I recently learned that the wavelength of a note is half that of the same note one octave lower. Do the wavelengths of the notes in a chord have some sort of similarities? Is there another reason that the notes sound good together?
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There’s a good Minute Physics video on it you can find [here](https://www.youtube.com/watch?v=1Hqm0dYKUx4).
To summarize it, starting from a certain key, every other key in the octave is a whole multiple of the same frequency. Because they’re whole frequency multiples, they stay in harmony with each other. All of your sound waves start and end at the same time, meaning you get a constant sound.
If you play notes that are not frequency multiples of each other, you end up with sound waves that waiver over time, with the peaks of the sine wave happening at different times making the sound warble over time.
> Do the wavelengths of the notes in a chord have some sort of similarities?
Yes, they do! Well… kinda.
A major triad includes a root, major 3rd, and a perfect 5th. A “pure” perfect fifth is exactly 3:2 the wavelength of the root note, and “pure” major 3rd is exactly 5:4 the wavelength of the root note.
These relationships aren’t just theoretical – the root note itself, like any other notes, includes more than just the one fundamental frequency. The note called “A” at 440Hz vibrates not just at 440Hz but also 2x that (880Hz) and 3x and 4x etc, in theory into infinity but they get progressively less loud as you go higher and you can only hear the first 8 or so depending on the instrument.
The other non-fundamental frequencies are called “harmonics” or “overtones” and the relationship between them called the “harmonic” or “overtone” series.
But, we don’t actually use pure intervals. Since the 1600s or something like that western music has been using “equal temperment” instead. This is what allows an instrument to play in any of the 12 keys. Prior to equal temperment there were different harpsichords for different keys and/or more complicated instruments with separate sharps and flats, because the Eb in the key of C minor was actually a different note than the D# in the key of E major, for example.
> Is there another reason that the notes sound good together?
Yes, there are many other reasons, and important amongst them is cultural conditioning. If this is the first you are hearing about equal temperment, and that the intervals you have heard you entire life are not “pure” – did they sound bad to you the whole time? Did you ever get the sense that the notes were out of tune?
The overwhelming response from honest people to those questions is “no.” To those of us who grew up listening to music from the classical era to today, nothing ever sounded out of tune. It’s only to people who grew up hearing the other tuning systems, to which our music would sound out of tune.
So, there must be more to it than the ‘mathematical’ reasons, or you might even say that these ‘mathematical’ reasons don’t matter at all to why they sound ‘good’ together, but there certainly are mathematical ways to describe why the notes of the chord are the way they are.
Western music traditionally used simple fractional ratios of pitches to tune their notes. A major chord is made up of three notes – a root, a major third, and a perfect fifth. The ratio of pitches between the major third and the root is 5:4. The ratio of pitches between the perfect fifth and the root is 3:2. The Ancient Greeks figured that these ratios of pitches would make the most pleasing harmony to the human ear, so that’s what forms the basis of our music system.
Of course modern music doesn’t use those exact ratios anymore, but the difference between pure intonation and equal intonation is a whole separate ELI5 topic.
Music notes on the 12 step scale (A to G plus accidentals) are all mathematical ratios to each other. But because it’s 12 steps with comparatively equal ratios, it’s not a simple rational ratio. The ratio between two notes a half step apart is 1:2^(1/12), or about 1:1.059. Replace the 1 in 1/12 with the amount of steps apart the notes are to get any specific note’s ratio; 3/12 (1:1.189) for a minor third, 4/12 (1:1.260) for a major third, 7/12 (1:1.498) for a fifth, etc. This turns into 1:2^(12/12) for an entire octave, which simplifies to 1:2 as you know.
These are the precise increments for technical definitions, but there are alternate tunings that take into account individual preferences and ears that might round it to more convenient numbers (such as 1:1.5 for the fifth). There’s also multiple scale definitions that use a different amount of notes that might not be equidistant (such as a major scale only has 7 notes).
The Western system of music breaks tones down into groups of 12 notes (counting the white and black keys on a piano).
