Music and Math

Music and Math

Music is very strongly tied to physics and math. Sound is fundamentally a wave, which makes music highly tied to the physics of waves. In this post I’ll look at the mathematical and physics origins and development of the major scales we use in Western music.

Wave examples

Waves have a couple of related properties: speed, frequency, and wavelength. All waves, whether light or sound or ripples over the surface of a liquid have these properties. Speed is how fast the wave propagates through a medium (like air). For sound, that’s the ‘speed of sound’. You may be familiar with watching some distant event that generates the sound and seeing the event before you heard the sound. That is because light (also a wave) travels much faster than sound in the air… so you see a firework explode before you hear it, or you see the lightning bolt before you hear the thunder.

Wave properties

Waves are repeating events where (for sound), air pressure increases (peak), then decreases (trough), then increases again. Each repetition is a cycle, and frequency (f) is the number of cycles that pass by a fixed point in a second. If you take a snapshot of a wave in time, wave length (λ) is the distance between (two peaks). If the wave travels through a medium at a fixed speed (s), then you can see that f x λ = s. From this formula we will be able to relate wavelength and frequency.

Making music with waves

So that’s a fair amount of background, what does that have to do with music? Well each musical note is a different frequency. To produce this frequency we typically need some sort of device that has the same length as the wavelength. So tight string, or a tube with an open or a closed end.The string will want to vibrate when plucked or struck so that the trough will stay at ends, creating what we call a standing wave (because it just stays in place). This in turn causes the air around it to vibrate at the desired frequency.  We can tighten or loosen the string to change the speed of sound in the string itself to change the actual frequency emitted. This is how a guitar or piano is tuned. For brass instruments, a similar thing happens within a tube. Brass instruments are tuned by adjusting the length of the tube. When we have a vibration that produces a standing wave, this is called resonance.

Now more than one type of standing wave can be produced with a single length. Not just the standing wave (where the troughs of peaks occur at the ends), but also where one trough of peak also occurs in the middle. In this case the wavelength is ½ the original wave. Additionally if the wave is ⅓ the original length it will produce another standing wave. This series of waves is called ‘harmonics’, with the original wave we described called the fundamental, or the 1st harmonic. The second wave is the second harmonic and so forth.

When we cut the wavelength by 1/N we increase the frequency by N. So the if the fundamental, is a note, C, then the 2nd harmonic is the note 2 x C, the third is 3 x C and the fourth 4 x C.  In western music, whenever you double a note, the resulting note is labelled with the same name or letter. So the center ‘C’ is called ‘middle-C’ and the note twice as high is called ‘high-C’. The span between C’s are called an octave (for reasons that will become clear below).

Now if I have a bugle, tuned so the fundamental is the ‘C’ 3 octaves below middle-C. Bugles change notes by creating more air pressure and switching to a different harmonic. So the lowest note it can play is that fundamental ‘C’ 3 octaves below middle-C. Increasing the pressure creates a note at the second harmonic, which would be the ‘C’ 2 octaves below middle-C. The third harmonic is not a ‘C’ but a note between the ‘C’ two octaves below middle-C and one active below middle ‘C’. The fourth harmonic is ‘C’ again. From 4^th through the 8^th harmonic there are 3 notes, and from the 8^th to 16^th harmonic there are 7 notes. If we list these in a table:

Note	Harmonic	Relative Frequency to previous ‘C’
C (-3 Octaves)	1st	1.0
C (-2 Octaves)	2nd	1.0 (2/2)
G	3rd	1.5 (3/2)
C (-1 Octave)	4th	1.0 (4/4)
E	5th	1.25 (5/4)
G	6th	1.5 (6/4 = 3/2)
B♭	7th	1.75 (7/4)
C (middle-C)	8th	1.0 (8/8)
D	9th	1.125 (9/8)
E	10th	1.24 (10/8= 5/4)
F	11th	1.375 (11/8)
G	12th	1.5 (12/8=3/2)
A	13th	1.625 (13/8)
B♭	14th	1.75  (14/8=7/4)
B	15th	1.825 (15/8)
C (+1 Octave)	16th	1.0 (16/16)

The 8^th through 16th harmonics (minus the 14^th) form our major scale. There are eight notes from C to C in the scale (including the two C’s), and so we call that stretch an ‘octave’ for eight.

