One-sentence summary: In this article I will explain the math for the Just intonation and Pythagorean tunings for the diatonic major scale (all the white keys on the piano).
In the preceding article The Physical Nature of Musical Sound I have talked about how air vibration affects a human eardrum and causes the perception of sound pitch. I used a piano as a case study, noting that there are metal strings inside the piano that vibrate at different rates. The vibration frequencies to which the strings should be tuned, is the subject of this article. Please refer to the piano keyboard diagram above, for the names and location of piano keys.
Although I’m using a piano as the case study, this applies to all musical instruments. Wind instruments, such as flutes, have holes in carefully calculated locations to produce sound at desirable frequencies. Wind instruments with no holes, such as trombones, require the player to remember how much to elongate an air pipe of the instrument so as to produce the desired air vibration. On a clarinet or a saxophone the player can alter the pitch by pressing or relaxing a reed. Even on a violin, it is not enough to tune the strings: there are no frets like on a guitar and a player must remember where to depress a string.
Thus, tuning a musical instrument is not only a theoretical exercise for the instrument maker. Musicians are expected to hear and memorize the way the musical notes and intervals should sound. Here is a demonstration of what tuning skill is required of a violin player, in which a violin player is asked to switch between three different tuning schemes. (Two of these schemes I discuss at length in the remainder of this article.)
However, not only it is hard to memorize the proper pitch, it is hard to determine theoretically what the frequencies should be. Attempts to find the right frequencies goes all the way back to Pythagoras, an ancient greek philosopher from the 6th century BCE. The Pythagorean tuning is still in use today.
In this article I will abuse terminology by saying that a piano key is tuned. Actually, what must be tuned is the corresponding string inside the piano. Remember that a piano key is connected to a hammer which hits a tight string. The sound you hear is the vibrations of that string. (Even more precisely, the vibrating string vibrates the air around it; then, that air propagates a compression wave; the wave vibrates your eardrum; this sensation is processed in the brain as a perception of sound.)
The mathematics of music began after Pythagoras drew some conclusions after experimenting with vibrating strings of different lengths. By touching lightly onto a tight vibrating string with a finger, one can force the string to vibrate in a standing wave pattern which will have a stationary node at the position of the finger.
Refer to the diagram below. Let’s say a string vibrates as shown in the top vibration form. If you now press with a finger at the centre of the string, you will get the second vibration form, with a stationary node forming at the centre. Or, if you instead pressed your finger at a third-in from the end, the string would vibrate as a stationary wave shown at the bottom, with two stationary nodes forming.
What are the relative frequencies of vibrations in each case? Remember that a frequency is an inverse of a period. So, what are the periods in each case? Relative to the first wave, the period of the second wave is twice as small, and of the third waveform is three times as small. Therefore, relative to the first wave, the second wave is twice as frequent, and the third wave is three times as frequent.
Of special interest is the frequency that is double the original. Pythagoras noticed that doubling, quadrupling, or even halving a frequency doesn’t significantly alter the resulting sound. The pitch is different, but the perception is that the sounds are the same in a fundamental respect. These double or half frequencies are called octaves.
What are some uses of octaves? One use case is for singers to sing octaves apart. It is easier for a man to sing lower pitches, and for a woman, higher pitches. Therefore, a woman sings an octave higher than a man. The song Summer Love from the movie “Grease” is an example. You can hear it clearly in the last 30 seconds of the song. Also, octaves can be used for melodic movement, rather than harmonic convenience. Listen to the first 10 seconds of the song My Sharona by the band “The Knack”.
Although an octave is an interval which could have any absolute frequencies as endpoints, the range of a musical sound spectrum has been divided in a particular way into a set of adjacent octaves. These intervals are referred to by number. The centre of the piano keyboard has octaves number 3 and 4, and they are the common range of most music. Normally, octaves are between each C key and the next C key. (Refer to the keyboard diagram at the beginning of the article.) However, in this article I will ignore the standard division, and consider octaves as between each A key. The reason is that piano is tuned based on the A key, not on the C key. Interestingly, the lowest key on the piano is not C, but is A.
