ZynAddSubFX tuning
x31eq
I’m looking at ZynAddSubFX’s tuning model now. I have a Scala file for a full octave tuning -- mapping from my Ztar with no repetition. It should map so that the tonic is MIDI pitch zero.

There are 112 notes defined in the “scale”. By modifying the source code to print out the note within the scale, I can get the “shift” calibrated. What works, for A as 69, is a shift of -43. 69+43=112.

The “shift” is defined modulo the octave. The shifts from the main page (“Master KeyShift and KeyShift”) do pitch shifts measured from the tonic of the scale, and don’t mess anything up.

For this mapping, it makes sense to define note 55 as middle C. That means a Shift of -57. It seems to work.

I found that Scala .kbm files don’t work right. I might look into that more.

Another thing I found from the source code is that the configuration files are all in XML format. Specifically gzipped XML. That’s good because I’d always worried about having the presets stored in a binary format. (I didn’t think of running “file” on them.) With a text format extractable, they might work with version control, and I can certainly read them to see what I did.

It also means I can write out tuning files, instead of writing Scala files and working out how they get applied.
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One commit, multiple version control systems
x31eq
I took a long time cleaning bugs and this is what I got:
commit() {
    svnchanged=`svn status -q 2>/dev/null`
    hgchanged=`hg status -mardn 2>/dev/null`
    gitchanged=`git status -s -uno 2>/dev/null`
    if [[ -z "$gitchanged$svnchanged$hgchanged" ]]
        then echo "No modified repositories here" && return 1
    fi
    if [[ -z "$*" ]]
    then [[ -z "$svnchanged" ]] || svn commit
         [[ -z "$hgchanged"  ]] || hg  commit
         [[ -z "$gitchanged" ]] || git commit -a
    else [[ -z "$svnchanged" ]] || svn commit -m    "$*"
         [[ -z "$hgchanged"  ]] || hg  commit -m    "$*"
         [[ -z "$gitchanged" ]] || git commit -a -m "$*"
    fi
}
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Notating Jove
x31eq
Jove is the deity name for the 7-limit neutral-thirds lattice. It’s explained at http://x31eq.com/lattice.htm and implies a planar temperament.

It has exact neutral thirds, where the neutral third is one of the generators of the temperament. So you can notate each row easily with half sharps relative to Pythagorean intonation. Then you have to reconcile the different rows.

Moving up 4 places gets you to sharp of 25:24. A Pythagorean sharp corresponds to 2187:2048. This is twice 81:80 sharper than 25:24, so to write a 5-limit major third, you lower the expected degree by two steps of 81:80.

Moving up 2 places gets you to a half-sharp. It isn’t the same half-sharp as the one describing the neutral thirds. It’s half of a 25:24. That makes it one step of 81:80 flat of the half-Pythagorean sharp.

Moving up one place is more complicated. You go from C to Gb -- or is it F#? Well, let’s say that going up a 7:4 and down by three neutral thirds gives you a flat. The interval is 7/4*4/3*9/11 = 21/11. The sharp must represent 22:21. It happens to be flat of the Pythagorean sharp by an interval of 45927:45056 or 33.1 cents. Or take the extended 7-limit neutral third of 49:40. 7/4*4/3*40/49 = 40/21 so the sharp could represent 21:20 and the discrepancy is 3645/3584 or 29.2 cents.


Okay, lets look at 7:4. Twice a 7:4 in Jove is the same as 5:4 plus a neutral third plus an octave. This interval could be 5/4*11/9*2 = 55/18. Twice 7:4 is 49:16. The interval between these is 441:440, which is a Jove comma. the same interval could be 5/4*49/40*2 which is also 49:16. You write it as a fifth plus a half sharp less a syntonic comma plus an octave. From that, we can define a half-sharp of

49/16 / 3 * 81/80 = 1323/1280

Lower it by a 2401:2400, and it comes out as 405:392. So we have a rational approximation for the half-sharp, which is nice, but how do we step between adjacent rows?

