overtone.

Because each overtone has a fre­quency, we can say that each overtone represents a note, and this has every­thing to do with our main subject of tuning. When we hear two notes that are in tune, we say that their harmonics or overtones coincide. Consonant intervals are those with many coin­cident overtones, while dissonant intervals have few or no coincident overtones (which is the acoustical property that produces the “beats” we have tried to eliminate in tuning, as discussed in my previous article).

Of all the notes of the harmonic series (shown on the staff above), recorder players use the first eight. Each interval is in relation to the fundamental or its successive octaves:

1. fundamental
2. octave
3. perfect fifth
4. second octave
5. major third
6. perfect fifth
7. minor seventh, low­-tuned
8. third octave

These intervals form the basis of all tonal harmony, and are perfectly recognizable audibly while playing an instrument. The first eight over­ tones, omitting the out­-of-tune over­tone 7, when sounded simultaneously produce a major chord. In fact, the harmonic series is often called the “chord of nature”.

Exercise 1: Close all the holes of the recorder and play that note (C on soprano or F on alto). By blowing a little harder, you can hear the second overtone (octave). By blowing a little more, you hear the third overtone (soprano G, alto D). Overblow even a little more to hear the fourth overtone. You can reach overtones 7 or 8 with a nice instrument and good technique.

Exercise 2: Find a friend who plays a stringed instrument (violin, ’cello, viol, guitar, ukulele, etc.), and ask that per­son to demonstrate the harmonics of the instrument. If he does not know, ask him to lightly place a finger in the middle of a string, without pressure, and play. This will sound an octave above the note that is produced by that open string. If you divide the string into three equal parts, and place a finger on one of the divisions, it will sound the next note in the harmonic series (the 12th above the fundamental, or octave plus a fifth). With a violin or ’cello, it is easy to hear many harmonics that are produced by dividing the string in smaller and smaller equal parts.

Many guitar players use harmonics to tune the strings of their instruments. This is an excellent auditory exercise, and is the next topic I will discuss.

How we tune the notes of a scale: a brief history

A musical scale has seven notes, but if we consider sharps and flats (accidentals), we have 12 notes in each octave. We know that the note A (the one above middle C on a piano —in other words, the lowest A on an alto) has a frequency of 440Hz. Its overtones would be: A=880Hz, E=1320Hz, A=1760Hz, C#=2200Hz, etc ), following the harmonic series, which applies with any note as a fundamental.

We can also apply reverse thinking, and find the frequencies of notes lower than A, by dividing 440 by 2, by 3, by 4 and so on, resulting in A=220Hz, D=146Hz , A=110Hz, F=88Hz, D=73Hz.

For those who did not follow the math involved, I first multiplied the number 440 by 2 (880), then 3 (1320), etc, to produce the frequencies of higher notes. Then I divided 440 by 2, by 3, etc, moving through the intervals associated with those numbers in the harmonic series. If we think about a stringed instrument, we multiply the frequency as we divide the string into equal parts (the shorter the string, the higher the note). We divide the frequency as we stretch the string (the longer the string, the lower the note and its frequency).

Musicians, philosophers, physicists and mathematicians have understood these principles for centuries. The Greek philosopher Pythagoras (c 570–c 495 BC) discovered the basis of acoustics, math and proportions. Legend has it that, while listening to a blacksmith hammering an anvil, he noticed the musical intervals produced. He suggested that consonant sounds should be represented by simple numerical ratios derived from the tetractys (a triangle with 1, 2, 3 and 4 points per row, considered to be a mystical sequence of numbers representing the natural harmony of the universe). To define musical intervals mathematically, as I did with the multiplication above, 2:1 corresponds to the octave, 3:1 to the octave plus a fifth, 4:1 produces two octaves, 3:2 a perfect fifth, 4:3 a perfect fourth, and 5:4 the major third.

In music theory, the circle of fifths is a visual represen­tation of the relationships among the 12 tones in an octave, an important basis for tuning in tonal harmony (It also shows the order in which sharps and flats are added to key signatures See http://en.wikipedia.org/wiki/Circle_of_fifths ).

The perfect fifth is the most consonant non­-octave interval, and it is also important when used aurally in com­posing harmonious music. Following the acoustic principle of the interval ratios above, we might imagine that we need only follow the circle of fifths to find the 12 notes of the scale, right? However, our ears will prove that this is not true, and there is also an intrinsic problem with the math involved in mapping the musical scale in this way.

