Scheibler's Musical Scale

Helmholtz did a calculation of the frequencies of the musical scale. Like him we can start out with the ratios of Ptolemy's scale. Using the numerators, n, and denominators, d, below the ratios are n/d. If we want to adjust these numbers so the all the denominators are the same, the smallest denominator that will accomplish this task is D = 24. The corresponding numerators are N = 24, 27, 30, 32, 36, 40, 45, 48. These numbers are the harmonics of a fundamental frequency whose value is 1/24th that of Middle C and are part of a harmonic series.

On multiplying these numbers by the number 11 we find that the frequency corresponding to the note A is exactly 440 cycles per second. A440 is now the standard frequency for the musical scale. Helmhotz arrives at the same set of numbers in On the Sensation of Tone and attributes the scale to Scheibler.

The Beat of a Different Drummer?

For Christmas I got a set of tuning forks from scientificsonline.com.


The pitches marked on the forks are,

C: 256, D: 288, E: 320, F:341.3, G: 384, A: 426.6, B: 480, C:512
which are not the usual pitches used in music. The label on the box doesn't help much by way of an explanation of the difference.


The notes indicate that it is a diatonic scale and the proportions (fork pitch/C256) are as follows,

1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2
indicating a Ptolemaic diatonic scale. One may ask, "Why is this particular scale is used?" It presented a bit of a puzzle and further study was indicated.

I consulted Rayleigh's The Theory of Sound and found the C256 "is usually adopted by physicists and acoustical instrument makers, and has the advantage of simplicity." But he says nothing of its origin. Helmholtz in On the Sensations of Tone is not much more helpful. He does refer to "The theorectical English pitch, c' = 256" which indicates an English origin. Through a Google book search I came across Ellis' History of Musical Pitch which mentions C512 and a report by the Society of Arts. Another book search led to an entry in the Journal of the Society of Arts, Jun 10, 1859 which discusses the need for a "uniform pitch or diapason." C512 was favored by Mr. Hullah. The signatures there indicate that members of the Royal Society were present. One also get the impression the C512 scale predates this discussion. A paper, The origin of the tuning fork, that I found at the National Institutes of Health indicates that John Shore and Handel used this scale.

So this may be a partial explanation for why this particular scale is used by the scientific community.

Founders Tree

In Humboldt Redwoods State Park near Weott, CA along an old route of Hwy 101, also known as The Avenue of the Giants, is Founders Grove in which one can find one of the tallest living trees in the world. It's known as The Founders Tree and is dedicated to the founders of the Save-The-Redwoods League. It is 346.1 feet or 111.0 meters tall.


Founders Tree
It's not the tallest tree in the park. That title is claimed by Stratosphere Giant. The tallest redwood currently known is Hyperion in Redwood National and State Parks.

Supplemental (Dec 27): The Founders Tree coordinates on a
USGS topographic map (CA Weott 102418 1969 24000 geo)
are 40° 21.121'N, 123° 55.426'W.

Dark Winter Nights

Some of you may have noticed how dark it is at night these days. One reason is that the Winter Solstice was on Dec 22 at 5:30:00 UT. The US Naval Observatory's Multiyear Almanac gives the times of the Winter Solstice for when the Sun crosses Right Ascension 18 hours. Another reason is that the New Moon also occurs this week on Christmas Eve at 18:06 UT. I used the Almanac to compute the angular distance between the New Moon and the Sun at the time of the Winter Equinox to see how often they occur together between the years 2000 and 2050.


The plot above shows that this is a rather rare event astronomically but in 2014 they will come rather close together, less than 1.5° in RA apart. The probability of this happening is about once in 240 years. The average number of days in a month is the Moon's synodic period of 29.5 days. So the probability that the New Moon will occur on the same day as the Winter Solstice is about once every 30 years.

Greenland Geological Record Atypical?

When looking at the geological record for a given locality we have to look at extraneous factors factors which might influence conditions. There is a Croll quote from Lyell's Antiquity of Man referring to a warmer Greenland in the past and evidence of plant life there. However, Greenland may be atypical because of the presence of "numerous hot springs" on Disko Island, the location mentioned. This may be a contributing factor for Sequoia langsdorfii being able to grow at such a high latitude.

It would be interesting to see what radiocarbon or some other form of radiometric dating would tell us about the age of the fossils found. One has to be careful about the assumptions one makes when extrapolating the proxy temperature records and not read too much into them. Paleontology and the fossil record may provide some additional information over a longer span of time.

Ancient Sequoia

Both Croll and Lyell discuss the change in climate over time for a given locality. An example is the Sequoia which is now only found in California and Oregon on the west coast of the United States. The fossil species Sequoia langsdorfii is nearly indistinguishable from the modern species Sequoia sempervirens.