What frequency should we assign to the A keys, then? Taking 440 Hz as the starting vibration frequency we can form octaves by multiplying or dividing this frequency by 2, repeatedly. We would get this sequence of frequencies for the A keys: 27.5, 55, 110, 220, 440, 880, 1760, 3520. It is not difficult to tune all the A keys to this sequence, using an electronic tuner device.
But, what about the other keys, the ones between the A keys? We can already be sure that the frequencies would double and halve for their namesakes in the other octaves. For instance, if a certain key E has frequency X, then the other keys “E” will have frequencies 0.25X, 0.5X, 2X, 4X. Therefore, if we can determine how to tune strings within a single octave, it would be a simple matter to translate them to other octaves. We would only need to multiply their frequencies by 2, or divide by 2, in order to fill other octave ranges. This is also called transposing up or down by an octave.
In this article, I will denote each piano key by a name followed by a tuning frequency. For instance, A-220 is a piano key called “A” tuned to emit 220Hz vibration. I will also use the word “key” and “note” interchangeably. (The difference is that notes are what is written in the musical score, while keys are what you press to voice the notes.)
The task then simplifies to find the intermediate notes between A-440 and A-880. The earlier diagram with standing waves gives the frequency for one key between these two A keys: it is the key “E”. Recall that we have already concluded that the third standing wave has a vibration 3-times more frequent than the first wave. If the original frequency is X, then this one will be 3X. The pitch interval between them is bigger than an octave, because the multiplicative factor of 3 is greater than 2. If we transpose 3X down one octave, we get 3X/2 for the new vibration. The ratio between frequency X and frequency 3X/2 is 3/2. If we use 440 as the value for X, then multiplying 440 by 3/2, gives 660. Thus, we will assign 660 Hz frequency to the key “E”.
In summary, the key E, which is an intermediate key between A-440 and A-880, has been determined to have frequency 660 Hz. The ratio of E to the lower A is 3/2. What is the ratio of the upper A to E? This is easy to calculate by using the absolute frequencies 880 and 660 of these notes. It is 880 / 660 or 4/3.
Just as the interval 2-to-1 had a special name of “octave”, the intervals of 3-to-2 and 4-to-3 are so ubiquitous that they too have names. In European terminology, the interval 3/2 is called a quinta, and the interval of 4/3 is called a quarta. These are latin words.
The American English names for quinta and quarta are “fifth” and “fourth”. These names correspond to the order of the intervals as observed in a basic scale — the diatonic major scale — which we are about to construct. It’s important to understand that “fifth” in this context doesn’t mean 1/5 — it means the “fifth interval of a scale”. Likewise, the “fourth” doesn’t mean 1/4, but it means the “fourth interval of a scale.” Since in this article I discuss fractions, I will use latin names to refer to intervals whenever there’s a possibility of misinterpretation.
So, we have split an interval of an octave into two intervals. Let’s formalize this with a formula. Let’s say we have two frequencies “a” and “2a”, corresponding to an octave, and we wish to split it into two intervals L1 and L2. We want a*L1 to give an intermediate frequency X, and then X*L2 to give the frequency 2a. Simplifying this, gives the relationship of L1 * L2 = 2. The solution we found is L1 = 3/2 and L2 = 4/3.
We can generalize further by stating that to split any interval L, we must find L1 and L2 whose product equals to L. Let’s use this method to split the quinta (3/2), by finding a solution to the equation of L1 * L2 = 3/2. There are many potential solutions, and one of them is L1 = 5/4 and L2 = 6/5.
These intervals of 5/4 and 6/5 too have names. The interval of 5-to-4 is called a major tertia, and the interval of 6-to-5 is called a minor tertia. In American English, these intervals are called major and minor thirds.
Pictorially, we did this:
The piano key that goes in place of the question mark above, is C#-550. I have calculated 550 by multiplying 440 by 5/4.