In meantone, a 7:4 could be an augmented sixth or a minor seventh less a diesis. That diesis could be identified with a half-sharp, so we have two distinct spellings. In Jove, it could be written relative to either with an additional accidental.

Let’s deal with the augmented sixth first. It’s 10 steps on the spiral of fifths so in Pythagorean terms, it’s 3^10/2^15 or 59049:32768. 59049/32768*4/7 = 59049/57344 which is 50.72 cents sharp of a 7:4. That discrepancy’s equal to around 2.36 syntonic commas, so we can say that the 7:4 above C is A sharp less 2.36 commas. The 0.36 commas is 59049/57344/(81/80)^2 or 225:224. That makes 7:4 an augmented sixth less two syntonic commas and a marvel kleisma.

Now for the minor seventh less a half sharp. In rational terms, that’s 16/9 * 392/405 = 6272/3645. Divide 7/4 by this to get 3645:3584, 29.22 cents or 1.36 commas flat of a 7:4. Hey, there’s a .36 commas again! 3645/3584*80/81 = 225/224. The minor seventh less a half sharp is a syntonic comma and a marvel kleisma flat of 7:4.

Adding the augmented sixth to the minor seventh gives 59049/32768 * 6272/3645 = 3969/1280. This is exactly 81:80 sharper than 49:16, which is right because one interval was sharp of 7:4 by two commas and something and the other was flat of 7:4 by one comma and something. The half-Pythagorean interval above (an octave plus a fifth plus a half-sharp) also came out as a comma sharp of 49:16.

Notating Jove, then, requires half sharps and half flats, comma shifts, and marvel kelisma shifts. The Pythagorean sharp, 225:224 and 81:80 are linearly independent in Jove, so there’s no simpler notaton. Being linearly indpendent, though, they do constitute a basis. So there’s some redundancy. We should have expected that because we’ve already seen two ways of writing the same interval.

A Pythagorean whole tone is 3 syntonic commas and two marvel kleismas flat of 5 half-sharps. A Pythagorean comma is 3 syntonic commas and 2 marvel kleismas flat of a half-sharp. An octave is 21 syntonic commas and 14 marvel kleismas flat of 31 octaves. The matrix for converting from 11-limit vectors to half-sharps, syntonic commas, and marvel kleismas is

[<31 49 72 87 107]
<-21 -33 -49 -59 -72]
<-14 -22 -32 -39 -48]>

That describes 31&21e&14c.

http://x31eq.com/cgi-bin/rt.cgi?ets=31+21e+14c&limit=11

The optimal 11-limit tuning gives a half-sharp of 53.528 cents, a syntonic comma of 17.870 cents, and a marvel kleisma of 6.000 cents.

Given these numbers, setting a syntonic comma equal to exactly three marvel kleismas seems reasonable. It means you don’t need to worry about strange combinations of commas and kleismas because you’re only chaining one modifier to the split-Pythagorean half-sharps. It all amounts to tempering out

(225/224)^3*80/81 = 703125/702464

The result is called Tertiaseptal:

http://x31eq.com/cgi-bin/rt.cgi?ets=171_31&limit=7

http://x31eq.com/cgi-bin/rt.cgi?ets=31_171p&limit=11

Hugely complex, but that doesn’t bother us because we know how to notate it. Accurate to around a cent.

Miracle temperament’s a notable follower of Jove. To notate that, you throw away the kleismas.

Then there’s Harry. It should give about the same accuracy as Tertiaseptal with about half as many notes. The half-sharp comes out as 3 half-octave periods less 21 generators, the syntonic comma maps to 1 period less 7 generators, and the marvel kleisma 5 periods less 36 generators. It means the comma divides the half-sharp into three equal parts, so this is a “twelfth-tone” notation consistent with 72-equal, but with the septimal kleismas as independent modifiers. You can’t get rid of the kleismas, but you don’t need to chain them either. If you ever have a pitch modified by two kleismas, you can re-spell it in terms of sixth-sharps from a different position.
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Mavila in Tripod Notation
x31eq
Here are three 9 note Mavila MOS scales in tripod notation:

yab tad eb meb pud seth leth hod dov yab’

yad tan ed med pip seu leu hov dou yad’

yan tau eth meth piu sej lej hou doj yan’

They use Yan Tan Tethera names with “b” and “j” for semitoe shifts, “d” and “u” for inch shifts. The target is 41-equal, and so Magic. All semitoes are the same and all inches are the same.

