The Syntoniq pitch calculator provides some useful information about scales and pitches. This section is very mathematical. If you are not interested in the math behind scales and pitches, feel free to skip.
The pitch subcommand computes the product of its arguments as a pitch. The arguments may be either pitches in Syntoniq pitch notation or note names in the "JI" generated scale. For details on both, see Pitch and Note Primer. This command shows the canonical pitch of each value separately, and then it shows the final product in various ways. If the final pitch is in the range of MIDI notes, information about the MIDI note number and pitch bend is given. Otherwise, the number of octaves and cents are given. This way, you get the most useful information based on whether you're looking at a relative note in a scale or at a final pitch. Here are several examples.
See how the generated note E is interpreted:
syntoniq calc pitch E
---
E 5/4
final pitch 5/4
frequency 1.250
octaves 0.322
cents 386.314¢See how the generated note E is interpreted in 41-EDO:
syntoniq calc pitch E!41
---
E!41 ^13|41
final pitch ^13|41
frequency 1.246
octaves 0.317
cents 380.488¢Find the pitch of the 5th step of 19-EDO relative to middle C as defined to be 9 steps below A 440 in 12-EDO:
syntoniq calc pitch 440*^-9|12 ^5|19
---
440*^-9|12 220*^1|4
^5|19 ^5|19
final pitch 220*^39|76
frequency 313.978
MIDI note 63.158
MPE (hex) 3f, 201bFind the value of the generated note jI (the syntonic comma!)
syntoniq calc pitch jI
---
jI 81/80
final pitch 81/80
frequency 1.012
octaves 0.018
cents 21.506¢Follow a chain of transpositions:
syntoniq calc pitch E A2!17 C p!31
---
E 5/4
A2!17 ^2|17
C 3/2
p!31 1/2*^28|31
final pitch 15/8*^11|527
frequency 1.902
octaves 0.928
cents 1113.316¢The near subcommand can help you find ratios or scale degrees in an equally divided scale close to a given pitch. This command is complex and powerful, and there are some subtle aspects to its behavior.
Arguments:
--pitch — the pitch in Syntoniq pitch notation--interval — the interval to divide when showing pitches. If specified, the pitches presented will be divisions of this interval. If not specified, the default value depends on the pitch. If the pitch is a ratio, it defaults to 2 (an octave). Otherwise, the output contains ratios. See examples for clarification.--max-denom — the maximum denominator to use in output. For rational output, this represents how far to go in the harmonic series. For divisions of an interval, this represents the maximum number of divisions to consider. If not specified, the software picks what it considers to be a sensible default.--tolerance — specified as a Syntoniq pitch; indicates how close a match must be to be shown. The default value is ^1|75, which is 16¢. This is approximately the amount by which a 12-tone minor third differs from the ratio 6/5.The output consists of rows sorted by how close they are to the desired pitch. The columns are
Show equal divisions of the octave near the ratio 5/4, a major third. The output is shown as equal octave divisions because --interval was not specified and the pitch is rational. Here are things to notice:
^10|31. 31-EDO is known to have a very good major third. You can also see that 53-EDO has a very good major third.^1|3 as well as ^4|12 (and many others) in the output. The ones that are not in simplest form are shown as equal to the simplest version.syntoniq calc near --pitch 5/4
---
== 5/4 ≈ 1.250 ==
pitch value Δ cents
^9|28 1.250 -0.599¢
^10|31 1.251 0.783¢
^17|53 1.249 -1.408¢
^11|34 1.251 1.922¢
^8|25 1.248 -2.314¢