Scales and tuning

Below is how we represent the C major scale in written music.

A song which uses just these 7 notes (C, D, E, F, G, A, B) plus any of these notes shifted by 1 or more octaves is said to be written in the key of C major. If we are skilled enough, we should be able to play any such song on our bugle. Now let’s take our bugle and tune it so the 1st harmonic is ‘D’. If we then try to do the same thing with our D bugle and play a scale (going from D to D), we find that our notes won’t match all the notes in our original ‘C’ bugle:

                                                     

Note	Bugle D	Bugle C
D	1.125	1.125
E	1.26562	1.25
F	1.40625	1.375
G	1.54688	1.5
A	1.6875	1.625
B	1.82812	1.875
C	1.05469	1
D	1.125	1.125

What we would like is to be able to have bugles tuned to different notes, but to have the bugles be compatible. This was fixed by the ancient Greeks, notably Pythagoras (the same Greek philosopher/mathematician that gave you the Pythagorean Theorem in geometry). He added 5 more notes and then set the frequency of each note so that the 5th note in every scale was always exactly 3/2 (1.5). You may notice that the 5th note corresponds to the 3rd harmonic, and is the second most common note that our bugle can play. The calculation is based on what is called the circle of fifths. We start with the key of ‘C’ and then go to the fifth note in the ‘C’ scale, which is the key of ‘G’. We then select the fifth note of the key of ‘G’ and use that scale. The additional 5 notes are added between two existing notes and are named either note-sharp (written note-#) or note-flat (written note-♭). Sharp means the note is slightly higher in frequency and flat means the notes are slightly lower in frequencies. These notes are added between C and D (called C# or D♭), between D and E (D#/E♭), between F and G (F#/G♭), between G and A (G#/A♭), and between A and B (A#/B♭). Notice that our bugle already has a note between A and B which we called B-flat (B♭).

To calculate the frequency of a note relative to another note, count the number of notes between your starting note and your ending note clockwise and then for each position multiply by 1.5 (3/2). If the result is greater than equal to 2, divide by 2. With twelve notes. If you calculate the relative frequency of the note to itself, you get (1.5)¹² = 129.746, which is very close to 2⁷ = 128 making the relative distance pretty close to 1. With this method our D bugle would still start at 1.25 times C, but our scale is now D, E, F#, G, A, B, C#, D. These will match the same notes on our C Bugle. If I create an instrument that can play all 12 notes, that instrument can play in any key, and thus any song. If we lay the notes out in order of frequency to (C ,C#/D♭, D, D#/E♭, E, F, F#/G♭, G, G#/A♭, A, A#/B♭, B). We say the distance between each note in our 12 note system is a half-step. So our scale is full-step, full-step, half-step, full-step, full-step, full-step, half-step. In Pythagorean tuning (the system we just looked at), the ratios of each of these different sets are not fixed, but vary slightly between different notes. In the 19^th century both Europeans and Chinese developed a system where each half-step had a constant ratio – the 12th root of 2 which is approximately 1.059463. This is called Equal Temperament and is the system that is normally used in tuning a piano, while Pythagorean tuning is usually used in orchestras. Below is the comparison between the three systems we talked about, with the %difference of each note. For sharp and flat notes in harmonic tuning, I simply give the average of the two adjacent notes:

                                                     

Note	Harmonic	Pythagorean	12TET	HvP	PvE	HvE
C	1	1	1	%0.0	%0.0	%0.0
C#/D♭	1.0625	1.06787	1.05946	%0.5	%0.8	%0.3
D	1.125	1.125	1.12246	%0.0	%0.2	%0.2
D#/E♭	1.1875	1.20135	1.18921	%1.2	%1.0	%0.1
E	1.25	1.26562	1.25992	%1.2	%0.5	%0.8
F	1.375	1.35152	1.33484	%1.7	%1.2	%2.9
F#/G♭	1.4375	1.42383	1.41421	%1.0	%0.7	%1.6
G	1.5	1.5	1.49831	%0.0	%0.1	%0.1
G#/A♭	1.5625	1.60181	1.5874	%2.5	%0.9	%1.6
A	1.625	1.6875	1.68179	%3.7	%0.3	%3.4
A#/B♭	1.75	1.80203	1.7818	%2.9	%1.1	%1.8
B	1.875	1.89844	1.88775	%1.2	%0.6	%0.7