41 is a Mavila temperament, keeping the best approximation to 5:4, but 3:2 and 6:5 a step out. 41=32+9, 32=23+9, 23=16+7. The “inch shifts” move by one step of 41-equal, and so the amount by which a perfect fifth needs to be flattened to Mavila-ize it. A semitoe is exactly two inches. You could say this defines some kind of “double magic” mapping.
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Zhenjiang riots, 1889
x31eq
The British Consulate building in Zhenjiang is now part of the museum.



It's a solid brick building, and it shows the date of 1890. A board inside the museum explains that the former structure was burnt down in 1889. (The batteries had run out in my camera when I found it, so I don't have a picture.)

The British library has an electronic newspaper archive going back way before this date, so it's possible to find articles about the riots. You have to pay to read most of them, but The Graphic is free. Zhenjiang was then known as "Chinkiang" (some vinegar bottles still show that name) so that's the thing to search for.

The first report is from March 23rd, where it says

Life in the Chinese Treaty Ports is not altogether peaceful for foreigners even now, as the recent riot at Chinkiang showed. The foreign community carried on extensive business, and appeared perfectly safe, even though numbers of disbanded soldiers hung about the native city with little to do. However, a misunderstanding between a Chinaman and a Sikh policeman on the foreign Concession caused a street-row early in February, the crowd grew infuriated, and an attack was made on the police station, whose inmates fled for their lives. Then the mob attacked neighbouring foreign houses and offices and set fire to the British Consulate, the Consul with his family escaping by a back door. His wife fled barefoot, not having time to put on her shoes. The other foreign residents followed suit, some seventy-five in all, with no chance of saving any property, and managed to get on board the hulk Cadiz in the harbour, whilst the mob sacked various European chapels, clubs, &c. The Chinese tried to board the hulk, but were beaten off, and the refugees were subsequently transferred to a mail steamer which arrived in the nick of time. The Chinese authorities did little to allay the riot, the chief officials were away on a holiday, while the small force of soldiery sent to the Concession to keep order either ran away or fraternised with the rioters. A British gunboat was summoned from Shanghai, but did not arrive very quickly. Happily only one foreigner was injured, although another is reported missing.

That mostly agrees with what I remember of the account given at the museum. One discrepancy is the original argument is described here as involving a Sikh policeman, but English soldiers at the museum. They also point out at the museum that the soldiers were wrong to beat the Chinese pedlar. I entirely believed that story. I could imagine drunk English soldiers getting into a squabble with a Chinese man, who was probably walking very slowly in front of them, or trying to sell them something they didn't want. But no, it was a Sikh all the time, who was very unlikely to be English I would have thought.

There are more details in an illustrated article of May 4th. Why don't I transcribe that one as well?