^16|50 1.248 -2.314¢ (= ^8|25)
^12|37 1.252 2.875¢
^15|47 1.248 -3.335¢
^13|40 1.253 3.686¢
^14|43 1.253 4.384¢
^7|22 1.247 -4.496¢
^14|44 1.247 -4.496¢ (= ^7|22)
^15|46 1.254 4.991¢
^16|49 1.254 5.523¢
^13|41 1.246 -5.826¢
^17|52 1.254 5.994¢
^6|19 1.245 -7.366¢
^12|38 1.245 -7.366¢ (= ^6|19)
^11|35 1.243 -9.171¢
^16|51 1.243 -9.843¢
^5|16 1.242 -11.314¢
^10|32 1.242 -11.314¢ (= ^5|16)
^15|48 1.242 -11.314¢ (= ^5|16)
^14|45 1.241 -12.980¢
^1|3 1.260 13.686¢
^2|6 1.260 13.686¢ (= ^1|3)
^3|9 1.260 13.686¢ (= ^1|3)
^4|12 1.260 13.686¢ (= ^1|3)
^5|15 1.260 13.686¢ (= ^1|3)
^6|18 1.260 13.686¢ (= ^1|3)
^7|21 1.260 13.686¢ (= ^1|3)
^8|24 1.260 13.686¢ (= ^1|3)
^9|27 1.260 13.686¢ (= ^1|3)
^10|30 1.260 13.686¢ (= ^1|3)
^11|33 1.260 13.686¢ (= ^1|3)
^12|36 1.260 13.686¢ (= ^1|3)
^13|39 1.260 13.686¢ (= ^1|3)
^14|42 1.260 13.686¢ (= ^1|3)
^15|45 1.260 13.686¢ (= ^1|3)
^16|48 1.260 13.686¢ (= ^1|3)
^17|51 1.260 13.686¢ (= ^1|3)
^9|29 1.240 -13.900¢This output shows some approximations of 5/4 in equal divisions of the tritave (ratio 3). Here, we cap the maximum number of divisions to 27 and specify a tighter tolerance of 12¢ (1/100th of an octave). Notice 3^4|19: One 1/19 of a tritave is very close to 1/12 of an octave, and the major third can be found at the degree 4 in both cases.
syntoniq calc near --pitch 5/4 --interval 3 --max-denom 27 --tolerance ^1|100
---
== 5/4 ≈ 1.250 ==
pitch value Δ cents
3^1|5 1.246 -3.737¢
3^2|10 1.246 -3.737¢ (= 3^1|5)
3^3|15 1.246 -3.737¢ (= 3^1|5)
3^4|20 1.246 -3.737¢ (= 3^1|5)
3^5|25 1.246 -3.737¢ (= 3^1|5)
3^5|24 1.257 6.263¢
3^4|19 1.260 8.895¢This example shows some ratios close to degree 7 in 17-EDO. That scale degree is very close to a perfect fourth, ratio 4/3. We see ratios in the output because no interval was given and the pitch is not rational.
syntoniq calc near --pitch ^7|17
---
== ^7|17 ≈ 1.330 ==
pitch value Δ cents
4/3 1.333 3.927¢
41/31 1.323 -10.091¢
37/28 1.321 -11.600¢
33/25 1.320 -13.472¢
29/22 1.318 -15.858¢This example combines --interval with a non-rational pitch. Here, we are finding pitches close to the third degree in 12-EDO (approximately a minor third) in divisions of the tritave. Notice that 3^2|13 is less than 5¢ below the 12-EDO minor third. The scale made up of 13 equal divisions of the tritave is the Bohlen-Pierce scale.
syntoniq calc near --pitch ^3|12 --interval 3 --max-denom 27
---
== ^3|12 ≈ 1.189 ==
pitch value Δ cents
3^3|19 1.189 0.195¢
3^4|25 1.192 2.721¢
3^2|13 1.184 -4.664¢
3^4|26 1.184 -4.664¢ (= 3^2|13)
3^3|20 1.179 -9.279¢
3^1|6 1.201 10.721¢
3^2|12 1.201 10.721¢ (= 3^1|6)
3^3|18 1.201 10.721¢ (= 3^1|6)
3^4|24 1.201 10.721¢ (= 3^1|6)
3^4|27 1.177 -11.501¢The equal-scale subcommand shows you information about scales made up of equal divisions of an interval. The output columns are as follows. The examples should provide additional clarification.
pitch — the scale degree shown with divisions as the exponent denominator even if this is not the simplest formsimplified — the canonical representation of the scale degree's relative pitch in Syntoniq pitch notationvalue — the scale degree as a floating point number to three decimal placescents — the scale degree shown in centsnote — a note that would give this pitch in a generated scale with this many divisionsΔ scale degree — how far off the generated note's pure ratio is from the desired pitch in scale degrees. This is shown as the note ± a fraction of scale degreesΔ cents — how far off the generated note's pure ratio is from the desired pitch in centsThis table shows 12-EDO. Observe the following:
2^4|12.