We can express the proportions 5/4 and 6/5 concisely as 4:5:6. We can also express the proportions of the scale spanning the whole octave as 4:5:6:8. Notice that 8/4 equals to 2/1 and this is the octave interval. The corresponding notes in the scale derived so far are A, C#, E, A with frequencies 440, 550, 660, 880.
Let’s use the 4:5:6:8 proportions to determine the frequencies of keys in a lower octave that is in the range between A-220 and A-440. Multiplying 220 by 5/4 we get 275, which is the lower C# key. Multiplying by 220 by 6/4 we get 330, which is the lower E key. So the scale A, C#, E, A in this lower octave has absolute frequencies of 220, 275, 330, 440.
We can express the scale over the range of two octaves as: 4:5:6:8:10:12:16. The ratio 4:16 is the two-octave interval from A-220 to A-880. Notice that each frequency in the lower octave is half of the corresponding frequency in the higher octave. The notes in the two-octave range scale are A, C#, E, A, C#, E, A with corresponding frequencies of 220, 275, 330, 440, 550, 660, 880.
To summarize, we were able to create a scale of 3 notes which repeats every octave. Can the intervals be split further in order to create more notes? A hint comes from the following observation: the order of the sub-intervals doesn’t matter, such that an octave can be either a quinta, followed by a quarta, or a quarta followed by a quinta. Visually,
The frequency of the note a quarta ahead of A-440, is 440 * 4/3 which equals to 586.66… . I will round it to 587. It is the key D on the piano.
If we can begin an octave with a quarta, and end it with a quarta, we observe an “unfilled” interval in the middle:
The missing interval is the degree by which a quinta is higher than a quarta. Let’s calculate it. Let “a” denote the frequency of the root note of the octave. The note higher by a quarta is (4/3)*a, and the note higher by a quinta is (3/2)*a. The proportion between them is (3/2) / (4/3) which works out to 9/8. The interval 9/8 is called a tone. On the piano, it is the interval between the piano keys D-587 and E-660.
In summary, a quinta is a quarta followed by a tone. And, an octave can be seen as a (major tertia, minor tertia, quarta), or as a (quarta, tone, quarta).
Can we subdivide further? Let’s subdivide major tertia, which has a ratio of 4:5. A possible subdivision is 8:9:10, so that the two ratios are 9/8 and 10/9. The first frequency ratio we recognize to be the tone interval, already seen. The second ratio of 10-to-9 is new, and it is called a minor tone interval. Note that we can order the two intervals in any way (because the product of ratios is the same), but for reasons that are hard to explain at this point, I will order them as a tone, followed by a minor tone.
If you are a musician, you may be puzzled at this point. You know that a major third (major tertia) is two whole tones, both of which are the same. Yet, here I am telling you that the first tone is a little larger than the second. Indeed, musical instruments, particularly piano, are tuned so that both intervals exactly the same. But this is a calculated error. The purpose is to enable modulation, which is discussed later in this article.
Let’s split a minor tertia (minor third) by separating it into a tone and something else, that we will call a semitone. The minor tertia has the proportion of 6/5. It is higher in pitch than a tone by the factor of (6/5)/(9/8), which works out to 16/15. The frequency ratio of 16-to-15 is called a semitone interval. Again, we have freedom with the order of the intervals. We need to take it in the order of (semitone, tone). The reason will be apparent later; it has to do with having a quarta as the fourth interval in the final scale. Also, notice that two semitones are not equal to a tone: 16/15 squared is larger than 9/8.
Summarizing, our octave is now divided into the scale of relative intervals of major tertia, which consists of (tone, minor tone); followed by a minor tertia, which consists of (semitone, tone); followed by a quarta.
What about that last quarta? We haven’t yet attempted to separate a quarta into smaller intervals. We can just guess a combination, and check that the product of ratios equals to 4/3. How about a major tertia and a semitone? Let’s check: 5/4 * 16/15 equals … 4/3. (Of course, I cheated by looking at a piano.) And, we already know the expansion for major tertia to be (9/8, 10/9). A quarta is therefore a combination of tone, minor tone, and a semitone. Let’s order the intervals this way: (minor tone, tone, semitone).
