TAMNAMS/Appendix: Difference between revisions

Extending the spectrum builds on the central spectrum and relies on a few key observations.

The next observation is that the large values of L/s can be a lot more consequential than the ones close to 1/1 due to the fact that small steps are guaranteed to be smaller than large steps and that we don't know how many small steps there are compared to large steps, and therefore the ''hard'' end of the spectrum is more vast, and analogously, L/s values close to 1/1 will tend to be inconsequential and for very close values likely impractical to distinguish—in the extremes only serving small tuning adjustments rather than melodic properties. This leads to another observation: mos patterns with periods tuned to step ratios, while related to temperaments, ''are not'' temperaments—instead forming a sort of amalgamative superset of temperaments if you want to force a temperament interpretation, and thus their main function is in melodic structure, with temperaments informing potential harmonies and microtunings. Thus, the spectrum should be kept minimal and simple so that it is both generally hearable and not too specific.

=== Extended spectrum ===

|+ style="font-size: 105%;" | Extended spectrum of step ratio ranges and specific step ratios

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! colspan="3" | Central ranges

! colspan="2" | Extended ranges

! Notes

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=== Terminology and final notes ===

A perhaps useful (or otherwise mildly amusing) mnemonic is ''2-soft is too soft to be hard and 2-hard is too hard to be soft'', representing that {{nowrap|2-soft {{=}} 2-hard {{=}} 2/1 {{=}} '''basic'''}}.

== Reasoning for mos interval names ==

# If either ''x'' or ''y'' is equal to 1 (base cases):

## If both ''x'' and ''y'' are equal to 1, then the final scale is "Ls".

# If neither ''x'' nor ''y'' is equal to 1 (recursive cases):

## Let ''k'' be the greatest common factor of ''x'' and ''y''.

## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then:

### Let ''prescale'' be the mos string for ''z''L ''w''s. Recursively call this algorithm to find the scale for ''z''L ''w''s; the final scale will be based on this.

### If {{nowrap|''x'' < ''y''}}, reverse the order of characters in the prescale. This is only needed if there are more L's than s's in the final scale.

Using 3L 4s as an example, this is LsLsLss (brightest) and ssLsLsL (darkest). To find the large sizes of each ''k''-mosstep, consider the first ''k'' mossteps that make up the mos pattern for the brightest mode. Repeat this process with the mos pattern for the darkest mode to find each ''k''-mosstep's small size. To make these sizes more clear, we can denote the mos intervals as a sum of large and small steps {{nowrap|''i''L + ''j''s}}, where ''i'' and ''j'' are the number of L's and s's in the interval's step pattern; this is to say that the order of L's and s's doesn't matter, rather the amount of each step size. The large and small sizes should differ by replacing one L in the large size with an s.

# If either ''x'' or ''y'' is equal to 1 (base cases):

## If both ''x'' and ''y'' are equal to 1, then the generator is "L" and its complement is "s".

# If neither ''x'' nor ''y'' is equal to 1 (recursive cases):

## Let ''k'' be the greatest common factor of ''x'' and ''y''.

## If ''x'' and ''y'' share a common factor ''k'', where ''k'' is greater than 1, then recursively call this algorithm to find the generator and complement for {{nowrap|(''x''/''k'')L (''y''/''k'')s}}; the intervals returned this way will apply to the period rather than the octave.

## If ''x'' and ''y'' don't share a common factor that is greater than 1 (if ''x'' and ''y'' are coprime), then:

### Let ''gen'' be the scale's generator and ''comp'' be the generator's octave complement for the mos ''z''L ''w''s. Recursively call this algorithm to find these intervals for ''z''L ''w''s; the final scale's generator and complement will be based on this.

### If {{nowrap|''x'' < ''y''}}, reverse the order of characters in ''gen'' and ''comp'', then swap ''gen'' and ''comp''. This is only needed if there are more L's than s's in the scale.

#### If {{nowrap|''x'' > ''y''}}, every instance of an L in both intervals is replaced with one L and ''u'' s's, and every s replaced with one L and ''v'' s's. This produces the final scale's generator and complement.

#### If {{nowrap|''x'' < ''y''}}, every instance of an L in both intervals is replaced with ''u'' L's and one s, and every s replaced with ''v'' L's and one s. This produces the final scale's generator and complement.

! Can be non-octave? !! Etymology

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| Yes || The simplest valid mos pattern.

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! Can be non-octave? !! Etymology

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| Yes || Opposite pattern of 2L 1s, with broader range. Shortening of ''anti-trial''.

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| Yes || From tri- for 3.

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! Can be non-octave? !! Etymology

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| Yes || Opposite pattern of 3L 1s, with broader range. Shortening of ''anti-tetric''.

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| No (octave-only) || Blackwood[10] and whitewood[14] generalized to 2 periods.

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| Yes || From tetra- for 4.

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! Can be non-octave? !! Etymology

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| Yes || From Latin ''ped'', for ''foot''; one big toe and four small toes.

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| Yes || Common pentatonic; from penta- for 5.

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| Yes || Opposite pattern of 2L 3s.

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| Yes || From Latin ''manus'', for ''hand''; one thumb and four longer fingers.

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! Name

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| rowspan="5" | biwood<br />''(formerly unnamed)''

| rowspan="2" | citric<br />''(formerly lemon)''

| lime<br />''(formerly dipentic)''

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| lemon<br />''(formerly antidipentic)''

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| rowspan="3" | malic<br />''(formerly antilemon)''

| ekic<br />''(formerly echidnoid)''

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| rowspan="2" | subaric<br />''(formerly antiechidnoid)''

| taric<br />''(formerly antidimanic)''

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| jaric<br />''(formerly dimanic)''

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==== Machinoid (5L&nbsp;1s)====

This temperament is supported by {{Optimal ET sequence| 5, 6, 11, 12, 16, 17, 22, 23, 27, 28 and 33 }} equal divisions, many of which correspond to both simple tunings ({{nowrap|L:s {{=}} 2:1}}, 3:1, 3:2, etc) and degenerate tunings ({{nowrap|L:s {{=}} 1:1}} or 1:0) for 5L&nbsp;1s. Non-patent val tunings include {{nowrap|5 + 5 {{=}} 10e|5 + 10e + 12 {{=}} 21be|and 5 + 5 + 5 + 5 + 6 {{=}} 26qe}}; these are mentioned here for demonstrating virtual completeness of the tuning range, as is 33edo to show 11edo's strength as a tuning.

==== Sephiroid (3L&nbsp;7s) ====

This temperament is supported by {{Optimal ET sequence| 3, 10, 13, 16, 23, and 26 }} equal divisions, with non-patent val tunings including 6eg, 7e, 19eg, 20e, 29g, 32egq, 33ce, 36c. Like with that of 5L&nbsp;1s, these represent both simple and degenerate tunings for 3L&nbsp;7s. Extreme tunings, such as 7e, may lie outside the mos's step ratio spectrum, although such tunings are generally not considered good tunings.

==== Dicoid (7L&nbsp;3s) ====

==== Armotonic (7L&nbsp;2s) ====

==== On the term ''diatonic'' ====

Although the term ''diatonic'' has accrued a variety of exact meanings over time, both within and outside the contexts of xenharmonic music theory, in the context of TAMNAMS and moment-of-symmetry scales, the term ''diatonic'' exclusively refers to 5L&nbsp;2s.

[[Category:TAMNAMS]]