Early in February last, a terrible riot occurred at Chinkiang, a port on the Yangtsze River. For some time past it appears that bad blood has existed between the Sikh Police (who are employed by the Municipal Council of the Foreign Concession, ad are nicknamed by the populace "Red Heads," on account of their red turbans), and the inhabitants of the native city. Some of these policemen were accused of ill-treating a man who is variously described as a street-beggar and an interpreter employed at the American Consulate. The man fell down as if dead, but on being examined by a doctor and a police inspector was pronounced to be shamming. However, the mob declared that he had been killed, and at once commenced a furious attack on the Station. The few constables who were within fled for their lives ; whereupon the infuriated crowd poured in, and pulled the building to pieces, scarcely leaving one stone upon another. Then, after smashing the windows of the houses of some Members of the Municipal Council (the occupants all having escaped), the mob turned towards the British Consulate, which is on a bluff overlooking the Settlement. The Consul, Mr. Mansfield, his wife, and two young children, had barely time to fly, when the building was in flames, the rioters piling up the inflammable stuff all round it. Everything was destroyed, the building and its contents being reduced to a heap of ashes. The American Consulate was next attacked, but as it was surrounded by Chinese houses it was not set on fire. It was, however, stripped of everything moveable. The local authorities seem to have behaved with great apathy, for they sent only a few unarmed soldiers to quell the riot, and these men are said to have sympathised with the mob, and joined in the work of destruction. By this time the disturbance had reached enormous proportions ; private houses, chapels, and warehouses being set on fire. Meanwhile the foreign residents, among whom were a dozen ladies and some twenty children, fled for their lives. They were hotly pursued by the mob, but managed to get on board a foreign hulk lying in the river, and from thence were transferred to a foreign steamer, which had opportunely arrived. All this occurred on February 6th. Assistance shortly arrived: H.M.S. Mutine had been telegraphed for from Shanghai, but before she came the ladies and children had been transferred in a Chinese steamer to that city. The authorities now poured troops into Chinkiang and order was soon restored. We may mention that the town was full of famine-refugees, but they are not supposed to have had any share in the riots. Indeed, Mr. Mansfield had been most active in collecting funds, and distributing relief. . .

So, there may have been more than one of them involved, but both Graphic accounts identify the policemen as Sikhs.

There's no mention of Chinese casualties. The Chinese account is that the crowd was fired on while they were gathering outside the Consulate. We have to assume some of them were hurt.

It's clear that there was also an American Consulate at the time. At Pearl Buck's Former Residence, they say that the American Consulate is more recent than the British one, and it's now a museum building, in the dominating position as seen from the gates (and the Chinese settlement):



I think that building is more recent than the British one of 1890. I can't see a fire from that building spreading to any Chinese houses. There must have been another American Consulate there in 1889. (Probably in the same place.) It must have been smaller than the British Consulate in 1875, judging by the following piece in The Graphic from June 26th.

Some Chinese soldiers having insulted the United States Consul and his wife at Chinkiang, two of them were arrested and confined to the British Consulate. A mob collected round the house and attempted a rescue, but the disturbance was ultimately quelled by the authorities. The British and American Consuls at Shanghai and their respective vessels of war have gone to Chinkiang, but the demands for reparation have already been partially satisfied by the authorities.

It's interesting that they link the paying of reparations to the arrival of gunboats in the city. There were extensive reparations paid after the 1899 riots that paid for the new building for the British Consulate. We can imagine the political wrangling that must have gone on. The treatment of this unnamed man, who may have been shamming, by the policemen was intended to show that they were in charge. The behaviour of the crowd, the riots, and the fact that the Chinese authorities seem to have turned a blind eye or two, shows that the foreigners were inherently vulnerable, surrounded by overwhelming numbers of native Chinese. Then the British authorities ask for reparations and the Chinese authorities have to give in. The reality is that the foreigners were in the stronger position. The Graphic, on July 6th 1889, describes the ceremony that demonstrated foreign dominance to the local people.

China has made formal amends to England and the United States for the late riots at Chinkiang by paying homage to the British and American flags on the site of the disturbance. All the Chinkiang officials went in Sate to the American Consulate, which was looted but not burnt during the riots—more fortunate than its next door neighbour, the British Consulate. The English and American Colony received the officials, together with a detachment from the British man-of-war Swift, and the British Standard and the American Stars and Stripes were then solemnly hoisted aloft amid three volleys from the Chinese troops and salutes from the fort. Celestials and foreigners afterwards toasted their respective nations in an amiable repast at the United States' Consulate.

I don't know what the legal basis for reparations was. Whether a Treaty Port, a Concession, or a Colony, it was clearly under foreign administration. That probably means Chinese law didn't apply inside it. I don't think the Chinese invited the Sikh policemen over, anyway. So how does a riot inside it become the Chinese government's responsibility? The presence of those warships must have been decisive.