Putting all the pieces together, we can write out the octave as a list of these adjacent intervals: (9/8, 10/9, 16/15, 9/8, 10/9, 9/8, 16/15). Or, in words: (tone, minor tone, semitone, tone, minor tone, tone, semitone). These seven intervals define an 8-note scale that divides an octave into smaller intervals. We can also express the same scale with intervals relative to the root note, instead of intervals relative to the preceding note in the sequence. To calculate it, accumulate the product of ratios in the original sequence. In this case, the ratios will be: (1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2).
What we have reinvented is the just intonation scale discovered by Claudius Ptolemy in the 2nd century CE. Note that he lived 600 years after Pythagoras, who started research into scales, and in whose name we have another scale, the Pythagorean scale.
Unlike Pythagoras, who constructed a scale solely based on transpositions of quintas and octaves, Ptolemy used tertia intervals as the basis. Note that in the Ptolemy’s scale — which became the basic scale of Western music— the second note is a tone interval, the fourth note is a quarta interval, and the fifth note is a quinta interval. This explains the American English names of “second”, “fourth”, “fifth” for those intervals.
Ptolemy’s scale is classified to be in the category of diatonic scales. The word “diatonic” is derived from Greek language, and the prefix “dia” means “through” in Greek. The idea is that the scale is mostly made out of tones, which are relatively large, pleasing intervals. The two semitones, which are smaller and harsher intervals, are not close to each other because they are padded with at least two tones.
In contrast, a scale that has two adjacent semitones is not diatonic because it has a chromaticism. Chromatic intervals are unstable, in that the listener is expecting for them to be short lived and to be leading somewhere else. They could be leading to a stable, pleasing interval, or to another chromatic interval, which in turn would lead elsewhere. You can hear an example of chromaticism in the Flight of the Bumblebee interlude by Rimsky-Korsakov.
Let’s recap. The Ptolemy’s just intonation scale is: (1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2). The intervals shown are relative to the first note. (The first note is also known as a tonic. I call it a “root” note.)
The intervals have these names and frequency ratios according to Plotemy’s system:
An octave interval can be split into interval combinations of :
- quarta, quinta
- quarta, tone, quarta
- quinta, quarta
- major tertia, minor tertia, quarta
- (tone, minor tone), minor tertia, quarta
- (tone, minor tone), (semitone, tone), quarta
- (tone, minor tone), (semitone, tone), (minor tone, tone, semitone)
- 9/8, 10/9, 16/15, 9/8, 10/9, 9/8, 16/15
The combination in line #7 is the diatonic major scale. Notice that the diatonic scale is derived by expanding the combination #3 above, that of division of the octave into a quinta followed by a quarta. However, notice that the first three intervals (tone, minor tone, semitone) in line #6 add up to a quarta, leaving a tone leftover. This is identical to line #2.
Remember that I promised you an explanation for choosing a particular ordering for the quarta interval at the end of the octave? The first quarta is (tone, minor tone, semitone), but the last quarta is (minor tone, tone, semitone). If we were to order it as the first quarta, then the end of scale would have these intervals between the root and each note: (…, 27/16, 270/144, 2). The seventh interval 270/144 is not a harmonic overtone of the root note. Contrast this with the seventh interval of 15/8 in the correct scale. This interval is more consonant with the root note, because it is the 14th overtone, transposed 3 octaves down.
So far we have discussed all frequencies relative to A-440 and divided the piano keyboard into octaves from one key A to the next key A. I chose to use A, because A-440 is the note from which all tuning starts. However, now that we understand that a scale can be seen as relative intervals, the absolute frequency of the first note of the scale can be anything. In particular, it is convenient to talk about the diatonic scale when the octave is taken between some C key and the next C key, because in this case the diatonic scale is all white piano keys.