We can see that Harmonic and Pythagorean (HvP) notes vary above 2% only in the G#/A♭-A-A#/B♭ range, since these are usually very high harmonics, that doesn’t seem to cause a problem. Pythagorean and Equal Temperament (PvE) are extremely close only off by more than 1% in F and A#, and Equal Temperament and Harmonic  (HvE) are off by more than 2% in F and A. These tunings are all relative so a specific note (in this case ‘C’). Orchestras usually tune to whatever the 1st chair violin declares ‘C’ to be. It’s useful to have a standard, however, so the A above middle-C is usually set to 440 cycles/second. Every other note is based on the tuning you are using, so for 12TET middle-C is 440/1.68179=261.6 cycles/second, and for Pythagorean middle-C is 440/1.6875=260.7 cycles/second^[1].

Other scales

While a large number of western pieces are written in a major scale, that is not the only scale you can use to write music. Minor scales, like A minor, have 8 notes running from A to A with no sharps or flats (whole-step, half-step, whole-step, whole-step, whole-step, half-step, whole-step, whole-step). Other combinations are also possible, including combinations with different numbers of notes. Instruments that can play the 12-tone system, either with Equal Temperament or with Pythagorean tuning, are able to play in most of these scales.

Future

In Future blog posts I’ll examine why instruments playing the same note still sound different from each other, and how these instruments work.

 

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[1] If middle-C is calculated going from A to C using Pythagorean tuning, it will be 1.01 off our calculated value (264 cycles/second). This shows the downside of Pythagorean tuning, The direction affects the eventual results because (1.5)¹²/128 != 1. No such inconsistency exists for Equal Temperament tuning.

Lindsey Vonn is an American Hero

Lindsey Vonn is an American Hero

I don’t know much about Lindsey Vonn, or down hill skiing. Just what we all know about the events from the last couple of days, which we’ve all seen. From that little bit, though, I know that Lindsey is an American hero, one that we need desperately right now.

We Americans lionize winning and trash losing. For any of us, the Patriots had a failed season because they didn’t win the Superbowl. We deal with failure, but denying it, or laying the blame elsewhere. We even see this attitude in our politicians on both sides of the aisle. We need to learn again that there are good failures.

Lindsey could have failed in her goal of winning more medals in the Olympics by just not trying. She was 41, which is considered old in the world of top competitor sports. She had knee surgery. Just weeks ago, she blew out her ACL. No one would have thought poorly of her if she decided not to compete. Our attitude to failure and success would consider this a wise decision, keeping her out of the spotlight. The thing is, this would 100% guarantee failure of her goal.

Lindsey could have made a run that was more conservative. We know she could, she did it a day earlier to qualify. She would wind up on the leader board somewhere outside the medals. We would all say, “Look at what she overcame just to get here”. This would be our ubiquitous participation trophy. You participated, good for you.

Lindsey did neither of those things. She was going for the gold, which meant skiing the most aggressive, risky line she could with the smallest margin of error. If she skied that and still didn’t get the gold, we may not be able to tell the difference between being more conservative and just not making it, but Lindsey could. That fact that something freaky happened, or that she was off her line more than she wanted was always a possibility. That is the cost of getting the gold. Some may criticize Lindsey for risking too much – that her failure is a lesson in risking too much, or living too close to the edge, but if the true goal is to win a medal, all other paths had 100% failure.

We seem to have forgotten that to have a chance at real success, we need to take reasonable risks. They don’t always pan out, but we guarantee they will never pan out if we don’t try. That is the real lesson we learn from Lindsey’s bravery. Not all heroes accomplish their goals, but they all inspire us. Thank you Lindsey for showing us to take reasonable risks to achieve the best we possibly can.