Some S. langsdorfii links are:

Fossil Record of the Redwoods - NPS

On the Change of the Obliquity - Croll

Elements of Geology - Lyell

Croll's Climate Change papers, 1864-7

It can be a little difficult to keep track of Croll's papers on climate change spread out over the years from 1864 to 1867 in Philosophical Magazine. A list might prove useful.

Aug 1864, p. 121, On the Physical Cause of the Change of Climate during Geological Epochs

Jan 1866, p. 26, On the Excentricity of the Earth's Orbit

Apr 1866, p. 301, On the Physical Cause of the Submergence and Emergence of the Land during the Glacial Epoch

Feb 1867, p. 119, On the Excentricity of the Earth's Orbit, and its Physical Relations to the Glacial Epoch

Jun 1867, p. 426, On the Change in the Obliquity of the Ecliptic, its Influence on the Climate of the Polar Regions and on the Level of the Sea

These papers can be found elsewhere and much of the content is reproduced in Croll's first book.

Another Croll Climate Change Paper From 1867

Earlier today I found another climate change paper by James Croll, On the Eccentricity of the Earth's Orbit, and its Physical Relations to the Glacial Epoch, published in Philisophical Magazine, Feb., 1867. It appears to have be written about the same time as the Geological Society of Glasgow paper on the change in the obliquity. The obliquity paper is rather interesting especially the part about 25 ft and 40 ft beaches at 11,700 years ago and 60,000 years ago respectively. Forests below the present sea level that alternated with mud flats point to rather large changes in sea level possibly due to the melting of a mile of the Antartic ice sheet and a shift of the Earth's center of gravity northward about 8 ft. The implication seems to be that the Arctic Ocean will rise somewhat when aphelion and the winter solstice coincide.

Croll's 1864 Paper on Climate Change

James Croll's first paper on climate change, On the Physical Cause of Climate Change during Geological Epochs, was published in 1864 in Philosophical Magazine, Vol 28. In it he starts out by reviewing earlier discussions of climate change and proposes an alternative theory. He notes that in past ages there were alternating periods of warm and cold which could be deduced from the species of sea shells that were present in the oceans around England. Glaciers and icebergs transported broken rock which ended up as breccias in the seas. A warm period was responsible for the coal formations in England. He suggests that the cold period may have been due to an absence of the Gulf Stream at that time. There is a discussion of trade winds and ocean currents. He proceeds to a discussion of Leverrier's work of the variation of the eccentricity of the Earth's orbit and the effect of the Earth's precession in alternating the temperature of the northern and southern hemispheres. Finally he correlates these temperature changes with the warm and cold periods in the two hemispheres.

Supplemental (Dec 14): Another paper, On the Change in the Obliquity of the Ecliptic, was published in the Transactions of the Geological Society of Glasgow in 1866.

Croll on the Heating Power of Ocean Currents

Capter 2 of Croll's Climate and Time in their Geological Relations uses the Atlantic Gulf Stream to show the heating power of ocean currents. The Gulf Stream flows along the eastern coast of North America from the Caribbean to northern Europe. The heat it carries with helps moderate the climate of northern Europe. His estimate of the amount of heat transported might be a little diffucult to follow and some illumination would probably help. He starts out with a simple channel-like model of the Gulf Stream.

His units are somewhat outdated. The foot pound is the amount of energy required to lift a pound mass 1 foot against the force of gravity. The rate of flow of sea water in the channel is Q = depth * width * speed of the flow. To compute the heat capacity of a cubic foot of sea water we need to know the density of sea water and its specific heat capacity. Croll uses the density of ordinary water and assumes we know the density of sea water¹. Multiplying these values by the change in temperature gives the heat per unit volume transported away from the Caribbean. The total heat transported per day is the heat per unit volume times the volume rate of flow. Doing the math gives us the numbers cited by Croll.

Croll also makes an estimate of the rate of solar heating or the amount of heat recieved by a unit of area in a unit of time for a point on the equator at one of the equinoxes. He starts out with a value close to Pouillet's for the rate of heating at the top of the atmosphere and uses an estimate that 22% is stopped by the atmosphere. Again we get the value cited for the heat flux at the surface.

Croll notes that for the current orbit and orientation of the Earth the southern hemisphere gets more heating than the northern hemisphere which experiences milder conditions. Since the northern hemisphere is colder there is some heat transport across the equator. If we look at the ocean currents in the Atlantic near the equator we see that there is a current from the Tropics off the west coast of Africa along the equator that transfers to the east coast of South America and up into the Caribbean some of which ends up in the Gulf Stream.

Supplemental (Dec 10): ¹The value used for the volume heat capacity appears to be the one used by Croll. I used the specific heat capacity for sea water at a slightly higher temperature (20°C or 68°F) and the density of fresh water. The error seem to be negligible. The specific heat capacity for sea water is slightly less than that of fresh water since there are fewer molecules for a given weight due to the presence of the dissolved salts. The salts also increase the density of sea water slightly to 1.025 gm/cm³.