In order to split the piano keyboard into octaves from C to C, we need to know the absolute frequency of at least one C key. By convention, we can take the middle C key to be a minor tertia interval ahead of A-440. This works out to 440 * 6/5 = 528.
The image below shows the intervals between the keys in the C-to-C octave. Note that this ordering is exactly the Ptolemaic just intonation scale. What is strictly Ptolemaic is the actual numerical ratios between note’s frequencies. They are slightly different in other tunings, but generally a semitone is about twice as small as a tone.
With reference to the C-octave, the peculiar arrangement of the black keys on the piano becomes understandable. If all the white keys on the piano are to be arranged to model the diatonic scale, then the only place that the black keys could go are between the white keys that are a tone interval apart.
What should the black keys be tuned to, and why do we need them anyway, if all we play is the diatonic scale (the white keys)? The black keys can be used for embellishing a melody through chromaticism. A black key can be tuned a semitone above the adjacent lower white key (called “sharp”), or a semitone below the adjacent higher white key (called “flat”). Remember that because in Ptolemy’s scale two semitones are not equal to a tone, a tuner has to choose relative to which white key and in what direction to tune the black key. For instance, C-sharp and D-flat would not be the same frequency.
When we have worked in A-octaves, the black key C# had to be included in the scale. But, since in the C-octave the diatonic scale is precisely all white keys, we can see that the piano keyboard layout was designed with the C-octave as the basis.
Yet, A is the leftmost key on the piano, and this note, also known as “la”, is named by the first letter of the English alphabet. For historical reasons, tuning of orchestral instruments began on the A note. We have used 440 for A as the starting tuning frequency, which is the standard today. However, A was also tuned to other nearby frequencies (Wikipedia).
Choosing an octave range, such as A-to-A or C-to-C requires to completely re-tune the piano keys according to the Ptolemaic just intonation ratios. This was an inconvenience which caused musicians to invent other types of tunings.
Let’s contemplate intervals some more, by looking at history. Ptolemy was ahead of his time, because the intervals of minor and major tertias (thirds) were not considered pleasing or consonant. On the other hand the quarta (fourth) was considered to be a beautiful consonance. The ancient Greeks based their music on tetrachords: a selection of 4 notes spanning the interval of a quarta. However, by the time of Renaissance, human perception changed to perceive a quarta to be an imperfect consonance, while tertias had gained in acceptance.
Contrast Ptolemy’s scale with the scale invented by Pythagoras. The scale of Pythagoras was a descending scale with these ratios between frequencies to the root frequency: (8/9, 27/32, 3/4, 2/3, 16/27, 9/16, 1/2). Note that all ratios are less than 1. Furthermore, all intervals can be created by product combinations of 8/9 (an interval we have derived), and also the interval 243/256 which got the name of hemitone (not semitone).
How is the hemitone different from the semitone 16/15 that we derived? If we divide 16/15 by 256/243, we get 1.0125. This value is close to 1, and thus we can see that the Pythagorean hemitone is similar in pitch to the Ptolemy’s semitone.
While Pythagoras used both ascending and descending quintas (fifths) to develop his scale, later medieval theorists modified the method. They used ascending quintas only, transposing down by octaves as necessary. Here is the resulting scale that they developed: (1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2). All those intervals can be created by combinations of a tone 9/8 and a hemitone 256/243.
I have said earlier that in antiquity a tertia was not considered a consonance. However, to be fair, the major tertia that Greeks knew was not the 5/4 frequency ratio of Ptolemy. That’s because Ptolemy came up with it only much later, in circa 130 CE. Instead, people knew the Pythagorean major tertia with frequency of 81/64. This interval is 81/80 times higher (about 1.01 in decimal) than Ptolemy’s.
The ratio of 81/80 is called comma of Didymus, in the name of Didymus Chalcenterus who lived a generation before Ptolemy. This small interval also goes under other names of “syntonic comma”, “the Ptolemaic comma”, or “diatonic comma”. Didymus noticed that if the Pythagorean comma is reduced in pitch by 81/80, a much more pleasing consonance results. Perhaps, Ptolemy took a tip from Didymus when choosing 5/4 to be the ratio for the major tertia in his scale.