The First Alphabet

Writing systems have been invented by humans independently at least 3 times, probably 4. We know this because they arose in locations that did not have contact with each other geographically (China, Mesoamerica, and either Egypt or Sumeria). It’s likely that the Egyptian and Sumerian writing systems are independent of each other based on their differences, even though there was probably communication between the regions. These writing systems were a mix of pictural (called ideograms where the character represents a word or idea), or phonetic (where the character represents a sound). Other groups develop writing systems after exposure to one of these systems for their own purposes. Sometimes they adopted the system wholesale, particularly if their languages were related, or they were close enough. Sometimes they used the characters to inspire their own values to write their own language. These new systems were almost always what we call syllabaries. A syllabary uses one character to represent a single syllable in your language. Many languages have restrictions on how a syllable can be formed (ending in a vowel, or a restricted number of constants), and additional restrictions on what kind of constants can start a word (for instance, in english, we never start a syllable with the ‘ng’ sound). These restrictions mean you can often express your whole language using a syllabary of only about 40 or 50 characters. We see examples of these in Japanese and Korean scripts.

The first alphabet

Sometime in the 19th or 18th centuries BCE Canaanite speaking people did something remarkable with Egyptian hieroglyphs. Instead of taking those characters and forming a syllabary or ideograms (which most Egyptian hieroglyphs are), they instead took a subset of those representing objects, and used their own names for them. So the Egyptian symbol for ox () became alp, or ‘ox’ in cannanite, the symbol for house () became bayt, and so forth. But instead of the symbols representing the full sound alp or the meaning of ‘ox’, they represented the same sound as the first sound of the Canaanite word. The Canaanites only represented the constants of a word^[1]. This system is easy to teach to others as there were only some 26 characters and they each can be remembered as mnemonic of their sound by what they represent. Scholars describe this system as proto-Sinaitic, since the first examples we see are in Sinai. Over the next 8-9 centuries the use of this alphabet spread throughout the Levant (what makes up modern day Israel), which the symbols becoming more abstract, but easier to produce over time. Since the people of those regions spoke related languages, the system could be easily adopted everywhere. Only the constants were represented. The vowels were implied.

Examples of the proto-Sinaitic script as proposed to the Unicode consortium for inclusion in the alphabet.

Character	Name	Meaning	Sound
	ʾalp	ox head	ʾ
	bayt	house	b
	gaml	throwing stick	g
	dalt	door	d
	ḥe	fence	ḥ
	ho	man calling	h
	hll	jubilate	h
	wāw	hook	w
	ḏayp	eyebrow	ḏ
	ziq	fetter/ anke chain	z
	ḥaṣir	mansion	ḥ
	ḫayt	thread	ḫ
	ṭab	good	ṭ
	ẓil	shade	ẓ
	yad	hand	y
	kap	palm	k
	lamd	goad/cattle prod	l
	maym	water	m
	naḥš	snake	n
	samk	fish (?)	s
	digg	fish	d
	ʿayn	eye	ʿ
	piʾt	corner	p
[	pu	mouth	p
	ʿayn	eye	ʿ
	ṣad	plant	ṣ
	qop	monkey	q
	ṣirar	bag	ṣ
	qaw	cord/line	q
	raš	head	r
	šamš	sun	š
	ṯad	breast	ṯ
	ṯann	composite bow	ṯ
	taw	owner’s mark	t
	ġinab	grape	ġ
	ṯa ?	?	ṯ?
	?	?	?
	šin ?	>	š?

As you can see from the table, not all symbols are fully decoded. Scholars think that the actual alphabet has 26 characters, with some characters changing over time to produce the 30 in this table.

By the end of the 10th century BCE, early Hebrew and Phonician formalized the script, consistently writing from left to right, and reducing the characters used to 22.

Even though Mesopotamia had its own writing system, cuneiform, which was highly optimized for their writing material (clay tablets), there are instances of the Mesopotamian languages being written in this alphabetic script. This isn’t surprising since their languages were highly related to those in the Levant.

