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The velocity of a traveling wave in a stretched string is determined by the tension and the mass per unit length of the string. for a string of length cm and mass/length = gm/m. For such a string, the fundamental frequency would be Hz. Any of the highlighted quantities can be calculated by clicking on them.

http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/string.html

The ancient Greek and even earlier civilizations have developed a theory of intervals by the monochord. How did they count and measure the waves and wavelengths without the units of time like seconds and measures of length?

I‘m breaking my head about this ...

What I am missing?

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    They don't need time units to measure the lengths of standing waves on strings, and they had smaller units of length in Ancient Greece. My understanding of Ancient Greek mathematics is that is was more often proportional than absolute, meaning they would be more likely to think and talk about one wavelength being 1/2 or 1/3 of another wavelength than to think or talk about a precise wavelength measured by some unit. Limiting though that may seem to us today, they developed algebra with such thinking. – Todd Wilcox Sep 23 '19 at 8:35
  • But could you help me, if I'd ask differently: Would they have had the means and utils to make a standard pitch, if needed? – Albrecht Hügli Sep 23 '19 at 10:02
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    Oh yes, quite possibly. All it takes is any resonant idiophone (e.g., something like a tuning fork) that can be reproduced to a certain pitch and then distributed. A resonant air column of a certain length (e.g., pitch pipes) is a slightly less stable but still perfectly usable pitch reference. Whether the Ancient Greeks could have produced consistent idiophones I don't know, but they definitely could and did produce consistent wind instruments that could have been used as tuning references, similar to the oboe in the modern orchestra. – Todd Wilcox Sep 23 '19 at 15:20
  • What they probably couldn't have done is analyze the pitch reference itself as a frequency or wavelength value. If they wanted to communicate the pitch reference without actually sending a set of aulos (as one possible example instrument), then they would have had to communicate instructions to construct a near-identical set of aulos. I doubt they could have either measured or communicated a specific frequency or wavelength. The might have been able to measure tension, and therefore could have used a tense string of fixed length as a reference, but aulos would be much easier. – Todd Wilcox Sep 23 '19 at 15:23
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    Never mind about the tags. I realized out that you must have typed tonhöhe, so I'll fix them myself. – phoog Sep 23 '19 at 15:55
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How did they count and measure the waves and wavelengths without the units of time like seconds and measures of length?

What I am missing?

They didn't count or measure waves or frequencies. They measured wavelengths proportionally relative to the length of the monochord they were using at the time.

They had no way to put an absolute number on the frequency of any pitch, but they didn't need that: it doesn't stop them from developing a theory of intervals based on the monochord. The theory of intervals is based on the ratio of various lengths of string with identical mass and tension. The frequencies of standing waves in such lengths of string are proportional to their lengths.

So, they found that a monochord vibrating along 2/3 of its length sounded the pitch a fifth above that produced when the same monochord vibrated along its entire length, and when it vibrates along half its length the result is an octave higher, and so on.

You don't even need units of length to do this (other than the unit system based on the length of a given monochord), but the ancient Greeks did indeed have units of length.

They did not need to divide, for example, 880 cycles per second by 440 cycles per second to get the ratio 2:1, nor did they need to divide a length of 60 cm by 30 cm. They just needed to observe that the lower pitch was produced by a string that was twice as long as the string producing the higher pitch.

Don't forget that the ancient Greeks were very good in mathematics, especially geometry.

Measurement of frequency would not have been possible without a sufficiently precise and accurate way to measure time. As far as I can tell, having looked into this for another question recently, this wasn't possible before the invention of the pendulum clock in the 17th century. This allowed the measurement of time from an accuracy of 15 minutes per day to 15 seconds per day. I haven't found out when it was actually done for the first time, but it was certainly before the early 19th century. I would guess it was done in the 18th century.

I am also a bit unsure about the precision side of it: how do you know that something has vibrated 440 times during your second of timing rather than 441 or 439? A brief look at Helmholtz suggests that this may have been done by precisely controlling the rate of revolution of a siren and then tuning another pitch to match it. I would be surprised if people were doing this much before the 18th century.

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  • The two last sections of your answer are exactly what I am occupied too! The question remains: how and when and by whom came concert pitch up, and how was it measured before. I think this are interesting questions. Also you're mentioning the invention of the pendulum. Wouldn't it be nice if this pendulum had provided a sound? ;) Concerning the ancient Greek: I think they had a kind of standardisation of pitch derived from range of the human voice for their modes. But if they had asked me: I would have had a splendid idea! – Albrecht Hügli Sep 23 '19 at 18:07
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    @AlbrechtHügli "a kind of standardization" also existed in 18th-century Europe. Pitches varied, but not entirely arbitrarily. But even today there are orchestras that tune to 415 or 430 or 443. Have you read Helmholtz? – phoog Sep 23 '19 at 18:14
  • I am still reading. I am reading a lot. But I only understand a few. And the most I forget! But it helps me to find more links and information reading your questions and answers. I've encountered Helmholtz in all this research but I would lie if I said I know him. – Albrecht Hügli Sep 23 '19 at 20:45
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There is nothing to break your head about here.

Small children learn to sing "in tune" (more or less) without knowing anything about music theory or physics. They just copy what they hear.

If you think ratios are interesting (which the Greeks did, as is demonstrated by their writing on mathematics), you can invent a system where the notes of a scale correspond (more or less) to simple ratios of the lengths of a string like 2:1, 3:2, 4:3, without knowing anything except how to use a ruler to measure lengths.

Of course that theory has no particular relevance to constructing real instruments like harps or lyres, because there is no reason why the tension of every string should be the same, even if you have the technology to make strings with consistent mechanical properties (which means metal strings, not gut). It is also irrelevant to constructing wind instruments, which is obvious if you spend five minutes actually looking at the finger hole positions on a recorder, rather than believing some half-remembered bit of theory about where they should be positioned.

A lot of nonsense has been written about Greek musical theory with the "benefit" of a one or two millennia of hindsight. In fact modern research (based on the few surviving artefacts such as wind instruments with finger holes, which preserve a record of the original intonation) suggests that the whole medieval theoretical edifice of "modes with Greek names" has little or no relationship to what ancient Greek music actually sounded like.

A final comment: not every document claiming to be written by somebody famous, like Pythagoras or Aristotle, was actually written by them - and some of the documents might have been written centuries after their alleged authors had died!

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    They biggest nonsense I've read the other day in this context was: The C-major scale is the most natural scale as it can be played on only white keys! That's why it has so much benefit for meditation and therapy! (I'd say: Especially for the musician who plays a piece in major C instead one with 7 sharps :) – Albrecht Hügli Sep 23 '19 at 9:43
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    As far as I understand it, even ancient Greek theoretical writings make it clear that the medieval modes were not the same as the ancient Greek modes. – phoog Sep 23 '19 at 13:44
  • I know that the modes were not the same. Reading Boethius, Hucbald, Glarean and others - and of course all the newer books about history of music tell that the modes have been exchanged. But the tetrachords remained identical and what surprises me: The lowest note was G! That's about the lowest note of human (men) voices ... – Albrecht Hügli Oct 12 '19 at 15:36

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