2^4|12 means the 4th degree of a scale that divides ratio 2 (the octave) into 12 equal pieces^1|3 is the simplified version of the pitch. We usually omit 2 when it's the base, and 4/12 simplifies to 1/3.E represents the ratio 5/4. That is the closest "simple" ratio to this scale degree.E! + 0.137° means that this scale degree is 0.137 scale degrees higher than E!. The ! "coerces" the pitch into a pure ratio. Since one scale degree is a half step in 12-EDO, this is telling us that this scale degree is 0.137 half steps sharper than the pure generated note E. This information can be useful if you are using generated notes to create intervals that are "portable" across different equally divided scales.E! + 13.686¢ shows us the same information in a different way: this scale degree is 13.686¢ sharper than a perfect major third. Because this is 12-EDO and scale degrees are 100¢, we see this is about 100× the value in scale degrees.syntoniq calc equal-scale --divisions 12
---
pitch simplified value cents note Δ scale degree Δ cents
2^0|12 1 1.000 0.000¢ A A! + 0.000° A! + 0.000¢
2^1|12 ^1|12 1.059 100.000¢ R R! + 0.010° R! + 1.045¢
2^2|12 ^1|6 1.122 200.000¢ I I! - 0.039° I! - 3.910¢
2^3|12 ^1|4 1.189 300.000¢ F F! - 0.156° F! - 15.641¢
2^4|12 ^1|3 1.260 400.000¢ E E! + 0.137° E! + 13.686¢
2^5|12 ^5|12 1.335 500.000¢ D D! + 0.020° D! + 1.955¢
2^6|12 ^1|2 1.414 600.000¢ Cq Cq! + 0.030° Cq! + 3.000¢
2^7|12 ^7|12 1.498 700.000¢ C C! - 0.020° C! - 1.955¢
2^8|12 ^2|3 1.587 800.000¢ Be Be! - 0.137° Be! - 13.686¢
2^9|12 ^3|4 1.682 900.000¢ Bf Bf! + 0.156° Bf! + 15.641¢
2^10|12 ^5|6 1.782 1000.000¢ Bi Bi! + 0.039° Bi! + 3.910¢
2^11|12 ^11|12 1.888 1100.000¢ Br Br! - 0.010° Br! - 1.045¢
2^12|12 2 2.000 1200.000¢ A' A'! + 0.000° A'! + 0.000¢Let's take a look at 19-EDO. If you're used to looking at scales defined in terms of cents (as in Scala files), the value 63.158¢ will likely look familiar to you: it is the step size in cents of a 19-EDO scale. You can see it in a lot of other ways as well. Here are some things to notice:
# or % accidental to force the pitch to go one way or another.2^5|19 shows that F is less than 0.2¢ away from the scale degree. Since F is 6/5, which is a minor third, this confirms that 19-EDO has an extremely precise minor third.G% for 2^4|19. The % here is unnecessary since G is closer to ^4|19 than to ^5|19, but Syntoniq suggests a % or # if the note is more than 0.2 scale degrees away. This just gives you an extra visual indicator that the ratio is a slightly worse approximation of the pitch.syntoniq calc equal-scale --divisions 19
---
pitch simplified value cents note Δ scale degree Δ cents
2^0|19 1 1.000 0.000¢ A A! + 0.000° A! + 0.000¢
2^1|19 ^1|19 1.037 63.158¢ Y Y! - 0.119° Y! - 7.515¢