Traveling around the circle of fifths (here starting arbitrarily on C, which has no sharps or flats in its key sig­nature), we would cover a musical distance of eight octaves. However, the mathematical equivalents (the number of fifths at a ratio of 3:2 to return to C vs the math used to move eight octaves at 2:1 frequency ratio) do not match as we see above.

The figure shows that if we start from C, always tuning each pair in pure fifths—i.e., C, G, D, A, E, B, F#, C#, G#, D#, A#, E# (or the last four would be Ab, Eb, Bb, F in the more familiar enharmonic labels shown), the B# reached last will not have exactly the same pitch as C. Instead, it will sound much higher, a difference called the “Pythagorean Comma”.

We could take another approach, tuning pure thirds and using the mathematical ratio of the fifth harmonic in the “chord of nature”. With C again as our starting note, we would have in the space of only one octave:

This shows C,E,G,B (enharmonic for C), with a major third between each two successive notes in the octave. We can see that in this case the B# would be lower than what we expected!

These issues began to be discussed by theorists of the 14th century, when temperaments emerged.

Temperaments

Temperaments were created to solve the problems mentioned before, in order to keep the scale tuned as harmoniously as possible, and to avoid the use of “bad” or impurely tuned intervals.

In the Middle Ages, the most common was the so­-called Pythagorean Temperament, based on pure tuning of fifths Pythagoras used a base note of D, the note considered the center of the scale, especially for stringed instruments. As the repertoire at that time contained only simultaneous intervals of fourths and fifths, with thirds considered dissonances, this temperament was very appropriate to the music.

In this temperament, after “stack­-ing” fifths as they are encountered in the circle of fifths, the fifth between Eb and Ab (or G#) is out of tune — the interval is too small, for mathematical reasons discussed before. Therefore, the repertoire of this period rarely uses these notes. The beats pro­duced when these notes are played together sound like the howl of a wolf, the so­-called “wolf interval”.

In the Renaissance (around 1500 AD), musicians and theorists sought a better way to divide the scale, always seeking a better match to the current music repertoire. Polyphony was becoming more popular, and Pythagorean Tuning did not meet the Renaissance aesthetic requirements.

Meantone Temperament, favor­ing thirds instead of fifths, was intro­duced. Thus, some fifths are “narrowed” so that the major thirds remained pure, allowing them to be sonorous when sounded simultaneously. This was the first step in the direction of the development of tonal harmony, which was necessary to create the music of a few centuries later.

In both Pythagorean and Mean­ tone Temperament, the tuning of the note G# is not the same as that of Ab. Thus it is not possible to transpose keys, because some keys are more in tune than others, and enharmonic intervals are always different — i.e., C# and Db have different pitches. For this reason, these temperaments are categorized as linear.

The figure on left is from The Modern Musick-master of 1730-1731, where Peter Prelleur talks about “The art of playing on the violin”. We can see, for example, that the notes D# and Eb on the fingerboard are not equal (Eb should be slightly higher than D#). Looking more closely, we can see that all sharps should be lower than the corresponding flats — this would be impossible to produce on a keyboard instrument, but it is perfectly reason­able on the violin or recorder. This book can be found in a free digital format at http://imslp.org/wiki/The_Modern_Musick-Master,_or_The_Universal_Musician_(Prelleur,_Peter)

Beginning with the Baroque era (about 1600­1750), the musical aesthetics demanded a greater variety of keys and colors, and embraced tonal harmony (music organized around a tonal center, although that can change during a piece, and employing chords based on thirds) as the standard. Music theorists created different tempera­ments, based on Meantone but extend­ing its possibilities: Vallotti, Kirnberger, Werckmeister and Young are examples of these tuning systems. These temper­aments all favor some keys that were used often, up to three sharps or three flats — ones that would sound more in tune — while there are compromises in keys with more accidentals, which would sound harsh or out of tune.

This is part of the aesthetic of the Baroque period, as composers would use the qualities of a key to create different emotions in each composition. Some temperaments of this period can now be categorized as circular — they allow musicians to play in all 12 keys, but each key has a different sound and relationship among its notes.

J S Bach composed The Well-Tempered Clavier to demonstrate that temperament enabled melodies to be played in all keys, but this does not mean that all keys sound alike; each would have a different affect.

From 1730 on, the aesthetics start to demand more “equal” temperament, because of the demand for modulations (a song may begin in C major and modulate to G major and A minor, for example). This need increases with the Romantic period (from the mid­ 1800s onwards), when tonal harmony reaches its limits and composers make use of very distant modula­tions (A major with three sharps moves to C minor with three flats, for example). This would be impossible in a Mean­tone Temperament, such as that used in the Renaissance.

