Regarding week days and music.

Gregory wrote: “we accept the law concerning the octave which cleanses and circumcises because once time represented by the number seven comes to a close, the octave succeeds it.

Indeed. In Judaism the circumcision is done on the 8th day after the birth.

]]>The Egyptian hieroglyph representing "star" had five points, while the "star" sign in Mesopotamian cuneiform had eight.

Sopdet, the Egyptian personification of the star Sirius, is always shown with the five-pointed star hieroglyph on her head.The Sumerian sign DIĜIR [see symbol at link] originated as a star-shaped ideogram indicating a god in general, or the Sumerian god An, the supreme father of the gods. Dingir also meant sky or heaven …

By the time of Late Babylonian/Assyrian the cuneiform got simplified to a cross-like symbol with an extra wedge — i.e. four or maybe five points, not seven.

Star of David has six points "A derivation of the Seal of Solomon, which was used for decorative and mystical purposes by Muslims and Kabbalistic Jews, …"

Starfishes have five points usually.

"In alchemy, a seven-sided star can refer to the seven planets which were known to early alchemists, and also, the seven alchemical substances: fire, water, air, earth, sulphur, salt and mercury." (There's some more coincidences with 7.)

]]>*beyond what the evidence will support*

I see a couple of parallels:

In Historical Linguistics, there are 'lumpers' who think they can 'extend' the Comparative Method to reconstruct Proto-languages. There's a even a team claiming to reconstruct some vocabulary of what they call Proto-Sapiens. The trouble is, no-one can point to counter-evidence. We just can't tell — is the point.

In music, there's been some celebrated copyright cases where artists have been accused of 'stealing' ideas: George Harrison, Robin Thicke/Pharrell Williams, Mariah Carey, Katy Perry, …. Putting these cases in front of a jury (people who only listen to music, and that probably only a few genres) produces outlandish damages awards. Yes those songs do sound similar; that's because they sound similar to a whole tradition of styles, memes, chord changes. Professional musicians are outraged at these awards: You can't own the Minor scale (at less than a minute in, Adam's found a parallel from Katy Perry to a rapper to a Bach Violin Sonata. Wait for the take-down starting 6:15.)

Similarly, you can't own tuning-by-fifths. It's inevitable you'll exhaust the 'interest' in octaves pretty quickly; 'use your ears' you'll find fifths by overblowing a horn or stopping a string; muck about with this for a while, you'll find other resonances. Yes you need some sophisticated maths to regularise the tuning when you get to tones that don't resonate. That doesn't mean you must have taken the maths from another culture.

]]>Please point to the evidence for this transmission. AFAICT from your paper, the angular harp got only as far as the Tarim basin/ illustration in the DunHuang Caves. That was not China at the time. (Many of us think it should not be China today.)

There is no archaeological proof that the specific type of angular harp excavated in Xinjiang made its way deeper into China. Nevertheless, music archaeologist B. Lawergren suggests that the Xinjiang harps had an influence on the design of later Chinese stringed instruments. [p 9]

It's a long way from "influence" to 'adopted the same tuning system' — especially since the Chinese *qin* appears to have 10 strings vs the angular harp's 7 (heptatonic). (I don't see the shape of the *qin* being very similar — even as vaguely as Lawergren's "inspired the shape".) He does concede

[The from-Mesopotamian harp] ended up with five strings, and this probably implies it had severed any association with the Mesopotamian tuning theory. It arrived in Xinjiang without the theoretical framework.

There follows some discussion of earlier 5-string harps in China. That would be consistent with a pentatonic scale, that would need only a few steps of fifths up/down. Not enough to realise it forms a cycle.

There's also mention of 9-string lyres, based possibly on Mesopotamian models. Richard Dumbrill, who has also analysed CBS 1766, is of the view the 9-string lyres used a different tuning system, not heptatonic.

The trouble is we can't know how those other instruments were tuned. Did the 9 or 10 strings span more than an octave? (Dumbrill thinks so — with an alarming level of interpolation into the fragments of writings.) The 25-string Chinese *se* presumably had/has more than 7 tones within an octave.

If all that remained of the 'theoretical framework' after dispersion through the Silk Road was 'use the second harmonic' (fifths), that's really no different to 'use your ears' and listen for the resonance. Figuring out the cycle/Pythagorean comma could have been independently rediscovered.

But then many in the field (including Lawergren) want so hard for Mesopotamia to be the one source of all music, they're pushing claims beyond what the evidence will support.

Adding all this Numerology/Astrology about days of the week is subtracting from credibility.