The alphabet goes west

In the west, Phonecian traders brought this script to the Mediterranean. There the Greeks adopted it. Unlike the Semitic languages of the Levant and Mesopotamia, the character names didn’t have a meaning for the Greeks. They took the Phonecian names for the letters, modified the sounds to fit their sound system, and treated them as abstract names for their letters. So the proto-Sinaitic alp became alef in Phonecian and alpha in Greek. For the Greeks, these letters only represented the sounds. When the Greeks adopted the alphabet, they added vowel sounds by repurposing some of the characters that represented consonants that were not in Greek. Greek had a lot more ambiguity between words when the vowels were removed, for instance you couldn’t tell the difference between the word for ‘we/us’ and ‘you (plural)’. They also added a few new letters (you can tell because the name was fully Greek like omicron or omega - little-o and big-o respectively). The Romans took the Greek letters and adapted them for their language. These gave rise to the Latin alphabet, used by most of Western Europe, and often adopted by those who in modern times wish to create a new written version of their language (aided first by the proliferation of the Latin movable print systems, making printing easier, and later by typewriters and computers). The Greek alphabet was also used by the Byzantine monk Cyril in the 9th century CE to write Slavic languages, which gave rise to Cyrillic.

The alphabet goes east

The original Canaanite script continued to spread eastward as well. Modern Arabic derived from it, as well as most Indian scripts, Burmese, Thai, and Khmer. In these scripts, the essential ‘constant only’ nature of the script was preserved and vowels were handled either by ‘double duty’ of some of the ‘vowel-like consonants’ (alef for ‘a’, yod for the latin ‘i’, w for ‘o’ - much like english will sometimes use ‘y’ and ‘w’ as vowels), or more commonly, by diacritics around the consonants (pointing in Hebrew or Arabic, characters before, behind, above or below in Thai).

Use in the Hebrew Bible

Today, the Hebrew Bible is written in a Modern Hebrew square script. This script comes from Babylon, and is more stylized than the early old Hebrew script. In its modern version it is written with full vowel pointing, so the correct pronunciation is no longer ambiguous. The original Hebrew writings, however, at least those written before the Exile, were written in the paleo-Hebrew script, or its ancestor the proto-Semitic alphabet. They were available for the earliest part of the written Hebrew scriptures. We have an example from the 6th century of the Biblical blessing of Yahweh written on a metal scroll (see below). This blessing is written in the old Hebrew script. Other archaeological inscriptions from the late Bronze Age and early Iron Age are in this form. After the return, the Babylonian script became the predominant form for both religious and administrative texts. The old script did not die out completely. During the Maccabees, when Judah was briefly independent from the Hellenistic kingdoms, the old Hebrew script saw a resurgence. We have some examples of text in the script from the Dead Sea scrolls. There is indication that the text used in some of the Septuigint translations (the pre-Christian translations of the Hebrew scriptures into Greek) were translated from a text written in the paleo-Hebrew script (some of the translation errors were best explained by the confusion of similar paleo-Hebrew letters which were not similar in the Babylonian Square script). The Samaritans, even today, use the pelo-Hebrew script when writing their copies of the Hebrew Torah.

 

One of the most remarkable instances is the Ketef Hinnom scrolls, two small silver  dating from the 6th Century BCE and contains a blessing of Yahweh found in Exodus, Deuteronomy, and Numbers.

Today the Samaritans* still copy their scriptures in paleo-Hebrew.

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The Ketef Hinnom 2 scroll with the image on the right, a rendition with the paleo-Hebrew script in the middle and the modern square Hebrew on the left.

Image from Tamar Hayardeni, Attribution, via Wikimedia Commons

The first alphabet today

Today, the unicode consortium includes code points for most of the daughter alphabets that is descended from proto-Sinaitic, as well as code points for its parent, Egyptian Hieroglyphs, but proto-Sinaitic is not yet included as code points in the standard. A proposal written in 2019 to include the script. The table I concluded in this article comes directly from this proposal.

The invention of the script was a clever way to create a written language that could be taught in a matter of minutes as was easily remembered. As the script spread and was stylized, and particularly when it was picked up be other language groups, the easy identification of the character to is sound was lost, but the ability to express your language in something closer to 20 characters than 50 was a big win and it was adopted by many groups, eventually becoming the dominant way of expressing written language in the world today.

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[1] The first sound of alp is the glottal stop, the sound that the apostrophe represents if you pronounce Hawai’i as a native Hawaiian would. We don’t have that sound in our sound system in English, but we unconsciously make it whenever a work begins with a vowel.