Equal Tempered Tuning

Although it was known long before the present, this is the temperament utilized by any modern electronic tuner. It is widely used in classical and popular music nowadays. Only with the emergence of electronic musical instruments and tuners was it possible to carry out in practice and to disseminate to all musicians.

What we call Equal Tempered Tuning, or Equal Temperament, actually means that each note of the scale is equally out of tune. Since antiquity, music theorists considered the merits of Equal Temperament — Vincenzo Galilei (c. 1520­-91), father of Galileo Galilei, advocated Equal Temperament — but there was no practical means to fine­-tune all instruments by this method.

In addition to this, Equal Temperament was not considered appropriate, because instead of favoring the most commonly­-used scales and keys, and assigning a disadvantage to the less­-used ones, Equal Temperament detuned all intervals equally, so that the music could be transposed to all keys. For this reason, although theorists in our musical history were already aware of it for several centuries, musicians simply ignored this temperament.

To standardize the scale, the octave is divided into 12 equal parts mathematically. Instead of using pure proportions, a rational number is computed and used to calculate the pitch of each note of the scale. While it is not absolutely necessary to understand the math behind this calculation, it involves an equation computing a ratio that is the twelfth root of 2, since 2:1 is the acoustical ratio that produces a pure octave containing 12 notes:

In Equal Temperament, it turns out that the perfect fifths are very close to the pure fifths of the Pythagorean temperament — only two cents narrower than the pure fifth (the octave has a value of 1200 cents, with each half­-step measured at precisely 100 cents) The problem of “wolf interval” is divided equally across the scale.

However, the thirds, both major and minor ones, are very different from pure thirds (see table of all intervals, at the end); some theorists often say that thirds are more “brilliant” when produced in this way, but we might also describe them as just out of tune.

Thus, if we calculate the frequency of the middle C from the A=440Hz using the different methods, we can find different results (image on the left). We also can see that the math behind equal temperament is much more complex.

In the link following, you can download a CDF Player that allows you to view a demo illustrating the differences among Pythagorean, Meantone and Equal Temperaments:

Pythagorean, Meantone, and Equal Temperament Musical Scales

Just Intonation

At this point we reach the most important part of this article: Just Intonation. Up to this point, my discussion has covered models of fixed pitch — i.e., those used in instruments that have a fixed pitch during performance (like the piano, for example).

Some instruments allow a player to tune the notes while performing — as with the recorder, violin, and almost all woodwind and stringed instruments.

Musicians and theorists of all time periods, both in Western and Eastern music, were aware of the idea of Just Intonation. While much sought after, it is impractical in fixed pitch instru­ments (piano, for example) because it demands a certain pitch flexibility on the part of the musician. In this model, the proportions of the harmonic series rule the tuning system. When using Just Intonation, the problems mentioned above, such as the “wolf interval,” are avoided.

You may watch a demonstration of the differences between Just Intonation and Equal Temperament above. Pay close attention to the audio, listening to the differences between intervals. In Just Intonation there are no beats — i.e., there are no oscillations when playing two or more simultaneous sounds. In Equal Temperament, the beats are always present, as if there were a kind of vibrato even when no vibrato exists.

Considering all of the models pre­sented, Just Intonation is the easiest to accomplish by ear: we always seek the absence of beats. Moreover, this model is not fixed — i.e., a G is not always tuned to the same fixed pitch, because the pitch varies with harmonic function; it depends on the notes that are played simultaneously (as mentioned in my article on tuning in a group).

Each note of the chord should be tuned according to the table. In general, the upper note of a fifth is slightly high (a wider interval), major thirds low, and minor thirds high.

Da Capo: Playing in a Group

When playing in a group, we must define what we want in terms of pitch. In always aiming for this goal, we should remain aware of the reasons behind the way we tune. Also, there are criteria that we must use to define which model of tuning is best suited to the group’s situation:

• Any group containing piano, guitar or another instrument of modern fixed pitch – use Equal Temperament (all intervals equal);
• Any group containing historical instruments of fixed pitch, such as harpsichord, lute and theorbo – match the older temperament (Meantone, Vallotti, etc ) of the fixed instrument and be consistent with the chosen repertoire;
• Groups with instruments using untempered tuning, like recorder consorts, string groups – use Just Intonation (tune intervals accord­ing to function in the harmony);

Pure vs. Tempered

In the table, you can see all intervals, showing the differences between Just and Equal Tempered tuning. This reference is relative, so although the examples almost always start with the note C, they can also be used starting on any note

The first column gives the interval names. Note that some of these inter­vals may have more than one way to tune. I can include only some of them (you can probably find other possible proportions). I chose the simplest pos­sible radios for each interval.