]]>Please consider this friendly suggestion: in future, skip a line between paragraphs.

]]>First, to illustrate the gearing of 7 and 5, write the numbers 1 through 7 five times, as shown below. Now, circle the first number and every subsequent fifth number (count by including, each time, the last circled number):

1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,6,7…

You come out with the sequence 1,5,2,6,3,7 (an inversion of the sequence 4,1,5,2,6,3,7). If we were to continue on writing numbers and circling them, the sequence would be generated indefinitely.

Now, here’s how to illustrate the gearing of 7 and 12. Do the same thing as above, but write the numbers 1 through 7 twelve times instead of five. Now, circle the first number and every subsequent twelfth number (count by including, each time, the last circled number):

1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,6,7,1,2,3,4,5,6,7…

Once again, you come out with the sequence 1,5,2,6,3,7,4.

So, this is where the sequence 4,1,5,2,6,3,7 comes from, but how does it relate to music? And why do I start it with the number 4?

An octave is a natural harmonic phenomenon: when a high voice and a low voice sing the same melody together, the two voices naturally sing an octave apart. When doing this, the two voices blend seamlessly – but they are not singing the same notes, instead every note sung by the higher voice is the higher octave of the lower voice.

To play an octave on a musical string, one simply plays the whole string (the lower voice, described above) and then shortens the string exactly in half. Then, playing the half string one hears the higher octave (the higher voice, described above). From this, a mathematical relationship emerges: octave = 1/2.

Now let’s divide the string into the next possible smallest number of parts: three parts. If we play 1/3 of the string, we hear a new note. If we play 2/3 of the string, we hear this new note again, but this time an octave lower. (Why? Because 1/3 is one half of 2/3, so the two notes are an octave apart.) Today, we call the note sounded by 2/3 of the string, the ‘fifth’, because if the whole string plays the first note of a major scale (C, for example), playing 2/3 of the string will sound the fifth note of the same scale (G, to continue the example). Here, we have another mathematical relationship: fifth = 2/3.

The ancient Chinese sanfen sunyi method (very probably transmitted from Mesopotamia with the angular harp, circa 1000 BC) make use of just these two ratios, 1/2 and 2/3, to generate a chromatic scale, and it is in this gearing of the cycles of the octave and the fifth that the numbers 7 and 12 appear. Here’s how:

If one generates consecutive fifths [the first fifth having 2/3 of the length of the whole string; the second fifth having 2/3 x 2/3 (or 4/9) of the length of the whole string; the third fifth having 2/3 x 2/3 x 2/3 (or 8/27) of the length of the whole string, and so on…] the twelfth fifth will have a string length of (2/3)12 or 4096/531441 = .0077073.

The string length of the twelfth fifth (.0077073) is important to note, because it is very close to the string length of the seventh higher octave of the whole string: (1/2)7 or 1/128 = .0078125. In other words, twelve fifths span seven octaves, with a very small discrepancy (.0078125/.0077073 = 1.0136433), a difference known as the ‘Pythagorean comma’.

Twelve fifths span seven octaves, but it’s only on a relatively modern instrument – the piano (and its earlier relatives) – that we can hear seven octaves and therefore hear all twelve fifths. For our distant ancestors to hear all twelve fifths on a musical string, a fundamental realization had to be made (and this realization is incorporated into the ancient Chinese sanfen sunyi method, circa third century BC): that every fifth can be moved into the very lowest octave by doubling its string length, anywhere between one and seven times, depending on its original position relative to the first octave.

Moving the twelve fifths down through seven octaves, so that all twelve fifths are in the lowest octave (creating, by the way, the familiar pattern of frets on stringed instruments) is, in fact, an example of the gearing of cycles of 7 and 12. Consequently, when creating a 12-tone scale from twelve consecutive fifths (which is what the sanfen sunyi method does) the sequence 4,1,5,2,6,3,7 is generated – and it begins with the number 4. To learn more, watch the video: “The Origin of the Sequence 4,1,5,2,6,3,7” (https://youtu.be/7JEm0lJ83UE) or read from page 17 to page 46 in my SPP paper (linked above).

To recap, the sequence 4,1,5,2,6,3,7 is generated when gearing the cycles of the octave and the fifth. This is a mathematical truth, not an arbitrary human convention. As Andrew Usher pointed out, the mathematics behind the ‘planetary hours’ explanation of the week-day order is the same, but there is, nevertheless, one very important difference:

The octave and the fifth are innate mathematical/musical cycles while the division of the day into 24 (or two times 12) hours is completely arbitrary. The listing of seven planets is, of course, also arbitrary. These two cycles combine to create Dio’s ‘planetary hours’ explanation, but both are arbitrary human constructs – not mathematical truths.