The new British Consulate is a covert fortress. It's made of brick, so it won't burn, and there are narrow steps leading up to the office, at the top of the hill. If a crowd attacked it, they'd have to line up to go up the stairs, which could have been barricaded, and a few men with guns would have been able to hold them off. The office is also at a prime location at the top of the hill, with a view right across the river, so that you could sit there with a telescope and see what ships were going past. Unfortunately, what with my batteries being flat, I didn't get pictures to illustrate this. Here, though, is a picture from where I think the foreign settlement was:



The offices are right at the top of the hill. (They may be more recent buildings than the building I verified as 1890.) As an Englishman, I can see exactly why this spot was chosen. You can see the Chinese district, the foreign settlement, and the river. From behind the walls of the Consulate, you directly overlook the traffic going past from the docks, further up this street:


(Note the groove in the middle so you can pull a wheelbarrow up the hill. The same design works for bicycles.)

The American consulate, though next door, seems to be sited for a different reason: it looks imposing from the Chinese area.

Here, for atmosphere, is another picture from in or near the foreign settlement:



One of the banners is for the Hengshun vinegar shop. Another picture showing the work they're doing to turn the former settlement into a tourist area, and that they managed to attract a foreign tourist:



There was one board up around here revealing it to be the former British Concession. I don't think there was any mention of the Americans. Maybe the British were running that Municipal Council. The policemen were obviously imported by the British Empire.

I think that's it. If there's anything I forgot, maybe I'll tag it on when I write about these famine victims who kept turning up.

Power failure
x31eq
I found my laptop switched off and with no power. I tried to turn it on, but the battery was critically low and it refused. After investigating the situation with all the advantages of my years of experience with computers, I isolated the problem, pushed the plug into the wall, and switched on. I was pleasantly surprised to find the desktop restored to exactly where I left it. I was always worried about letting the battery run out (it’s old, and unpredictable) but in this case the OS detected the problem and put it into hibernation successfully. Isn’t it nice when things work the way they’re supposed to?

Blackjack on a Tripod
x31eq
The decimal scale (nominals of decimal notation) comes out like this in terms of the tripod scale:

9♯⇑ 1↿ 2↿3↿ 4 5 6 7⇂ 8⇂ 9⇂ 9♯⇑

where ♯⇑means to raise by a standard toe (or secor, approximating 16:15 or 15:14) and ↿ means to raise by a quomma or standard inch.

As I think I got this wrong somewhere before, take note that I’m right today. The inch that isn’t a syntonic (dydymic) maps to a quomma in miracle temperament. It’s defined as the difference between three toes and a pace (5:4). In tripod notation, the scale step that takes you from one foot to another is larger than a toe by this amount. In decimal notation, three secors plus a quomma approximate 5:4.

The tripod staff can be decimalized by adding a position for a zero degree. That position already exists as a dummy ledger line. I’ll call the pitch “dim” after the Welsh word for “zero”. Welsh is the nearest language to yan tan tethera that has such a word.

Using dim, and ^ and v for quomma shifts, the scale can be written in pure ASCII:

0 1^ 2^ 3^ 4 5 6 7v 8v 9v 0 (Tripod)

The blackjack scale can be written most simply as one of these in decimal notation:

0 1v 1 2v 2 3v 3 4v 4 5v 5 6v 6 7v 7 8v 8 9v 9 9^ 0v 0 (Decimal)

0 0^ 1 1^ 2 2^ 3 3^ 4 4^ 5 5^ 6 6^ 7 7^ 8 8^ 9 9^ 0v 0 (Decimal)

Either of these will require double quomma shifts in tripod notation, even the dim variety. But it’s also possible to extend the chain of secors symmetrically about 5:

5^ 6^ 7^ 8^ 9^ 0 1 2 3 4 5 6 7 8 9 0v 1v 2v 3v 4v 5v (Decimal)

In pitch order, this becomes

0 1v 1 2v 2 3v 3 4v 4 5v 5 5^ 6 6^ 7 7^ 8 8^ 9 9^ 0v 0 (Decimal)