The second column illustrates the given interval, listing specific notes as an Example. Now we can clearly see that intervals involving enharmonic pitches (C# and Db, for example) should not be tuned the same way.

The third column shows the ratio used to calculate the distance between the notes in Just Intonation. This ratio can be compared to the frequencies of the notes (or to the ratio of the size of the string, or the ratio of the air column length, or another precise measurement), but the proportion defines the interval.

The fourth column shows the size of that pure intonation interval in cents — that is, as it is widely expressed using electronic tuners.

The fifth column shows the size in cents of the Equal Tempered interval. Since each semi­ tone equals 100 cents, and the full scale is equivalent to 1200 cents, the difference between the fifth and the fourth column produces the amount we need to adjust the interval to have a perfectly tuned note — one without beats.

Thus, we can make these tuning adjust­ments for the following common intervals:

• Minor third, widen the Tempered interval by 15.6 cents;
• Major third, shrink by 13.7 cents;
• Pure fifth, widen by 1.96 cents;
• Minor seventh, usually a dissonance, and a relationship too distant to change during performance, so we can use the 16:9 ratio and reduce by only 3.9 cents;

For other intervals, just check the table at the end of this article.

Should I play the leading tone high?

You’ve probably heard the rule about playing the leading tone (seventh note of the scale) slightly higher, so that it “leads” to the tonic (home key) of the scale. This idea may cause us to think that we should play sharps higher than flats — exactly the opposite of what I suggest in this article!

This thought about the lead­ing tone was defended by a great ’cellist Pablo Casals at the begin­ning of the 20th century, and is based on a melodic principle that anticipates the resolution of the note by implying the direction it should take. Moreover, in Pythagorean Temperament, the sharps are higher than the corresponding flats, making the leading tone higher — and many players of violin, viola, ’cello, etc, use a variation of Pythagorean Temperament. However, none of these principles is compatible with Just Intonation, the main topic of this article.

This subject is very extensive; even striving to be accurate and mathematical, we have seen many creative possibilities regarding pitch. With a multitude of choices to make, each choice has its pros and cons.

As pitch is directly related to the tech­nique of an instrument, we cannot avoid thinking about it — we must learn it, even in our earliest steps toward playing our instrument, the recorder.

For those that would to know more about this subject, I recommend the following books:

• How equal temperament ruined harmony (and why you should care)
  Ross W. Duffin
  ISBN 978-0-393-33420-3
• Tuning and temperament – a historical survey
  J. Murray Barbour
  ISBN 0-486-43406-0

Intervalo	Exemplo	Proporção	Afinação Justa	Afinação Temperada
Semitom Cromático	C-C#	25:24	70,7	100
Semitom Sintônico	C-C#	135:128	92,2	100
Semitom Diatônico	C-Db	16:15	111,7	100
2.a Maior (baixa)	C-D	10:9	182,4	200
2.a Maior (alta)	C-D	9:8	203,9	200
3.a Diminuta	C#-Eb	256:225	223,5	200
2.a Aumentada	C-D#	75:64	274,6	300
3.a Menor	C-Eb	6:5	315,6	300
3.a Maior	C-E	5:4	386,3	400
4.a Diminuta	C-Fb	32:25	427,4	400
3.a Aumentada	C-E#	125:96	478,5	500
4.a Justa	C-F	4:3	498,04	500
4.a Aumentada	C-F#	45:32	590,2	600
5.a Diminuta	C-Gb	64:45	609,8	600
5.a Justa	C-G	3:2	701,96	700
5.a Aumentada (baixa)	C-G#	25:16	772,6	800
5.a Aumentada (alta)	C-G#	405:256	794,1	800
6.a Menor	C-Ab	8:5	813,7	800
6.a Maior	C-A	5:3	884,4	900
7.a Menor (justa)	G-F	7:4	968,8	1000
6.a Aumentada	C-A#	225:128	976,5	1000
7.a Menor (baixa)	C-Bb	16:9	996,1	1000
7.a Menor (alta)	C-Bb	9:5	1017,6	1000
7.a Maior (sensível)	C-B	15:8	1088,3	1100
8.a Diminuta	C-Cb	256:135	1107,8	1100
7.a Aumentada (baixa)	C-B#	125:64	1158,9	1200
7.a Aumentada (alta)	C-B#	2025:1024	1180,4	1200
8.a Justa	C-C	2:1	1200	1200

---

Articles ● English