The gaps between the notes of a musical scale are called “intervals”, and music theory is largely concerned with these intervals. The simplest interval to understand is the “octave” – where we go from one note – say a middle C – to the same note but higher up the scale – say the next C (one “octave” higher). That interval is defined by a doubling of the frequency of the note’s pitch and going up a full octave takes the full set of 12 (white and black) keys on a piano.
Other intervals between the notes in a scale can be described using a system of “just intonation”, where the ratios between the frequencies of notes on the scale are described in simple ratios such as 3:2 or 4:3. Certain intervals – with simple relationships like this – strike our ears as pleasant when the two notes are played together. This system proved impractical for tuning different instruments together and for transposing music between different keys, so since the 18th century we’ve generally used the “12-tone equal temperament” tuning system.
In this system, as you go up the 12 intervals in the scale, each note has a main frequency that is a fixed proportional amount higher than the frequency of the note before. Since it takes 12 intervals to double the frequency, each interval has a proportional increase in frequency of the 12th-root of 2.
The 12th-root of 2 is approximately 1.05946, so if we take the frequency (in hertz) of any note on the piano, the frequency of the next note (including white and black keys) will be 1.05946 times higher. Repeat this 12 times and we get a doubling of the frequency because we’ve gone up an octave.
This 12-tone equal temperament (“12-TET”) system loses the simpler, more natural, relationships between notes, and some people with very well-trained ears have a sense that some notes, played on well-tuned instruments, don’t feel quite right – they seem slightly sharp or flat.
There’s a video about this stuff here: https://youtu.be/bCYcS57eCqs
Yes. Harmony are integer ratios of base frequencies, like 2/1 for an octave, 3/2 for a quint, 4/3 for a quart, etc.
Those ratios sound pleasant to the ear, for possibly three reasons:
(1) the acoustic waveform remains strictly periodic only for integer ratios of frequencies.
(2) naturally created sounds are rich in overtones, the ratios between overtones are integer ratios.
(3) base frequencies in integer ratios have their corresponding overtones aligned to each other.
These are all mathematical features. Why they sound appealing is not explained by math, but it’s likely that our ear evolved around naturally produced sounds which do exhibit those overtone features. Chords are then artificial sounds that emphasize and exaggerate those overtone features.
The sounds that sound pleasing together create nice simple whole number ratios when their wavelengths are played together. If their wavelengeths don’t do this, they tend to create little spikes of high frequency noise that people find unpleasant.
Yes, in general simpler ratios between frequencies make them sound the more pleasant (in music we call this a consonant interval) and more complex ratios tend to clash (dissonant interval). The simplest ratio is for the octave which is 2/1. Other consonant intervals include 3/2 and 5/4. Meanwhile a dissonant minor second has a more complex ratio of 16/15. Chords are built by stacking intervals, for example a basic major chord has frequency ratios of 4/5/6. It is a little more complicated than that because about the 18th century we figured out that there are some problems with tuning our instruments to these pure ratios so instead we now use a system called equal temperament where distance between all notes is the same and ratios between them are an aproximation of those simple ratios.
Speaking of reason – no, there is no mathematical reason and can not be. Mathematical formulas can not be a reason for anything outside math. There is a physiological/psychological reason that some sound frequencies ‘work’ together and some do not. Mathematics, however, can help us to calculate these frequencies. Usually they are in fraction 5:4 or 3:2 one to another.
to raise a note by a semitone, you multiply its frequency (aka you divide its wavelength) by 2^(1/12), aka the twelfth root of 2, which is rougly 1.059
raising a note by an octave means raising it by 6 tones, or 12 semitones. Hence, it means you multiply it’s frequency by 2^(1/12), 12 times. In other words, you multiply it by 2^(1/12) to the 12, = 2^((1/12)*12) = 2
Not sure if that has to be called “a mathematical reason why the music notes sound good together”, but in short: two notes separated by a semitone, have a set ratio between their frequencies (that is, 2^(1/12) aka ~1.059)
In general, music notes follow a logarithmic scale. Am too tired to explain but look up what it is.
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