You can check that each step is either a quomma or a secor reduced by a quomma. That can be translated to the dim tripod by adding a ^ to degrees 1 to 3 and a v to degrees 7 to 9:

0 1 1^ 2 2^ 3 3^ 4v 4 5v 5 5^ 6 6^ 7v 7 8v 8 9v 9 0v 0 (Tripod)

By adding the dim and using inch shifts, tripod notation is almost as efficient as decimal notation in handling this scale. The problems are that you have to remember the asymmetry of the base scale and that not all modulations are as simple. There are two other efficient rotations:

0 1 1^ 2 2^ 3 3^ 4v 4 4^ 5 5^ 6 6^ 7v 7 8v 8 9v 9 0v 0 (Tripod)

0 1 1^ 2 2^ 3 3^ 4v 4 5v 5 6v 6 6^ 7v 7 8v 8 9v 9 0v 0 (Tripod)

Writing these scales using numbers makes the decimal and tripod notations confusingly similar. The distinctive appearance of the tripod staff should lessen this problem.

Mohajira Tricycles
x31eq
Tricycle notation is defined around marvel temperament because it started as an off-shoot of tripod notation, which was built for magic temperaments and marvel-tempered augmented triads. Tricycle notation doesn’t need to preserve this heritage, though. Let’s look at non-marvel tricycles consistent with mohajira temperaments.

I’ll talk about the “parity” of an interval as either null (staying on the same wheel), up (moving up a wheel) or down. I don’t know if it’s correct to talk about three values of parity, but I will anyway.

The rule for standard tricycle notation is that 5 has up parity, 3 has down parity, and 7 has null parity. That leads to 225:224 being tempered out.

Lilypond’s default pitch representation, based around common practice staff notation with Tartini-Couper accidentals, translates best to mohajira temperaments. That is, the geometry of half-sharps from the spiral of fifths gives a neutral third generator. 11:9 is associated with this generator (so half-sharps indicate 11-limit neutral thirds) leading to 243:242 being tempered out. The 7-limit is defined so that C to B-flat-and-a-half approximates 7:4. Unfortunately the mohajira mapping is inconsistent with the marvel one.

The 5-limit (didymic) subset is still isomorphic with just intonation. 25:24 has null parity: two fives on the top add up to have down parity, which is canceled by the 3 on the bottom. The half-sharp describes a new interval that lies outside the 5-limit geometry. As both 5-limit thirds have up parity, it makes sense for 11:9 to be the same. That means 243:242 is still tempered out. You could try a system where 11:9 and 27:22 are distinguished but it’ll probably work better as a bicycle than a tricycle.

Choosing the 7-limit mapping amounts to setting the parity of 36:35. This is the septimal diësis that separates 8/7 from 10/9 or 9/5 from 7/4. In mohajira, it’s written using the half-sharp.

If 36:25 has null parity, all half-sharps are equal. 7:6 has up parity along with 6:5 and 5:4 and the half-sharp works like the miracle quomma. Sadly, that doesn’t mean 2401:2400 is tempered out. That would lead to quarter-sharps because both 3- and 5- directions have to be divided. The comma that does arise is 31104:30625 ((36/35)^2 * 24/25). As a planar temperament, it doesn’t have a name. It’s consistent with Myna, Würschmidt, Mohajira, a Tetracot variant, Semiaug and Dicot.

http://x31eq.com/cgi-bin/uv.cgi?uvs=31104/30625

In the 11-limit, the simpler ratio of 176:175 gets tempered out. The result is consistent with Myna, Mohajira (surprise!), and not much else:

http://x31eq.com/cgi-bin/uv.cgi?uvs=176/175+243/242

So, that’s where 36:35 has null parity. Down parity would mess up the pitch clustering, so the alternative is up parity. Then, 7:6 has null parity so 7 and 3 have the same parity: down. That make sense, because it gives 8:7 up parity, so 7:6 and 8:7 have been reversed but nothing more dramatic goes out of shape.