Combine this with the fact that Dio actually gives two reasons for the week-day order, and that the first reason he cites is musical, and we have strong support for the theory that the week-day order is drawn, as Origen put it “from musical reasons…quoted by the Persian theology”.

In response to AntC: The Mandeans, native to Iraq, name the days of the week using their ancient Babylonian names. (see page 129-130 of my paper). Also, thanks for quoting Falkenhausen, who is, by the way, very supportive of my thesis (see quote from him in my previous comment).

In response to maidhc: Tablet CBS 10996 (Babylon 700 BC) lists fourteen intervals. Seven of these (six fifths and one tritone) are referred to in the tuning instructions on tablet UET VII 74 (1800 BC) – instructions which derive the seven diatonic modes. The remaining seven intervals are all major or minor thirds (see Table 14 on pg 75 of my paper). PS. I have no idea why the old system collapsed in the Babylonian heartland.

My final comment:

The generation of the sequence 4,1,5,2,6,3,7 is a mathematical truth. A seven-pointed star resembling the one on tablet CBS 1766 can be derived from this mathematics. In other words, the sequence and diagram are derived from a rational exploration of the simple mathematics of music – an exploration which almost certainly was one of the first forays into abstract mathematics.

I believe that the discovery of this sequence and the related seven-pointed star in the mathematics of music inspired our ancestors to consider (con =with; sider=star) that music somehow played a role in cosmic order. Because, for them, there were seven planets. (By the way, I’m curious as to when a star shape became associated with a star in the sky. Could this date back to a belief that there is a connection between music and cosmology?)

I also wonder if the zodiac, which is an arbitrary division of the ecliptic into 12 constellations, which dates back to 5th century BC Babylon, could have been inspired by the much earlier (tablet UET VII 74, 1800 BC) knowledge of the circle of fifths. In other words, did the Babylonians attempt to map out the cosmos using a musical template? If this is the case, other factors may have contributed: for example, the fact that the ratio between the duration of the solar year and the duration of the “lunar year” (twelve full cycles of the phases of the moon) is 365.25/354.37 = 1.0307…a value reminiscent of the “Pythagorean comma”.

I will now let this topic lie, and have my paper speak for itself. I will soon post, however, at Victor Mair’s suggestion, a short description of the angular harps discovered in the Tarim Basin.

Thanks for the stimulating discussion,

Sara ]]>

It's true that both rules generate the same sequence by counting 4 against 7 — or, equivalently, decrementing by 3. But as Ant asks, where does the 4 or 3 come from?

In the planetary hours rule, the 3 comes from 24 modulo 7. In the tetrachord rule, it comes from treating planetary orbits as keys on a piano keyboard and counting off by fourths. Since the definition of a musical fourth has nothing to do with the number of hours in a day, that makes it a coincidence in my book.

Sara:

How did this first, musical reason get forgotten? Because scholars who didn’t understand it, simply dismissed it.

Or maybe they did understand it, and dismissed it on grounds of parsimony. If the sequence of weekday names can be adequately explained in terms of astronomical quantities (number of planets, length of a day), then what additional explanatory power is gained by bringing musical theory into it? None, as far as I can see.

]]>If you want to derive a major triad either from the circle of 5ths or from the harmonic series, you don't have to go too far to get the major third.

The big problem in the major/minor system is that you have to go a long way to get that minor third. There's not much justification for why it should be considered the equivalent of the major.

The modal system produces various minor modes, but they are not the same as the modern (western European) classical system. Not that there's anything wrong with that system, but you can't really claim it's based on physics. Just appreciate it for what it is.

**Sara de Rose:**

Harmonic systems based on the "Greek" modal model are still found in Western Europe, but in the regions closest to the Babylonian heartland, in klezmer, Balkan, Arabic, Persian and Indian music, things have taken a much different direction since the Muslim expansion of the 7th century or so. Why did the old system collapse?

Lothar von Falkenhausen is Professor of Chinese Archaeology and Art History at UCLA …

He has published copiously on musical instruments, …

]]>The specific musicological question I want to examine is: given that these tunings arise from physics/natural harmonics; could they have been invented only once in one place, then transmitted culturally; or could they have been independently discovered multiple times?

How could we investigate that? Especially amongst cultures remote from Mesopotamia and/or without writing — because musical practices could be transmitted alongside writing.