However, despite the fix in the tertia, the tertia interval was still considered not as pleasant an interval as the quarta. Ptolemy’s scale based on tertias fell into obscurity, all the while Pythagorean scale continued to reign. It took 1500 years (!) for Ptolemy’s scale to be rediscovered by Gioseffo Zarlino in the 16th century.
Let’s get back to comparing scales. The Pythagorean scale is compatible with a sequence of ascending or descending quintas. This is important, because much of the harmonic theory, since the days of Bach and Mozart, is based on the “Circle of Fifths”. It is a sequence of quintas: 3/2, (3/2)², (3/2)³, ….
How does the Ptolemaic scale fair with a list of ascending fifths? Here’s Ptolemy’s scale again: (1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2), and let’s look at a three-quintas interval. This interval has the ratio of (3/2) cubed, transposed down an octave. This works out to 27/16. The sixth note in the Pythagorean scale is exactly this ratio, but the sixth note in the Ptolemaic scale is different: it is a 5/3 ratio. Thus, the “Circle of Fifths”, a staple in harmonic music, is broken in the Ptolemaic scale.
If we can take a lesson from the material presented so far, it is that there’s no clear way on how to divide an octave into small intervals. To recap, we had got all our ideas so far from looking at the first two harmonic overtones (the diagram with three red waveforms above). What else can we learn from harmonic overtones?
Here are the first six overtones,
Observe the pattern: the overtone number (n-1) has a period of 1/n relative to the topmost vibration. This means that its frequency is n times higher than the base frequency. Let’s call the base frequency “a”.
The eighth overtone has a frequency of “9a”. If we transpose it down by three octaves (dividing by 2*2*2), then we get the ratio of 9/8. This is the familiar “tone” interval we had already seen. Similarly, other overtones transposed just enough octaves down give the ratios of quarta 4/3 and quinta 3/2. The fourth overtone transposed two octaves down is the major tertia 5/4 that we have seen. So far, the overtones are not telling us anything that we haven’t seen already.
We can also look at the intervals occurring between the overtones. Consider this diagram from Wikipedia article on harmonic overtones, showing just that. (Note that the notes are not transposed down to the octave of the base note.)
The diagram doesn’t give the ratios, only the names of the intervals. However, we can calculate them. Consider the interval that’s called “supermajor second” in the diagram. This interval is the ratio of frequency “8a” to “7a”, which equals 8/7. (Compare it to 9/8 which, in addition to a “tone”, is also known as a “second”.)
Of particular interest is the harmonic with frequency 7 times the base frequency. Transposing it down two octaves gives the ratio 7/4 or 1.75. This interval, known as the “blue” note, is used in the Blues. It is the “diminished seventh” used in the dominant chord that we designate by “C7” in modern music notation.
(In the below discussion of the blue note, I will assume a C based octave, instead of an A based octave. It is more familiar, because in this octave the diatonic scale is all white piano keys.)
Where does the blue note fit in the Ptolemaic scale? Here is that scale again: (1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2). Let’s express all these ratios in decimal form, in order to make easier the comparison with the blue note ratio of 1.75. In decimal form, the interval ratios are: (1, 1.125, 1.25, 1.40625, 1.5, 1.6875, 1.875, 2). Thus, the blue note is between the sixth and seventh note.
Let’s calculate the interval between the blue note and the sixth note of the Ptolemaic scale. If the base frequency “a”, the blue note is 7/4*a, and the sixth note is 5/3*a. The ratio between them is (7/4) / (5/3), which equals 21/20 or 1.05… in decimal. Compare it to the Ptolemaic semitone of 16/15 which is 1.066… in decimal. The blue note is a little lower in pitch (perceived as “flat”) than the Ptolemaic semitone. Thus, the blue note is found between the sixth note (A key on the piano) and the sixth note raised by a semitone (A# key on the piano).