64:63 is narrower than 36:35 by a syntonic comma of 81:80. Because 81:80 has up parity, 64:63 has null parity when 36:35’s is down. (Yes, 64:63 is a small interval pointing up but with down parity when 36:35 is null. That’s the way it goes.) The interval tempered out is now 25/24*(63/64)^2, giving 33075:32768. It covers Miracle, Mohajira(!), Worschmidt, Monkey and, a long way down, Sentinel:

http://x31eq.com/cgi-bin/uv.cgi?uvs=33075:32768

As 243:242 is a miracle comma, this is still consistent with it in the 11-limit, and 385:384 is also tempered out:

http://x31eq.com/cgi-bin/uv.cgi?uvs=385:384+243:242

This gives us a way of notating Miracle without either 225:224 or 2401:2400 being tempered out. The planar temperament (unnamed) has slightly lower error than Miracle, and so is the more accurate of the mohajira tricycles.

Note: Mohajira always seems to give a 7p-ness.
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Bicycle Notation
x31eq
Tricycle notation is one of those obscure ideas for microtonal notation. You divide the octave into three wheels, and each note of a triad sits on a different wheel. Otherwise, you follow meantone spellings, and the wheel indications lift it into 5-limit (didymic) just intonation, or an isomorphic planar temperament. Because the wheels are cyclic, you don’t accumulate comma shift accidentals with pitch drift.

(Sometimes I might get mixed up with pedestrian notations, and talk about feet instead of wheels. The idea’s the same, but pitches are tied more closely to feet.)

The question, then is why do we need three wheels? Two would be enough to distinguish 10:9 and 9:8. The simple answer is that two wheels don’t tell you which direction to apply the comma shifts. For example, with C on foot 1, D as 10:9 will also be on wheel 1, but D as 9:8 will be on wheel 2. If you see D moving from wheel 1 to wheel 2, you know you should raise it by a comma, and vice versa. The same applies to any other pitch. If C moves from wheel 1 to wheel 2, raise it by a comma. If C moves from wheel 1 to wheel 3 (== 0 (mod 3)), lower it by a comma. But if there are only two wheels, and C moves from one to the other, do you raise or lower it? In practice, the context will probably make it clear. Adaptive tuning algorithms work reasonably well without wheel indications after all. The simplest rule is to move to the nearest wheel boundary. Three wheels simplify the implementation but once you learn to balance on two they might be sufficient.

If pitches are supposed to cluster into two wheel per octave, two pitches a fifth apart must sit on different wheels. Let’s put C on wheel 0 and G on wheel 1. Octave complements sit on the same wheel (0-0=0, 0-1 == 1 (mod 2)) so F also naturally sits on wheel 1. For pythagorean intervals, an even number of factors of 3 put you on the same wheel and an odd number on the other wheel. C, D, and E are on wheel 0 and F, G, A, and B are on wheel 1. I’ll write it like this:

C D E f g a b C

Capital letters are on wheel 0, lower case on wheel 1.

With either meantone or schismatic temperaments, 5:4 maps to an even number of fifths (C-E for meantone, C-Fb for schismatic). If both sides of a 5:4 sit on the same wheel, the wheel indications become redundant. So if the wheels are to mean anything, 5:4 has to take you from one wheel to the other. Intervals are either even (sitting on one wheel) or odd (spanning the different wheels). Each factor of 3 or 5 in the ratio switches the parity of an interval.

Note that this is not Vicentino’s second tuning, probably intended for adaptive temperament. Each pitch belongs on one of a pair of manuals, but an interval of 5:4 sits on one manual.

If 3:2 and 5:4 both take you to the other wheel, 6:5 must leave you on one wheel. That’s consistent with two odd factors, or factors of 3 and 5 balancing in the numerator and denominator. 9:8 has two factors of 3, and so is an even interval. That’s why the two wheels divide the Pythagorean diatonic into strings of whole tones.