2^2|19 ^2|19 1.076 126.316¢ N N! - 0.031° N! - 1.982¢
2^3|19 ^3|19 1.116 189.474¢ J J! + 0.112° J! + 7.070¢
2^4|19 ^4|19 1.157 252.632¢ G% G! - 0.225° G! - 14.239¢
2^5|19 ^5|19 1.200 315.789¢ F F! + 0.002° F! + 0.148¢
2^6|19 ^6|19 1.245 378.947¢ E E! - 0.117° E! - 7.366¢
2^7|19 ^7|19 1.291 442.105¢ FN FN! - 0.029° FN! - 1.834¢
2^8|19 ^8|19 1.339 505.263¢ D D! + 0.114° D! + 7.218¢
2^9|19 ^9|19 1.389 568.421¢ DY DY! - 0.005° DY! - 0.296¢
2^10|19 ^10|19 1.440 631.579¢ Cy Cy! + 0.005° Cy! + 0.296¢
2^11|19 ^11|19 1.494 694.737¢ C C! - 0.114° C! - 7.218¢
2^12|19 ^12|19 1.549 757.895¢ Bfn Bfn! + 0.029° Bfn! + 1.834¢
2^13|19 ^13|19 1.607 821.053¢ Be Be! + 0.117° Be! + 7.366¢
2^14|19 ^14|19 1.667 884.211¢ Bf Bf! - 0.002° Bf! - 0.148¢
2^15|19 ^15|19 1.728 947.368¢ Bg# Bg! + 0.225° Bg! + 14.239¢
2^16|19 ^16|19 1.793 1010.526¢ Bj Bj! - 0.112° Bj! - 7.070¢
2^17|19 ^17|19 1.859 1073.684¢ Bn Bn! + 0.031° Bn! + 1.982¢
2^18|19 ^18|19 1.928 1136.842¢ By By! + 0.119° By! + 7.515¢
2^19|19 2 2.000 1200.000¢ A' A'! + 0.000° A'! + 0.000¢If you see a generated note in the table and you're not sure what it means, you can always use the pitch subcommand. For example, 2^12|19 is Bfn. What's Bfn?
syntoniq calc pitch Bfn
---
Bfn 65/42
final pitch 65/42
frequency 1.548
octaves 0.630
cents 756.060¢Let's take a fresh look at Bohlen-Pierce. Since this divides the tritave, we specify --iterval 3. Notice that the top note, A', is 1901.955¢. This is slightly above 1900¢ because the ratio 3 is just slightly sharp relative to an octave and a fifth in 12-EDO. Also notice that, as we climb past the octave, we start seeing notes that start with B. Since B is the ratio of 2/1, a B adds an octave to the pitch. This is an absolute ratio; it always means an octave regardless of the divided interval. In this case, A' is a tritave above A because the divided interval is 3.
syntoniq calc equal-scale --divisions 13 --interval 3
---
pitch simplified value cents note Δ scale degree Δ cents
3^0|13 1 1.000 0.000¢ A A! + 0.000° A! + 0.000¢
3^1|13 3^1|13 1.088 146.304¢ L L! - 0.030° L! - 4.333¢
3^2|13 3^2|13 1.184 292.608¢ F F! - 0.157° F! - 23.033¢
3^3|13 3^3|13 1.289 438.913¢ E# E! + 0.360° E! + 52.599¢
3^4|13 3^4|13 1.402 585.217¢ DT DT! - 0.011° DT! - 1.629¢
3^5|13 3^5|13 1.526 731.521¢ C# C! + 0.202° C! + 29.566¢
3^6|13 3^6|13 1.660 877.825¢ Bf Bf! - 0.045° Bf! - 6.533¢
3^7|13 3^7|13 1.807 1024.130¢ Bj Bj! + 0.045° Bj! + 6.533¢
3^8|13 3^8|13 1.966 1170.434¢ B% B! - 0.202° B! - 29.566¢
3^9|13 3^9|13 2.140 1316.738¢ BO BO! - 0.018° BO! - 2.705¢
3^10|13 3^10|13 2.328 1463.042¢ BG BG! - 0.026° BG! - 3.829¢
3^11|13 3^11|13 2.533 1609.347¢ BE BE! + 0.157° BE! + 23.033¢
3^12|13 3^12|13 2.757 1755.651¢ BD# BD! + 0.394° BD! + 57.606¢
3^13|13 3 3.000 1901.955¢ A' A'! + 0.000° A'! + 0.000¢