If that tablet hadn't mistakenly been taken in 1929 to be for day-naming/astronomical purposes, would we even be having this conversation?

The critical question is: did the Mesopotamians/Persians/Babylonians operate a seven-day week? If so, how did they name the days?

"A continuous seven-day cycle that runs throughout history … was first practiced in Judaism, dated to the 6th century BC at the latest … " I already quoted from wikipedia. (The wiki hypothesises/your video repeats the Judaic system was acquired during captivity with the Babylonians — but they had a variable-length fourth 'week' of the month to keep aligned with phases of the moon. Hence not a fixed cycle of seven. Hence naming for exactly seven heavenly bodies doesn't work.) The Judaic day-names don't relate to astronomy.

For me this is conclusive evidence the 7 comes from dividing a 28-day moon cycle. Nothing to do with tuning cycles.

Your video jumps from The Mesopotamian/Persian musical system ~2,500 BC; to the tablet ~1,800 BC; to Judaism adopting the 7-day week C6th BC to Origen of Alexandria/Mithraists third Century AD — after adopting Judaic/Christianity's seven-day week.

You do not show causation/cultural transmission. Only just-so stories.

That's a huge span of time for the significant number 7 to acquire all sorts of mystical associations/noted coincidences which were simply no consideration in 6th century BC.

The association of the days of the week with the Sun, the Moon and the five planets visible to the naked eye dates to the Roman era (2nd century). [wiki]

To repeat myself: if not for the coincidence the Romans could see only seven heavenly bodies (excepting the 'fixed stars'), would we be even having this conversation?

It's a mathematical coincidence that if you repeatedly exponentiate tripling (being the first 'interesting' ratio after doubling), you'll eventually come to a number close to a power of two, so repeatedly halving will get back close to where you started. The number of exponentiations happens to be 7 triplings. Making the exponentiating ratio 3:2 (a 'fifth') and halving (moving down an 'octave') in alternating exponents is just mystification.

@Andrew * as both generate an arithmetic progression mod 7, increment 4 *

But why pick 7? Because of the coincidence between the already-established 7 days of the week and the happenstance of 7 visible bodies. And why pick 4?

If there weren't 7 heavenly bodies, I daresay Mithraists could have dreamt up some explanation involving the 7 hills of Rome. And they'd have found a 4 from somewhere.

I'm all done with names of the days. There's a much more interesting question of whether the Mesopotamian tuning system was adopted by the Greeks/Romans and/or by the Chinese.

]]>k_over_hbarc at yahoo.com

]]>Three intervals of a major third gives an octave — but this is another coincidence/a terrible approximation. The harmonically-derived third is also not a good fit to the third obtained via the circle of fifths.

At what stage did harmony and polyphony enter Western music, needing different notes to 'sound together'? Other musics I'm aware of have a single melodic line played over drums or drones accompaniment.

It's been observed (from memory this is traditions in Eastern Europe) that unaccompanied choristers will sing at the fifth and even at the (natural harmonic) third.

]]>*So the "Pythagorean comma" is sort of like the leap second?*

Well for some value of 'like'/'similar'. What we have in both astronomy and harmonics is a load of approximations/coincidences. And kludges to line things up.

There's no reason to expect the period of spin of the earth would have any relation to the period of moon cycles or to the period of earth revolving around the sun. Nor to the periods of planetary motion as viewed from the earth.

The period of the spin of the earth is much shorter than a moon cycle, so let's round that to 28 days; and very much shorter than revolving around the sun, so let's call that 365 days. 28 almost exactly divides into 365; 7 does divide exactly into 28. Those are all coincidences. (The Greeks rounded to 360 for degrees of a circle, so they had plenty of factors to divide into.)

How shall we punctuate our daily lives? A stretch over 28 days is too long to pay attention to in a mostly non-literate society. The Romans divided months at the Ides and the Nones — but this gives somewhat irregular punctuation. A persistent cycle of 7 days is a good fit mathematically.

I don't think we need any more explanation than that as to where 7 came from. Having chosen 7, we then see mystical coincidences. (In the same vein, the alleged spacing of the five planets for their orbits to fit inside a nesting of the Platonic solids is also just a mystical coincidence. Another coincidence with human female cycles gets them called 'menstrual' — that is, moon-based. Other mammals have 'menstrual' cycles varying between 9 and 37 days, depending on the species. Biorhythms include a 28-day cycle. It seems humans just can't help themselves wanting to explain numerical coincidences. See Kabbalah.)

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