In the Pythagorean medieval scale, the sixth note of the scale is 27/16. The ratio of the blue note to that sixth is (7/4) / (27/16), which equals 28/17 or approximately 1.037. But, the hemitone is 256/243, or in decimal, 1.053…. Again, we see that the blue note is higher than sixth, but lower than sixth plus a hemitone. (Again, it’s flat of A#.)
If the blue note is not included in the scale, and can’t even be produced through a combination of tones, minor tones, and semitones, how can a musician play it? All instruments have a key configuration for sixth+semitone (for the C diatonic scale, this would be the key for A#.) The musician then can dynamically lower the pitch enough to reach the blue note. For instance, on a saxophone, relaxing lips lowers the pitch.
A piano player relies on the listener to approximate the blue note. He may play A# hoping that the listener’s perception will actually hear A# flattened. Or, he can play both A and A# — a simultaneous semitone interval — hoping that the listener would perceive the blue note as the intermediate interval between the two.
The dominant chord like C7 has a significant role in modern music theory. It has a characteristic “driving” sound that wants to resolve to a major chord a quinta below. (A major chord has three notes, corresponding to interval frequencies based on tertias (thirds). They are (1, 5/4, 3/2) with octave transpositions of each.)
However, it is not clear why the dominant chord works the way it does. The great mathematician Leonard Euler proposed his own music theory, but his theory was criticized for failure to explain the dominant chord. One suggestion he made to redeem his theory was that we don’t have the right frequency for the dominant chord.
Let’s investigate numerically. The dominant chord such as C7 has A# as its 4th note (sixth+semitone). In Ptolemy’s scale the sharp-sixth has ratio of (5/3) * (16/15) which works out to 16/9. This interval can be derived also from the harmonic overtones. It is formed by two descending quintas translated an octave up: (2/3)(2/3)*2.
Or should the sharp-sixth in the C7 chord be replaced by a flat seventh (a seventh minus a semitone)? That would be (15/8)*(15/16) which works out to 255/256. It is a too distant, 254th overtone. (Notice that in the computation I used an inverted semitone 16/15, because we are descending.)
However, I wonder, should we be playing the blue note (which is 7/4) in the dominant chord, instead of the conventional sixth+semitone? That would explain why it sounds so good, because it’s just the seventh overtone.
All of the scales we have discussed are based on ratios between frequencies. It was Pythagoras who originated the method of tuning through ratios.
Pythagoras lived in the 6th century BCE, and during that time ratios had a special status in mathematics. Ratios were seen as relationships between quantities, but not quantities themselves. Mixing rations and quantities was seen as mixing “apples and oranges.” For instance, it was considered illogical to add a number 5 to a ratio of 4/2.
Pythagoras rejected the existence of any quantities that can’t be expressed via whole numbers, and he permitted only ratios of whole numbers. The idea of not using ratios to come up with a tuning scheme probably had never occurred to him.
However, tuning an instrument based on a ratios creates an inconsistency between quintas and octaves. If a quinta is a 2:3 ratio and an octave is 1:2 ratio, then a note 12 quintas away from the root is dissonant with a note that is 7 octaves away. The dissonant interval between them is called the “comma of Pythagoras”, and has a frequency ratio of approximately 1.013:
Notice that this is a very distant 531,440th overtone, translated about as many octaves down. Another music theorist, Philalaus of Tarentum (5th century CE), noticed that two hemitones 256/243 and this unusual interval combine to a tone of 9/8.
There are several other scales that were in used in the past, such as the Quarter-Comma Meantone and Well Tempered scales. These scales, and the Equal Temperament scale (a scale in which all intervals are equal), where designed to alleviate harsh dissonances like the comma of Pythagoras. More importantly, these scales had workarounds to allow modulation: transposing music by less than an octave without retuning the piano keys.
I am taking up modulation in the next article of this series Problems with Ancient Musical Scales. The goal of this article was just to show the general pattern of how mathematics becomes relevant in devising musical scales.