9:8 and 10:9 are on different wheels. 10:9 has one factor of 5 and two of 3, giving three factors in total. That’s good because the wheels wouldn’t be much use with meantone if they couldn’t distinguish the different whole tones. But it leads to an apparent paradox: C and Eb are naturally on wheel 0 but D can be on wheel 1. The wheels don’t partition the octave correctly.

In practice, you wouldn’t naturally have 10/9 and 6/5 in the same scale. The interval between them is 27:25 whereas the interval between 9/8 and 6/5 is 16:15. If anything, the wheels are helping you with the tuning here.

In C major, then, E, F, and G are on wheel 1 and A, B, and C on wheel 0

C D e f g A B C

with D movable. (Other pitches may move, and are more likely to the closer they are to the wheel boundary.) In A minor, A, B, and C are on wheel 0 and D, E, and F are on wheel 1, with G movable.

A B C d e f g A

The 25:24 chromatic semitone is two syntonic commas flat of the pythagorean apotome. Hence the # and b symbols will usually take you to a different wheel whichever meaning they have. This isn’t a universal law: 25:24 can be raised by 81:80 to give the even interval 135:128. But it’s consistent with harmonic minor. G# is naturally on wheel 0 in A minor (16:15 is an even interval) and F is on wheel 1.

A B C d e f G# A

Change F to F# and you have F#, G#, and A on wheel 0.

A B C d e F# G# A

With A dorian, G has to be on wheel 0. That’s allowable because G is movable.

A B C d e F# G A

With marvel tempering, 16:15 and 15:14 are equated, and so should have the same parity. 16:15 is even and for 15:14 to be so, factors of 7 must have even parity. 8:7 and 7:4 are even intervals, 7:6 is an odd interval. The marvel wheel boundary still lies between 6:5 and 5:4. Not everything has to be marvel tempered, of course. You might want the wheels to disambiguate marvel equivalences, and so preserve the rule that odd primes have odd parity.

Another way of defining the wheels is in line with mystery (29&58) temperament. There, pythagorean intervals all have even parity and higher primes have odd parity. Such a notation would break the pitch-clustering of wheels. Roots and fifths would have even parity with thirds odd parity. It looks like it’ll divide the octave into four wheel-clusters. You could call it “quad bike notation” where the mixture of 2- and 4-ness has some sense to it. It does the basic job of distinguishing 10:9 (odd) and 9:8 (even). 16:15 has odd parity. C major will look like:

C D e F G a b C

with D movable. I don’t see any contradictions here. It’ll function like trojan or HEWM notations where comma shifts indicate a departure from a pythagorean chain, or a different rung on the lattice. The difference is that the comma shifts don’t accumulate with pitch drift.
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Latin Names for 12 Division
x31eq
I’m reading Jan Gullberg’s “Mathematics From the Birth of Numbers” now. It’s an interesting book. It’s good that he expects you to read it passively, instead of doing a load of exercises like most mathematics books require. I bought it second hand and I hope the minor errors have been fixed in later editions. Anyway, what I’ll write about are the names for a division into 12 parts. Here they are (p.16):

1 uncia
2 sextans
3 quadrans
4 triens
5 quincunx
6 semis
7 septunx
8 bes
9 dodrans
10 dextans
11 deunx
12 as

The could be used for equal divisions of the octave. In English, fractions can have the same name as ordinal numbers and as diatonic degrees are named as ordinals that could be confusing.

There are plenty of names for sets of twelve notes, but these do well for intervals. They’re relevant because categorical perception will follow equal temperament for people familiar with it and diatonic names and ratios don’t capture this.

The other thing, as mentioned in Trippy Trojans, is that there are names for smaller divisions. They can be used for the shifts of Trojan notations. Here they are as fractions of both the uncia and as:

semuncia 1/2 1/24
duella 1/3 1/36
sicilicus 1/4 1/48
sextula 1/6 1/72
scripulum 1/24 1/288

There’s nothing for 60 or 84 divisons of the as. But as “sextula” is derived from “sextus” for “sixth”, the same pattern would turn “quintus” into “quintula” and “septimus” into “septimula”.
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